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CLEP College Algebra/Intro to Algebraic Proofs

To certain students, math feels like a memorized set of rules. However, the only things that students may ever need to memorize in algebra are the basics of what is true about the object being studied and some notation. Any and all properties that students may feel like they need to remember are actually something someone can derive from the foundations they actually should remember.

In this page, we will introduce laws of logic and properties of conditional statements. We will be applying the laws of real numbers and algebra to prove some often-used theorems. Finally, we will use algebra to demonstrate a mathematical statement as true.

Introduction to Sets
The basic understanding of anything related to mathematics begins with learning about sets. While this will not be a substitute to a discrete math class (or the CLEP College Mathematics exam), some concepts you usually learn in there will be introduced much earlier on. As such, we will attempt to make this introduction intuitive and easy-to-understand. Finally, learning about sets and logic can easily help you in real world situations (moreso logic), especially in the context of probability.

Most people will imagine a set as an aggregation of objects (called elements) that which is defined. However, there needs to be a restriction.

Here is a question: can a set contain itself? If there is such a thing as the set that contains all sets, then the set that does not contain itself would be in the set. However, that is impossible; that set would not be in itself but can not be defined as such because this set contains all sets. Since it is impossible to define whether it is a set under this such definition, the set that contains all sets cannot be possible (this paradox is known as Russel's Paradox). The restriction that a set must be well-defined is a necessity for this reason.

"Well-defined" is vague for now, but the restrictions for what constitutes "well-defined" is beyond the scope of this text. For those that are curious, understanding Zermelo–Fraenkel set theory would be important (although a little high level for what we want to accomplish in College Algebra).

A set can contain any number of objects. For example, we could have a set that contains all of the positive, even, single-place numbers, $$A$$. We could also have a set that contains nothing (we call that the empty-set, $$\O$$). We could also have the set of common breakfast drinks, $$D$$. Note how we define the set. The way we define it is with curly braces, $$\{ \}$$ with the elements denoted and separated by commas. Sets are also unordered, meaning that the order of the elements in the set does not change whether one set is equal to another. Thus, if $$A$$ is defined as earlier, and $$B=\{0,4,2,8,6\}$$, then $$A=B$$. This tells us something very important about sets.
 * $$A=\{0,2,4,6,8\}$$
 * $$\empty=\{\}$$
 * $$D=\{\text{coffee},\,\text{milk},\,\text{water},\,\text{orange juice},\,\text{apple juice}\}$$

The Elements Define a Set
By what we defined, a set is a collection of elements. This means the objects define the set. Remember, this is true by definition, so this is sound logic.

If there are two sets, $$A$$ and $$B$$, then one of the sets can contain the other as an element, such as $$A\in B$$. If otherwise, then $$A\notin B$$. Consequently, the question of whether or not a set is contained within another is a question of conditional logic: it is "true" that $$A\in B$$ or it is "not true" that $$A\in B$$. There is no in-between option.

A set can only ever equal another if they contain the same elements. The order of the elements do not matter, so long as the element of each corresponds through its definition, and the number of the elements within each set is also identical. This therefore means a set is uniquely determined by its elements.

Let $$A=\{0,2,4,6,8\}$$ and $$B=\{0,1,\{2,4\},3,6,4\}$$.
 * (a) For all elements found in the both sets, write them as elements in the set $$S$$.
 * (b) For all non-common elements, write all such "$$e\in A$$ but $$e\notin B$$" or vice-versa, where $$e$$ is an element found in one and only one set but not in the other.

Answers:

(a) $$S=\{0,4,6\}$$.

(b) $$2,4,8\in A$$ but $$2,4,8\notin B$$ AND $$1,\{2,4\},3\in B$$ but $$1,\{2,4\},3\notin A$$.

Explanation for (a) and (b):

These are mostly self-explanatory. Keep in mind that any set within another set is defined as the element of the set. However, if that element contains any objects, then it is not part of the "parent" set. For example, let $$E=\{2,4\}$$. Here $$B=\{0,1,E,3,6,4\}$$. We know $$E\in B$$ and $$2,4\in E$$. However, any element in $$E$$ does not define $$B$$. The elements only define its own set.

How to define a set contains an element is done in two different ways, explicitly and implicitly.

The explicit way of defining a set is what we have been using throughout the entire section. However, this can also tend to be very useless when there are an infinite number of elements in the set. The next definition should held us in that aspect.

A more robust definition of a set is thus the following:

Because elements define the set, it is often important to know the size of the set. This is known as the cardinality or size of the set. Because we want to keep this part of the subject intuitive, we will use size from here on.

Let $$D$$ be the set of common breakfast drinks. A non-comprehensive list of the most common breakfast drinks in the U.S. are given: Coffee, Milk, Orange Juice, Apple Juice, Water.
 * (a) Write the set $$D$$ in set notation.
 * (b) Find the size of the set. Write it in official notation

Answers:

(a) $$D=\{\text{coffee},\,\text{milk},\,\text{water},\,\text{orange juice},\,\text{apple juice}\}$$.

(b) $$|D|=5$$.

Explanation for (a) and (b):

These are mostly self-explanatory. The number of the items in the list is five, so the size of the set is five.

Check your Understanding
Directions: Some questions will require you to select from among five choices. For these questions, select the BEST of the choices given. Some questions will require you to type a numerical answer in the box provided. Some questions will require you to select one or more answer choices.  {Let $$A=\{0,\{0\},1,\{0,1\},2,\{0,1,2\},3,\ldots,100\}$$ and $$B=\{0,1,\ldots,99,A\}$$. Which of the following statements are true? Select all that apply. + $$0,1,2,3,\ldots,99\in A$$ and $$0,1,2,3,\ldots,99\in B$$. - $$\{0,1,2,3,\ldots,99\}\in A$$ and $$\{0,1,2,3,\ldots,99\}\in B$$. + $$0,1,2,3,\ldots,99\in \{0,1,2,3,\ldots,99\}$$ and $$\{0,1,2,3,\ldots,99\}\in A$$.
 * type="[]"}
 * Because sets are unordered, and this set is implicit, any and all numbers below 100 and above 0 are in set $$A$$. Further, as listed in $$B$$, those are elements of the list.
 * While it is true $$\{0,1,2,3,\ldots,99\}\in A$$, it is not necessarily true that it is in $$B$$. It is true that $$A\in B$$. However, elements of $$A$$ do not automatically become objects of $$B$$.

{Let there be a set $$B=\{A\}$$. Find $$|A|$$ and $$|B|$$. If unknown, write unknown. $$|A|=$$ { unknown | Unknown _10 }, and $$|B|=$$ { 1 _10 }
 * type="{}"}
 * Although it is unknown how many elements are found in the set $$A$$, it is known that there is one set $$A\in B$$. Therefore, we know the number of elements within set $$B$$ is $$1$$.

{Use the following information to answer items 3-4 below. It is known that $$a\in A$$, $$a,b\in B$$, and $$c\in C$$.}

{Which of the following is true of $$D=\{A,B,C\}$$? - $$a\in D$$. - $$a,b\in D$$. - $$c\in D$$. - $$a,b,c\in D$$. + None of the above. More questions added later.
 * type=""}

Properties Define an Element, Ergo, the Set
An object can have properties that would be important to write down. If the object itself changes because of it, then it is important to have that defined. These are referred to as conditionals (the more common one is propositional functions, but this is avoided in the interest of providing a definition of functions that is not treated as tautological when the definition of a function is non-tautological).

Since a set is defined by its objects, the conditional also defines the set. The notation for a given set that has a conditional is
 * $$S=\left\{x\mid C(x)\right\}$$ or $$S=\left\{x:C(x)\right\}$$

where such a set is read as $$S$$ is the set of all $$x$$ such that $$C(x)$$ is true. The term "such as" originates from either the vertical bar ( $$|$$ ) or the colon ( $$:$$ ). In this textbook, we will use the vertical bar $$|$$ to refer to "such as" since this is standard in most College Algebra courses.

A special type of notation is used when a set can better represented through conditionals. Take the set of all Fibonacci numbers, for example, $$F$$:
 * $$F=\{0,1,1,2,3,5,8,13,21,34,55,89\ldots\}$$.

This set can be listed implicitly. However, when there is a pattern to the set's elements, then a set can become much more useful. Let $$f(k)$$ represent the index of the Fibonacci sequence, where $$k$$ is an index starting at zero. The set can be equivalently written as the following:
 * $$F=\{f(k)|f(k)=f(k-2)+f(k-1),\, f(k),k\in\mathbb{Z}^{+},\, f(0)=0,\, f(1)=1\}$$.

While this notation seems to hurt rather than help (and seem more confusing), in actuality, this tells us a lot of information about the set, including the pattern used, what initial conditions are necessary for the set to exist, and what set $$f(k)\text{ and }k$$ need to belong to for this pattern to work.

We can now easily read sets. Looking back at the set $$F$$ (the set of Fibonacci numbers): $$F=\{f(k)|f(k)=f(k-2)+f(k-1),\, f(k),k\in\mathbb{Z}^{+},\, f(0)=0,\, f(1)=1\}$$. $$F$$ is the set of all $$f(k)$$ such that $$f(k)=f(k-2)+f(k-1)$$, $$f(k)\text{ and }k$$ belong to the set of positive numbers, and there are two initial conditions, $$f(0)=0\text{ and }f(1)=1$$.

Write each of the following using set-builder notation and implicit or explicit notation.
 * (a) The set of all even numbers.
 * (b) The set of all odd numbers.
 * (c) The set of all prime numbers.
 * (d) The set of all real solutions to the equation $$\frac{12(x+2)(x\sqrt{2}-1)(x-1)(x-2)}{(x+2)-(x+4)}=0$$.
 * (e) The set of all natural number solutions to the equation $$\frac{12(x+2)(x\sqrt{2}-1)(x-1)(x-2)}{(x+2)-(x+4)}=0$$.
 * (f) The set of all integer solutions to the equation $$2x+5=8$$.

Answers:

(a) $$\{2n|n\in\mathbb{Z}\}=\{n\in\mathbb{Z}|n\text{ is even}\}=\{\ldots ,\,-4,\,-2,\,0,\,2,\,4,\ldots\}$$.

(b) $$\{2n+1|n\in\mathbb{Z}\}=\{n\in\mathbb{Z}|n\text{ is odd}\}=\{\ldots ,\,-3,\,-1,\,1,\,3,\ldots\}$$.

(c) $$\{n\in\mathbb{N}|n\text{ is a prime number}\}=\{2,\,3,\,5,\,7,\,11,\,13,\ldots\}$$.

(d) $$\left\{x\in\mathbb{R}| \frac{12(x+2)(x\sqrt{2}-1)(x-1)(x-2)}{(x+2)-(x+4)}\right\}=\left\{-2,\frac{1}{\sqrt{2}},\,1,\,2\right\}$$.

(e) $$\left\{x\in\mathbb{N}| \frac{12(x+2)(x\sqrt{2}-1)(x-1)(x-2)}{(x+2)-(x+4)}=0\right\}=\{1,\,2\}$$.

(f) $$\left\{x\in\mathbb{N}| 2x+5=8\right\}=\O$$.

Explanations:
 * Items (a)-(c)

These are mostly self-explanatory. Note that there are two possible answers we could write. $$n\in\mathbb{Z}$$ and $$2n$$ are both expressions. However, to make sure both are equivalent, we need a rule so that the expressions can the elements in the set are equal to the other. Similar for (b).
 * Items (d)-(f)

The pair of solutions is given by the expression of some $$x\in\mathbb{R}$$ or $$x\in\mathbb{N}$$. The rule is the equation itself since it describes how one finds the solution set. This is all that is needed for the set-builder notation. To obtain the solutions in the set, simply solve:
 * $$\begin{align}

\frac{12(x+2)(x\sqrt{2}-1)(x-1)(x-2)}{(x+2)-(x+4)} &=0 &\text{(Original equation)}\\ \frac{12(x+2)(x\sqrt{2}-1)(x-1)(x-2)}{-2} &=0 &\text{(Distributive property)}\\ (x+2)(x\sqrt{2}-1)(x-1)(x-2) &=0 &\text{(Division property of equality)} \end{align}$$ By the zero factor theorem,
 * $$\Rightarrow x=-2,\frac{1}{\sqrt{2}},1,2$$

This is where the explicit set notation comes. Because the natural numbers do not have negative numbers, fractions, or square roots, the natural number set of solutions is only $$1,2$$.

This type of analysis allows us to determine that one of the sets, see item (f), is empty, and is therefore equivalent to the empty set, $$\O$$.

Check your Understanding
Directions: Some questions will require you to select from among five choices. For these questions, select the BEST of the choices given. Some questions will require you to type a numerical answer in the box provided. Some questions will require you to select one or more answer choices.  {Suppose $$A$$ and $$B$$ are well-defined sets. Let $$A=\{a|1\le a\le n,\, a,n\in\mathbb{N}\}$$ and $$B=\{b|b\ge 1,b\in\mathbb{N}\}$$. If $$|B|=|A|+1$$, which of the following must be true? - If $$a\in A$$, and $$|A|<|B|$$, then $$A\in B$$. + If any $$a\in A$$, then $$a\in B$$. - If any $$a\in\mathbb{N}$$, then $$a\in A$$. - If any $$b\in B$$, then $$b-1\in A$$. - If $$b\in B$$, and $$|B|<|A|$$, then $$B\in A$$. More questions added later.
 * type=""}
 * This is only true if the set $$A$$ was an element of $$B$$. Based on all the information given, the set-builder notation gives all $$b\in [1,n+1]$$.
 * The set-builder notation implies that $$a$$ is between $$1$$ and $$n$$ (inclusive), while $$b$$ is between $$1$$ and $$n+1$$ (inclusive). Because any element in $$a$$ is necessarily contained within that of $$b$$, $$a\in B$$.
 * If $$a=n+1$$, then $$a\notin A$$. One counterexample can quickly disprove this.
 * If $$b=1$$, then $$1-1=0\notin A$$. Pay close attention to the set-builder notation.
 * The size of $$B$$ cannot be less than $$A$$ because the size is necessarily positive. Even if true, however, $$B\notin A$$ because the set-builder notation does not give us the directions to place a set anywhere in $$A$$.

Comparing Sets
Of course, many sets are similar to another. As such, many mathematicians find it very helpful to compare sets. Some vocabulary terms will be listed here.

The idea of the subset is very simple; however, it can be very powerful when comparing two sets. Nevertheless, many students may neglect to take care in their notation while trying to compare sets. The examples presented below are correctly stated. The reasoning is given as well. This type of reasoning will be required for you to identify subsets and non-subsets.
 * $$\{1,3,2\}\subseteq\{1,2,3,4,5\}$$.
 * Because $$1,3,2\in\{1,2,3,4,5\}$$, $$\{1,2,3\}$$ (the set) is a subset of $$\{1,2,3,4,5\}$$.
 * $$\{1,3,2\}\subseteq\{1,2,3\}$$.
 * Because $$1,3,2\in\{1,2,3\}$$, $$\{1,2,3\}$$ is a subset of $$\{1,2,3\}$$.
 * $$\{1,2\}\nsubseteq\{4,3,\{2,1\}\}$$.
 * Although $$\{1,2\}\in\{4,3,\{2,1\}\}$$, the elements of set $$\{1,2\}$$ cannot be found in the set $$\{4,3,\{2,1\}\}$$. In other words, $$1,2\notin\{4,3,\{2,1\}\}$$.
 * $$\{\{1,2\}\}\subseteq\{4,3,\{2,1\}\}$$.
 * Because $$\{1,2\}\in\{4,3,\{2,1\}\}$$, $$\{\{1,2\}\}\subseteq\{4,3,\{2,1\}\}$$. If one cannot see this, let $$A=\{1,2\}$$. Notice that $$\{4,3,\{2,1\}\}=\{4,3,A\}$$. Because $$\{2,1\}=A\in\{4,3,\{2,1\}\}=\{4,3,A\}$$, $$\{\{1,2\}\}\subseteq\{4,3,\{2,1\}\}$$.

One other definition that is useful to know is the superset. The definition is complex yet the intuition is rather simple.

Some examples below will be listed below, along with an explanation. Notice they are the same as above.
 * $$\{1,3,2\}\nsupseteq\{1,2,3,4,5\}$$.
 * Because $$\{1,2,3,4,5\}\nsubseteq\{1,3,2\}$$, $$\{1,3,2\}\nsupseteq\{1,2,3,4,5\}$$.
 * $$\{1,2,3,4,5\}\supseteq\{1,3,2\}$$.
 * Because $$\{1,3,2\}\subseteq\{1,2,3,4,5\}$$, $$\{1,2,3,4,5\}\supseteq\{1,3,2\}$$.
 * $$\{1,3,2\}\supseteq\{1,2,3\}$$.
 * Because $$\{1,2,3\}\subseteq\{1,3,2\}$$, $$\{1,3,2\}\supseteq\{1,2,3\}$$.
 * $$\{4,3,\{2,1\}\}\nsupseteq\{1,2\}$$.
 * Because $$\{1,2\}\nsubseteq\{4,3,\{2,1\}\}$$, $$\{4,3,\{2,1\}\}\nsupseteq\{1,2\}$$.
 * $$\{4,3,\{2,1\}\}\supseteq\{\{1,2\}\}$$.
 * Because $$\{\{1,2\}\}\subseteq\{4,3,\{2,1\}\}$$, $$\{4,3,\{2,1\}\}\supseteq\{\{1,2\}\}$$.

Check your Understanding
Directions: For each of the following conditions stated below, four to five lines will be provided. Identify the correct definition that would apply to the given set for the one provided in the problem. Some problems require multiple correct answers and as such multiple selections.  {Suppose $$A=\{4,5,10,20\}$$. -+- $$\mathbb{N}$$. +-- $$\{4,5\}$$. +-- $$\{5,10,20\}$$. --+ $$\{3,4,5\}$$. --+ $$4$$.
 * type="[]"}
 * Subset|Superset|Neither
 * The set of natural numbers is a set that contains only numbers. Applying the definition of superset here, for all $$a\in A$$ that can be found in $$\mathbb{N}$$. As such, $$\mathbb{N}$$ is a superset
 * $$4\in A$$ and $$5\in A$$, so this is a subset.
 * $$5,10,20\in A$$, so this is a subset.
 * While $$4,5\in A$$, $$3\notin A$$, so this is not a valid subset of $$A$$. From this, it cannot be a valid superset either since $$10\notin\{3,4,5\}$$, for example.
 * Although $$4\in A$$, $$4$$ is not a set. Because this is not a set, this could never be a valid subset nor superset.

{Suppose $$B=\{4,5,\{10,20\}\}$$. ++- $$B$$ +-- $$\{4,5\}$$ --+ $$\{5,10,20\}$$ +-- $$\{\{10,20\}\}$$
 * type="[]"}
 * Subset|Superset|Neither
 * Because all elements of $$B$$ can also be found in $$B$$, $$B\subseteq B$$. In a sense, it could also be a superset.
 * Since $$4,5\in B$$, $$\{4,5\}\subseteq B$$.
 * Although $$5\in B$$, $$10,20\notin B$$ (notice $$\{10,20\}\in B$$), so $$\{5,10,20\}\nsubseteq B$$. From this, $$\{5,10,20\}\nsupseteq B$$.
 * Notice that the set $$\{10,20\}\in B$$, so $$\{\{10,20\}\}\subseteq B$$.

{Suppose $$C=\mathbb{N}\cup\{0\}$$ (i.e., the set that contains the set of natural numbers including zero). -+- $$\{\sqrt{x}|x>0,\, x\in\mathbb{Z}\}$$ +-- $$\mathbb{N}$$ -+- $$\mathbb{Z}$$ +-- $$\{x^{2}|x\in\mathbb{Z}\}$$
 * type="[]"}
 * Subset|Superset|Neither
 * Notice how for every natural number and zero, it can be found $$\{\sqrt{x}|x>0,\, x\in\mathbb{Z}\}$$. Because of this, $$C\subseteq\{\sqrt{x}|x>0,\, x\in\mathbb{Z}\}$$, so $$\{\sqrt{x}|x>0,\, x\in\mathbb{Z}\}\supseteq C$$.
 * Because all elements of $$\mathbb{N}$$ can also be found in $$C$$, $$\mathbb{N}\subseteq C$$.
 * Notice how for every natural number and zero, it can be found $$\mathbb{Z}$$. Because of this, $$\mathbb{Z}\supseteq C$$.
 * Since $$\{x^{2}|x\in\mathbb{Z}\}=\{\ldots,9,4,1,0,1,4,9,\ldots\}$$, we can show that for all elements found in $$\{\ldots,9,4,1,0,1,4,9,\ldots\}$$, it can also be found in $$C$$. As such, $$\{x^{2}|x\in\mathbb{Z}\}\subseteq C$$.

{Suppose $$D=\{\mathbb{N},\{\mathbb{Z}\}\}$$. +-- $$\mathbb{N}$$ --+ $$\{\mathbb{N}\}$$ --+ $$\mathbb{Z}$$ +-- $$\{\mathbb{Z}\}$$ --+ $$\mathbb{R}$$
 * type="[]"}
 * Subset|Superset|Neither

Combining Sets
The process of combining sets can be very useful, especially when working in contexts of probability.

If you want to explain this a five-year-old: the "union set" is the set that contains all things in $$P$$ and $$Q$$.

A few examples are presented below, where $$A=\{\text{a},\text{b},\text{h}\}$$, $$B=\{\text{b},\text{d},\text{e},\text{f}\}$$, and $$C=\{\text{c},\text{e},\text{g}\}$$. These are all the possible combinations of the sets (excluding combinations with its own set).
 * $$A\cup B = \{\text{a},\text{b},\text{d},\text{e},\text{f},\text{h}\}$$
 * $$A\cup C = \{\text{a},\text{b},\text{c},\text{e},\text{g},\text{h}\}$$
 * $$B\cup C = \{\text{b},\text{c},\text{e},\text{g},\text{d},\text{f}\}$$
 * $$A\cup B\cup C = \{\text{a},\text{b},\text{c},\text{d},\text{e},\text{f},\text{g},\text{h}\}$$

If you want to explain this a five-year-old: the "intersection set" is the set that contains all things that are found in both $$P$$ and $$Q$$.

A few examples are presented below, where $$A=\{\text{a},\text{b},\text{h}\}$$, $$B=\{\text{b},\text{d},\text{e},\text{f}\}$$, and $$C=\{\text{c},\text{e},\text{g}\}$$. These are all the possible combinations of the sets (excluding combinations with its own set).
 * $$A\cap B = \{\text{b}\}$$
 * $$A\cap C = \O$$
 * $$B\cap C = \{\text{e}\}$$
 * $$A\cap B\cap C = \O$$

If you want to explain this a five-year-old: the "difference set" is the set that contains elements of $$P$$ but excluding what are also found in or are in $$Q$$.

Several examples are presented below, where $$A=\{\text{a},\text{b},\text{h}\}$$, $$B=\{\text{b},\text{d},\text{e},\text{f}\}$$, and $$C=\{\text{c},\text{e},\text{g}\}$$. These are all the possible combinations of the sets (excluding combinations with its own set and any combinations with all three sets).
 * $$A-B = \{\text{a},\text{h}\}$$
 * $$A-C = \{\text{a},\text{b},\text{h}\}$$
 * $$B-A = \{\text{d},\text{e},\text{f}\}$$
 * $$B-C = \{\text{b},\text{d},\text{f}\}$$
 * $$C-A = \{\text{c},\text{e},\text{g}\}$$
 * $$C-B = \{\text{c},\text{g}\}$$

Introduction to Logic
Logic is important in many fields. However, it is especially in mathematics since without this tool, we could not be able to come up with theorems nor algorithms. That is why it is especially important for us to teach this tool. While this may not be a philosophy class, we will be teaching as much as needed for us to be able to present proofs to you without too much trouble.

The process of deduction is simply the ability to "connect" statements through some related way.

For example, we know that Aristotle is a "grefunkle," and "grefunkles" are "prostireoni." Therefore, Aristotle is a "prostireonis."

Now, "grefunkles" and "prostireoni" are made up words; these things do not exist. However, the sentence we made is logically correct. Remember, logic does not require the correct information, so this sentence is logically correct while not essentially correct.

Statements
Logic must begin with "statements." After all, the example we presented could not exist without "statements."

Statements need a "verb" of some kind, mostly "declaratory verbs," like "is" or "are." For example, "Apples are blue" is a statement since it is either true or false (here, it is false for many apples). Questions are not statements. More examples are provided.
 * 1) Every even number is divisible by itself and two. True.
 * 2) All prime numbers are odd. False.
 * 3) Twelve more than twenty-two less than the initial means that the initial is twenty-two more than twelve less than the result. True.
 * 4) If a side of a parallelogram is $$p$$, then the area of the parallelogram is $$\frac{1}{2}p^{2}$$. False.
 * 5) The side length of any one side of an isosceles triangle is equal to the square root of the two remaining sides. False.
 * 6) If one and only one side of a quadrilateral is not parallel to its opposing side, then it is a trapezoid. True.

Statements #4 and #6 are a type of statement we will be discussing very closely in this section. Keep in mind the form of the sentence for future reference.

Notice how statement #4 contains a variable $$p$$. A statement may use variables so long as no matter what the variable stands for or what the variable will equal to, the statement will always either be true or false by simply the statement alone. We will discuss this more in this section.

Also the statements do not have to be sentences, as has been stated in the definition: Keep in mind that the mathematical expressions you use need to state something in the "declaratory verb" form we describe. For example, $$2\in\mathbb{N}$$ states in a more compact form $$2$$ is an element of the set of natural numbers, so this is a valid statement. What would not be a valid statement is simply $$\mathbb{N}$$ (the set of natural numbers). This tells us nothing.
 * $$\pi=3.141519265358979...$$ True
 * $$\pi=2.718281828459045...$$ False.
 * $$2\in\mathbb{N}$$ True
 * $$2\in\mathbb{Z}^{-}$$ False.
 * $$\{\}\subseteq \O$$ True
 * $$\{\}\supseteq \O$$ True

Statements can be used many times in a proof or even a math problem. This is why mathematicians have developed tools to be able to declare statements using only one letter, usually $$P,Q,R,S$$. Subscripts are used when more than four statements are necessary, although one may be allowed to use more than the letters provided (if somewhat unconventional).
 * $$P:$$ There are more than two different letters in the word "cranberry." True.
 * $$Q:$$ The sum of the digits of any three digit number is between $$1$$ and $$27$$. True.
 * $$R:$$ Any two consecutive primes have a distance of six between them. False.
 * $$S(x,y):\{x,\{y\}\}\cap\{\{x\},y\}=\O$$. True.

With $$S(x,y)$$, the parenthetical statement tells us that there are two variables within a statement. This is a notation because at times, the variables become very important to determine its truth value. This will be explained shortly.

As another rule of thumb, if a sentence does not describe anything about a subject, then it is not a statement. Here are a few examples of non-statements.
 * The set $$S$$ and $$P$$.
 * What is $$S\cup P$$?
 * $$2+2$$
 * $$\frac{1}{2}=2x-1$$.

The final bullet point is a tricky one for students unfamiliar with identifying statements. Some will say this is a statement because it is either true or false that $$2x-1$$ is equal to $$\frac{1}{2}$$. However, if you think about it, this statement is entirely dependent on the value of $$x$$. This is called an open sentence.

Look back at the open sentence. Let $$P(x):\frac{1}{2}=2x-1$$. If $$x=\frac{3}{4}$$, then $$P\left(\frac{3}{4}\right)$$ is true. Otherwise, it is false. This is why $$P(x)$$ is not a statement. It depends on the value of $$x$$. A statement needs to be either true or false but not due to a changing artifact.

One important topic to discuss when working with statements is the negation.

When we take a statement and we negate it, we make a true statement false or a false statement true. Take the following as a couple or examples: The second example required us to make the "all" into "some." More information about how and when this is done will be shown later.
 * Every even number is not divisible by itself and two. False.
 * Some prime numbers are not odd. True.

The examples above are from the beginning. Notice how we made these statements opposite their truth value from the beginning. Mathematicians use these so often that we have a notation for it.

Suppose $$R:\neg P$$. For $$R$$ to have a truth value of $$T$$ (true), it is required for $$P$$ to be false. Otherwise, $$R$$ is $$F$$ (false). This can be further shown using this simple truth table (a table that shows the truth values of each individual statement and the resulting column):

\begin{array}{|c||c|} \hline P & \neg P \\ \hline T&F\\ F&T\\ \hline \end{array}$$

Conditional Statements
If you want to have any hope of understanding mathematical theorems at a deeper level, then you will need to understand conditional statements.

Other ways in which this sentence can be written is presented below: Here are some examples:
 * "$$Q$$ if $$P$$"
 * "$$P$$ only if $$Q$$"
 * "Whenever $$P$$, then also $$Q$$"
 * "$$P$$ is a sufficient condition of $$Q$$"
 * "$$Q$$ is a necessary condition for $$P$$"
 * If you pass this final exam, you pass the course.
 * You can travel to Spain if you learn the Spanish language and culture.
 * Garfunkles are prostireoni only if at least 100 years have passed between garfunkles and prostireoni.
 * Whenever you lose power, then you also lose heat.
 * Composing good music is a sufficient condition of getting paid $100,000.
 * Getting famous is a necessary condition for writing the best calculus textbook.

Conditional statements have a sufficient condition (after the "if") – or a hypothesis – and a necessary condition (after the "then") – or a conclusion. This is why if our true hypothesis leads to a false conclusion (the result is different from our stated conclusion), then the conditional statement is false.

Mathematics is a big friend of the conditional statement. Here are some familiar statements using the conditional statement.
 * If a right triangle has leg lengths $$a,b$$ and hypotenuse $$c$$, then $$a^{2}+b^{2}=c^{2}$$.
 * $$A\cap\O =\O$$ if $$A$$ is a finite set.
 * $$f(x)$$ are one-to-one only if $$f(a)=f(b)$$ for some real numbers $$a\ne b$$.

Because these type of statements are so important, it is necessary to understand how they work. There is a short-hand notation for these statements, which is presented below.

The way we determine the truth value for $$P\Rightarrow Q$$ is to write a truth table. We can determine each row of the truth table by simply looking back at the definition of conditional statements.

\begin{array}{|c c||c|} \hline P & Q & P\Rightarrow Q \\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&F&T\\ \hline \end{array}$$ A conditional statement is true if the sufficient condition leads to the stated conclusion. That is why the second row gives us the result of a false truth value. This still explains the two final statements: if we have a false hypothesis but the conclusion is true, then either way, something true did happen but not as a result of the hypothesis. The hypothesis is not necessarily required for a conditional statement to be true so long as the conclusion is true. Finally, whenever a false hypothesis leads to a false conclusion, then that makes the conditional statement true because something false leading to something false is itself a confirmation of the conditional statement.

There are six rules to be able to enter or stay in a non-specific kid's music club. If at least one of these rules does not apply to the student, they will be placed in the teen's music club (covered by rule #5) or will be given a one month grace period if not over twelve years of age. The one month grace period will not be invoked if one is in the kid's or teen's music club. Items (a)-(d) below have a few situations. Determine if there is enough information to kick these people out (whether temporarily or permanently) of the club based on their descriptions. Provide a reason for each of the young people below. Otherwise, state that individual as allowed to stay.
 * 1) To enter, it is necessary for one to be twelve years or younger.
 * 2) If one is older than twelve years, then one can only teach the young musicians.
 * 3) Grade 3 music or higher gives you automatic acceptance only if you perform grade 3 or higher music thrice per month.
 * 4) Passing a mandatory music theory test at the end of the year is sufficient to maintain being in the kid's music club.
 * 5) Being in the teen's music club is necessary for one that is older than twelve years.
 * 6) Performing with "superior" rating on one's determined grade level is necessary to stay in the kid's music club.
 * (a) George is thirteen and plays grade 3 music or higher three times a month.
 * (b) Miranda is ten and failed the music theory test at the end of the year. Plays grade 2 music with "excellent" rating.
 * (c) Carlos passed the music theory test and is nine. Plays grade 1 music with "superior" rating.
 * (d) Paul is thirteen and is the head teacher of music theory for the young kids. Plays grade 4 music.

Items (e)-(f) ask you a couple of questions regarding the situation provided. Answer items (e)-(f).
 * (e) Is there a situation that would make it impossible to determine whether a student may stay in the kid's music club? What is that situation if possible?
 * (f) Rewrite rule #2 into an equivalent statement not in "if...then" form.

Answers:
 * (a) George is rejected from the kid's music club and hereby placed in the teen's music club for failure to be a teacher.
 * (b) Miranda is hereby rejected from the kid's music club for a duration of one month because of her failure to pass the music theory test and her failure to gain a superior rating for her grade 1 music.
 * (c) Carlos may stay in the kid's music club.
 * (d) Paul may stay in the kid's music club.

Explanation

More will be added later.

Biconditional Statements
Conditional statements also have its different, non-equivalent forms. These statements are not equivalent because their truth values can be different. Notice how the end result is different simply by switching the conclusion and the hypothesis.

\begin{array}{|c c||c||c|} \hline P & Q & P\Rightarrow Q & Q\Rightarrow P \\ \hline T&T & T&T\\ T&F & F&T\\ F&T & T&F\\ F&F & T&T\\ \hline \end{array}$$ By this brute force method, we were able to show that these statements are not equivalent. These statements are equivalent if both $$P$$ and $$Q$$ are true. However these two statements can be true depending on the statements. Real life examples are rare.

Here is a conditional statement that has a true converse: "If a phrase is chromatic, then it contains notes separated by half-steps." The converse of the statement is "If a phrase contains notes separated by half-steps, then the phrase is chromatic." The reason why is because a chromatic phrase, by definition, uses notes separated by half-step, so both $$P\Rightarrow Q$$ (the conditional statement) and $$Q\Rightarrow P$$ (the converse) are true.

In situations where both the conditional and converse is true, you obtain a biconditional statement.

Biconditional statements require that the conditional statement is true. Therefore, $$P$$ only if $$Q$$. However, the converse is also true, so $$P$$ if $$Q$$. As such, we say a statement is biconditional by stating that "$$P$$ if and only if $$Q$$."

Because it requires that $$P\Rightarrow Q$$ and $$Q\Rightarrow P$$ be true, it will mean that the middle rows will be false. Thus, the following truth table results:

\begin{array}{|c c||c|} \hline P & Q & P\Leftrightarrow Q \\ \hline T&T & T\\ T&F & F\\ F&T & F\\ F&F & T\\ \hline \end{array}$$ More detail will be provided as to why this is true by looking at the next section.

More will be provided later.

Related Forms of Conditional Statements
The related forms of conditional statements are of great importance. However, some proofs classes ignore the different forms of the conditionals statements. However, at times, proving a conditional statement involve proving all of the three related forms.

A mathematician's job is to test the limits of their statements and find what can be true or false.

There are some properties of the related statements that are not often taught. However, we feel it may be at least somewhat important to understand that these statements exist and that there may be some logical equivalences.

A good way to see this property is true is with a truth table:

\begin{array}{|c c||c||c|} \hline P & Q & P\Rightarrow Q & \neg Q\Rightarrow\neg P \\ \hline T&T& T&T\\ T&F& F&F\\ F&T& T&T\\ F&F& T&T\\ \hline \end{array}$$ Hopefully, it is clear why these are related statements. Because the negation of $$Q$$ and the negation of $$P$$, and the order of the hypothesis and conclusion require that the truth values consequently change. Thus, these statements are logically equivalent.

The same can be shown with the inverse and the converse.

\begin{array}{|c c||c||c|} \hline P & Q & Q\Rightarrow P & \neg P\Rightarrow\neg Q \\ \hline T&T& T&T\\ T&F& T&T\\ F&T& F&F\\ F&F& T&T\\ \hline \end{array}$$

More will be added later.

Combining Statements
There are many times in life we have to think about the truth of a statement which contains two statements. For example, "The light is on and the light is off." This statement is definitely false. How do we know? Because of the word and.

Let $$A$$: "The light is on and the light is off." Notice how we can break the one large statement into two small statements thanks to the conjunction and. As such, let $$P$$: "The light is on" and $$Q$$: "The light is off." According to the English language, "on" is opposite in meaning to "off." However, a lightbulb can only even be on or off. Thus, if $$P$$ is true, then $$Q$$ is false by default. Therefore, $$R$$ is false.

The process of combining statements will be very useful in proofs. Since these combinations are defined by themselves, we do not need to prove them. Finally, because this combination is based entirely on the english language, the truth of a statement can be easily determined.

Based on what we did with the introductory example, let us generalize what we did.

Suppose $$R:P\wedge Q$$. For $$R$$ to have a truth value of $$T$$ (true), it is required for both $$P$$ to be true and $$Q$$ to be true. If either one is false, or both is false, then $$R$$ is $$F$$ (false). This can be further shown using a truth table.

\begin{array}{|c c||c|} \hline P & Q & P\wedge Q \\ \hline T&T&T\\ T&F&F\\ F&T&F\\ F&F&F\\ \hline \end{array}$$

Suppose $$R:P\vee Q$$. For $$R$$ to have a truth value of $$F$$ (false), it is required for both $$P$$ to be false and $$Q$$ to be false. If either one is true, or both is true, then $$R$$ is $$T$$ (true). This can be further shown using a truth table:

\begin{array}{|c c||c|} \hline P & Q & P\vee Q \\ \hline T&T&T\\ T&F&T\\ F&T&T\\ F&F&F\\ \hline \end{array}$$

By using these conjunctions and truth tables, we can easily determine when a open sentence will be true and whether a statement will be true or false.

Read the open sentence below that someone claims to always be false for any $$(a,b)$$.
 * $$a$$ is negative and $$b$$ is positive but $$a-b\ne 0$$.

Answer items (a)-(c) below.
 * (a) Write this in symbolic notation.
 * (b) Find a counter example. If possible, how many possible counter-examples exist. If not possible, explain why.
 * (c) For this combination, how many possible false truth values are there?

Answers:
 * (a) $$(a<0)\wedge(b>0)\wedge\neg(a-b=0)$$
 * (b) Let $$b=2$$ and $$a=-1$$. $$-1-(2)=-3\ne 0$$, which is true. The open sentence is false in infinitely many instances so long as $$a<0$$ and $$b>0$$.
 * (c) 7.

Explanation

(a): Because $$a$$ is negative, $$a<0$$. Because $$b$$ is positive, $$b>0$$. Finally, because $$a-b\ne 0$$, it means that it is not the case that $$a-b=0$$, so $$\neg(a-b=0)$$. Knowing that these statements are in tandem with each other according to the English language, all statements are combined with an and. All a "but" means is an "and," with emphasis.

(b): Explanation already given in Answers.

(c): Because there are three variables, and each of either a $$T$$ or $$F$$ truth value, there are $$2\cdot 2\cdot 2=2^{3}=8$$ possible truth values for the final open sentence. The most organized way to check how many possible open sentences are false is through the truth table.
 * $$\begin{array}{|c c c||c||c||c|}

\hline P & Q & R & P\wedge Q & \neg R & P\wedge Q\wedge\neg R \\ \hline T&T&T & T & F & F\\ T&T&F & T & T & T\\ T&F&T & F & F & F\\ T&F&F & F & T & F\\ F&T&T & F & F & F\\ F&T&F & F & T & F\\ F&F&T & F & F & F\\ F&F&F & F & T & F\\ \hline \end{array}$$

More will be added later.

Logical Equivalences
More will be added later.

Symbolic Logic
More will be added later.

Quantifiers
More will be added later.

Applying Laws of Real Numbers to Verify and Prove Mathematical Properties
Verification is an easy task since all you have to do is rewrite the equation we see above and change only one side. Because only one side changes, a chain of transitive properties may be applied to that side, until one reaches the final conclusion: a simple statement as a property of equality to itself. This may be easier to show than explain, so simply follow along with us.

$$

According to the definition of division, $$\frac{a}{c}=a(c)^{-1}$$. Since this is true per each division found in the right expression, can be rewritten as the following:

$$

Notice how each term in the right-hand side of the equation has a factor of $$c^{-1}$$ therein. Because that is true, the following is true of the right-hand side of the equation.

$$

By the definition of division, the right-hand side of the equation is equivalent to $$\frac{a+b}{c}$$, so

$$

Notice how the chain of equations herein can connect to every equality. Since and  is true, we know the following equation below is true:

$$

Because and  is true, we know this below equation is true, and so on:

$$

$$

As shown in, we have verified the truth, and are therefore done with the problem. The "chain of equations" below shall perhaps show this chain of "change of one side only."
 * $$\begin{align}

\frac{a+b}{c}&=\frac{a}{c}+\frac{b}{c}\\ &=a(c)^{-1}+b(c)^{-1} & \text{Definition of fractions.}\\ &=c^{-1}(a+b) & \text{Factor }c^{-1}\text{ from }a(c)^{-1}+b(c)^{-1}\text{.}\\ &=\frac{a+b}{c} & \text{Definition of fractions.} \end{align} $$

It is important to take note the language of the directive verbs found within the prompt in the example above. By asking the examinee to verify, it, in effect, tests the understanding of the writer to communicate how a certain statement can be made true by only validly changing one side.

In the example above, we decided to change the right-hand side of Equation. Here is a question for the reader: "Why did we decide to change that side only?" If you can answer this question with precision before the next two examples, then it is safe to infer that this student understands how to verify literal equations.

''Note: despite being an important skill, it would be impossible to show this type of understanding on a multiple choice exam. Nevertheless, the journey of the mathematician is of the skeptic of statements made. You cannot fully regard a statement as true if you do not know the proof of it or the verification of it.''

This equation is not literal, so it is not necessary to change only one side to become the other. Nevertheless, it is not necessary one needs to solve the single-variable equation either. Instead, it may be easiest to simply substitute $$x=12$$ into the above equation. We shall do exactly that.
 * $$12(12)-12=11(12)$$
 * $$144-12=132$$
 * $$132=132$$

Because both sides of the equation are equal, we have verified $$x=12$$ is true.

One more example will be added later.

This problem is a little different from the previous ones because the examinee needs to show something is true by what they are given, as well as demonstrate something is true through the derivation of different formulae and equations from fundamental properties. The importance of mathematical communication is what is tested here. The mathematician needs to describe what is given, or else, the proof will not follow logically as a form of deductive reasoning.

We cannot follow the strategy we had for the previous two problems. This is because we are trying to communicate that a statement is beyond a shadow of a doubt true! With verification, we could assume the final statement is true, and thereby work backwards from where we started. However, when proving something, we have to show that when only given one statement, we can fully derive the other side without going backwards from where we started. Understanding this distinction is crucial!

Let us first start with rewriting the expression $$\frac{a}{b}\cdot\frac{c}{d}$$ into its equivalent: $$ Notice how we can apply a property of multiplication, that multiplying real numbers (and by extension integers) is associative. Therefore, we may rewrite as the following: $$ has the term $$b^{-1}d^{-1}$$. We can apply a property (proven in the exponents chapter) that $$b^{-1}\cdot d^{-1}=(bd)^{-1}$$. As such, we learn $$ With this, we are done with the proof.

There is a very subtle distinction here between a proof and a verification. If we were asked to verify the statement above, we would assume what we are given here is true and simply only change one side. However, notice how we started the proof in Example 3.1.d: "Let us first start with rewriting the expression..."

In a proof, starting with the conclusion is absolutely wrong and should not be allowed because such an argument is circular! Whenever working with deductive proofs, we start with what is given and try to derive a true statement from what we started. This is one of the fundamental properties of deductive proofs.

Usually, in proofs, especially in deductive ones, there are two principles in mind that are necessary in the development of further understanding: A "universal proposition," such as a theorem or definition, will imply a "singular proposition," such as a premise, conclusion, or intermediate conclusion. Either that or a singular proposition implies another singular proposition.

For this proof, the former is true (a universal proposition implied a singular proposition). By such an action, we can verify the validity of this proof. Although one may not call this a formal proof, per se, such proof did indeed follow from verified axioms, and since those axioms are true, and it led to a truth statement, this proof is valid.

As a student further explores mathematics, principles of proof and the formality of such proofs will be very important in the life of the mathematician. In exploration of such concepts, a deeper understanding will be essential, where further examination of logic will be in order, and different types of proofs will be categorized, explained, and used, sometimes with little delay in thought.

While this is not meant to be a further exploration nor explanation of proofs in general, this book does seek to further a student's foundation of such concepts before one gets into a discrete math class.

Let us first start with what we are given: $$\frac{a}{b}+\frac{c}{d}$$. One common trick done in proofs is to find a way to change the expression to make things easier on us without changing the meaning of the expression itself. For example, if one wanted to find the solution to $$2x+5=10$$, one may want to isolate the variable $$x$$. However, that cannot be done if the five is in the way, so one could subtract the five. This would change the meaning of the equation unless it is also done to the other side.

This same principle could apply to proofs. Here are some common ways in which a mathematician could make their lives easier. Let $$xyz$$ be an expression of interest: The last bullet point shall be the trick we employ for this proof. After all, we are working with fractions. However, this constant cannot simply be any number. We shall be clever and select a number that will eliminate certain terms within. This will make our life easier in the long run.
 * One could square $$xyz$$, then take the square root: $$xyz=\sqrt{(xyz)^{2}}$$.
 * One could take the natural logarithm of $$xyz$$, then have a base $$e$$ in the exponent: $$xyz=e^{\ln(xyz)}$$.
 * One could add a constant to $$xyz$$, then subtract that constant again: $$xyz=xyz+c-c$$.
 * One could multiply a constant $$xyz$$, then divide the constant: $$xyz=xyz\cdot\frac{c}{c}$$.

First, we begin by stating the Equality Postulate (an item must equal itself): $$ From there, we multiply $$\frac{bd}{bd}$$ to the right side. After all, this will simplify to, so nothing changes. $$ Next, apply the distributive property to the right hand side of. $$ Here is the special part of the proof. Notice how that there are two terms in which two fractions are multiplied against another. We can use the property we proved in Example 3.1.d. $$ As before, we colored all terms that are red as terms that can be canceled. We will leave this as a trivial exercise for the reader to prove this is true. This gives us our subsequent equation: $$ We can rewrite the rightmost side of as $$ad{\color{Blue}(bd)^{-1}}+bc{\color{Blue}(bd)^{-1}}$$. The factorable term is in blue. Therefore, we know the following can be rewritten as the final equation. $$ The entire argument above can be condensed into the following compact argument:
 * $$\begin{align}

\frac{a}{b}+\frac{c}{d}&=\frac{a}{b}+\frac{c}{d}&(\text{Equality postulate.})\\ &= \left(\frac{a}{b}+\frac{c}{d}\right)\frac{bd}{bd}&\left(\frac{bd}{bd}=1\right)\\ &= \frac{a}{b}\cdot\frac{bd}{bd}+\frac{c}{d}\cdot\frac{bd}{bd} &(\text{Distributive property.})\\ &= \frac{abd}{b\cdot bd}+\frac{cbd}{d\cdot bd}\\ &= \frac{ad}{bd}+\frac{bc}{bd}\\ &= ad(bd)^{-1}+bc(bd)^{-1}\\ &=(bd)^{-1}(ad+bc)&\left(\text{Factor }(bd)^{-1}\text{.}\right)\\ &= \frac{ad+bc}{bd}\blacksquare \end{align}$$

Applying Laws of Real Numbers to Derive New Mathematical Properties
Proofs are not only useful when it comes to algebraic statements but also mathematical properties in general. It may be the case that these such items are much more useful towards the latter than the former because the laws of real numbers are by themselves always true. Although empirical evidence satisfy scientists, mathematicians differ in that regard.

If at some point, we found a location in which an mathematical "property" were to be false, then we would need to modify that statement. From thereon, we need to do the same to other items that also use this property, which may include absolute statements, such as theorems. If a specific location fails and that specific location is integral to the proof, then the proof of that theorem is automatically invalid. We do not wish for this "domino effect" to occur, so we must prove statements are true through deduction (or induction, which we will not discuss).

Let us take a look at division properties again.

In both proofs (a) and (b), it relied on the property that $$a\cdot0=0$$. While there is a zero factor theorem, this does not necessarily mean that the products of two numbers, one being zero, equals zero. Therefore, this proof may be false. One needs to prove that any number times zero equals zero.

This may seem a little nit-picky. However, this can be very important when it comes to mathematical properties. If we assumed this were true, we may fundamentally be wrong about some solutions to rational equations, may fundamentally break some models of the universe, and may fundamentally conclude absurd ideas due to a wrong assumption. Therefore, mathematicians need to be picky!

Let us therefore start this section with a proof of this statement: any number times zero is equal to zero.

Let $$a\in\mathbb{R}$$ be multiplied by zero.
 * $$\begin{align}

a\cdot0 &= a\cdot0\\ &= a\cdot\left(0+1-1\right) & \text{(Proven later that }b-b=0\text{ in Exponents chapter.)}\\ &= a\cdot0 + a\cdot1 - a\cdot1 & \text{(Distributive Property.)}\\ \end{align}$$ Subtract $$a\cdot0$$ to both sides of equation.
 * $$\Rightarrow a\cdot0 -a\cdot0= a\cdot1 -a\cdot1$$

Suppose $$a\cdot0=y$$
 * $$\Rightarrow y-y=a-a$$

Already shown that $$b-b=0$$. Ergo,
 * $$\Rightarrow a\cdot0-a\cdot0=y-y=a-a=0$$

By transitive property,
 * $$\Rightarrow a\cdot0-a\cdot0=0$$

Factor $$a$$:
 * $$\Rightarrow a\left(0-0\right)=0$$

By the $$b-b=0$$ property,
 * $$\Rightarrow a\cdot0=0\blacksquare$$

In the mean time, the property we used for this proof will not be proven until the Exponents chapter. For now, this proof should be sufficient grounds to accept the properties of rational numbers as shown in Example 3.2.a.

The next property we will prove has usually been proven in Geometry classes due to its simplicity. However, will do these two proofs next because we want to show a semi-formal proof.

Let $$n\in\mathbb{Z}$$, $$m\in\mathbb{Z}$$, and $$k\in\mathbb{Z}$$. If $$N=2n$$, then $$N$$ is even. If $$N=2n+1$$, then $$N$$ is odd. Let $$a=2m$$ and $$b=2k$$.
 * $$\begin{align}

a+b &= 2m+2k\\ a+b &=2(m+k) \end{align}$$ Let $$m+k=p\in\mathbb{Z}$$.
 * $$\Leftrightarrow a+b = 2p$$

From this, if $$a,b$$ are even, then $$a+b$$ is even. $$\blacksquare$$

This proof will be finished later.

More will be added later.


text:
History of video games/How to Contribute

This section contains information related to the specific Wikibook History of video games.

For a general manual of style please read The Wikibooks Manual of Style.

Structure
Similar to the Wikibook Horticulture, this book is arranged in a mostly flat structure, going deeper when needed.

Local Manual of Style
Italicize the names of specific games, just as one would specific movies, and books.

Inline references, with a if used more then once. Read more about Inline References here. A surprising amount of common knowledge about gaming history is bunk, and is often based on rumors, sometimes a result of deceptive marketing or poor translations. Beyond being the right thing to do, citing sources allows claims to be evaluated should new information emerge, and allows those wanting to fact check a paper trail to do so.

Wikilinks should be used for internal book navigation. Wikilinks to other Wikibooks, Wikipedia, or other Wikimedia projects should make it clear that they navigate away from the book.

Images from Wikimedia Commons should be used if they help the reader understand the subject material better. Sometimes there won't be freely usable images. In these cases, link to a website that has the images at the bottom of the page, so as to give the reader a lead.

Device Pages
There have been thousands of models of different gaming computers and consoles designed and mass-produced since such the start of the industry. Attempting to dedicate chapters to each device unconditionally would result in a massive work worthy of its own book. However to ignore hardware would be a disservice to the reader, since a good deal of technological, marketing, economic, and design history derives from the hardware on which games are played.

In general a device should be added as a page if something about it would make it interesting to the reader, including as a topic of special interest. To aid the reader in this, on console pages the overall position a console held in the market is clearly marked. For example, most readers would care little for the Casio Loopy, an obscure game console from the 1990's that never left Japan, saw few games of note, and apparently didn't sell all that well. Yet in the intersection of Feminism and gaming, this console is an important subject as one of the few gaming devices marketed exclusively to women. That alone makes it a fascinating subject worthy of having it's history explored, albeit within its own niche.

Basic Game Info
Device pages should note a small number of their most prominent games by earliest release year regardless of territory. These games should have a small blurb about what made them important, their genre, and interesting facts about their development. Then link the Wikipedia article for that game so the reader can get more info if they choose.

Cross platform games without significant version differences are treated like games on device pages, except they go in timeline chapters.

The exception is listing all of games for a device if there were only a handful of games released for that device. If the games are very simple or self explanatory on the name, then a simple list suffices.

Detailed Game Info
It may seem strange that influential games are often not given more then a sentence or paragraph. The reason for this is that other Wikibooks like the Metal Gear Series book and the 0 A.D. book cover such matters in depth, so to avoid redundancy such material is only briefly explored when needed in this book. New in depth material for a series should go in an appropriate book - if no such book exists, and the topic is appropriate for Wikibooks you can make one!

There are exceptions for landmark titles. Ones which defined a genre or had major impacts on the industry.

Strategy guides for specific games
While specific novel mechanics may be mentioned and explained occasionally, as well as the subject of strategy guides themselves from a historical perspective, they are not a substitute for a strategy guide offering advice for a specific game. See Strategy guides for assistance creating strategy guides.

Every gaming fact ever
This book makes no attempt to cover all facts about the history gaming, simply the most important and most interesting ones. For notable topics in gaming, there is Wikipedia. This book is not an attempt at an all inclusive gaming encyclopedia, nor should it be. It's fine and important to mention these topics where relevant, but link to Wikipedia instead of remaking the wheel here.

In general the following are excluded
 * Detailed biographies of significant game developers. While facts about game developers may be brought up in relation to the topic at hand, full biographies are usually covered adequately on Wikipedia.
 * Chapters for specific companies. These are typically brought up briefly on the chapter concerning their first product instead.

A succinct history of gaming
There are already many books which cover the most important aspects of gaming history, the best selling consoles, and the people who lead them. Those stories also belong in this book.

However there are many more stories of failure, reckless ambition, and once popular games forgotten. For the benefit of the reader, failures, canceled projects, and other less glamorous subjects are not excluded, because they are also part of gaming history. Without them, it is hard for the reader to understand that the road to a successful gaming industry project has been historically wrought with peril and fumbles. In some cases, failures and limited releases have been important tools in a company strategy to succeed elsewhere, and these cases certainly should not be ignored.

In general, if a subject received significant citable coverage, it should go in regardless of its overall success or impact.

Expanding Chapters
When looking into the history of a topic in gaming there are a few things to consider.
 * Who was involved. Was there anyone important or well known as a major player? Why?
 * Where was an event held, or company located?
 * What technology was used. Was a design pedestrian or unusual? Why?
 * How expensive was it? How many sold? Did it turn a profit? How long was it on the market? What factors lead to these conditions?
 * How did it impact the industry? Did anyone try to copy it? What about specifically avoiding it? Does nobody remember it? Why?
 * What was its legacy? A failure? A success? Obscure? Why?
 * What is its color palate? Design language? Was it a bunch of puffy amebas or edgy angles? What typeface does it use? Were other things of the time like that?
 * What is it named that? What marketing campaign was used to sell it?
 * What organizations outside of the video game industry were involved? Who, where, and why?

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
User talk:Adrignola/2017

Un finnished book
Umm Atcovi told me you have ino on the highschool life science book. Im making it for now, but i realized the book has already been made, so what do i do? L3g3nd27 (discuss • contribs) 03:34, 15 March 2017 (UTC)
 * You are free to work on converting that book's PDF source material into content on the wiki. No one person owns or has claim on a book here.  Thanks for reaching out though, since we do still have to cooperate with one another if two or more people are working on the same book.  While I laid out the structure of the book with links based on the source material, I no longer have the time to finish it myself.

 * If you are instead writing a book on the same topic but not basing it on the PDFs linked from the main page of the book, you'll want to take a look at the source material and determine if your own end result would be superior or if it would be better as its own separate book alongside a wiki version of those PDFs. Alternatively, you may decide to combine both approaches and take the content of the PDFs, bring the content in to the wiki, and improve upon it from there.  Even the CK-12 books that I had fully brought in prior to this point are now living documents that can be made better and edited over time.  Anyone wanting a static document can just download the PDFs, but a wiki is a collaborative medium and so your contributions are most welcome.  – Adrignola discuss 09:36, 15 March 2017 (UTC)

Ok, thats good to know. I thought i had to write the entire book my self! L3g3nd27 (discuss • contribs) 12:55, 17 March 2017 (UTC)

Speedy deletion nomination of "Programming:WebObjects/Web Applications/Development/Examples/Login"
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title: User talk:Adrignola/2017


text:
User talk:Q28

Welcome, IN!

If you have any questions, you can ask in the assistance reading room or possibly contact me personally.--IN (discuss • contribs) 09:34, 4 February 2021 (UTC)

title: User talk:Q28


text: Shelf:Wikijunior early reader books

title: Shelf:Wikijunior early reader books


text: Category:Shelf:Wikijunior early reader books/all books


text: Category:Shelf:Wikijunior early reader books


text: Shelf:Wikijunior early reader books/ancestry


text: Category:Book:Wikijunior:The Rat and the Cat Can Run


text: Category:Subject:Wikijunior early reader books


text:
Wikijunior:Rush to the mart

This is an early reader phonics book to teach the digraph "sh".

Rush to the mart.

Rush to the mart.

A ship!

Pay with cash.

Rush to the mart!

A shell.

Pay with cash.

Rush to the mart.

Buy a fish.

Pay with cash!

Shall we rush to the mart again?

title: Wikijunior:Rush to the mart


text: Category:Subject:Wikijunior early reader books/all books


text: Category:Book:Wikijunior:Rush to the mart


text:
Wikijunior:Yeet

This is an early phonics reader to teach the digraph 'ee'.

Yeet

Dee.

A bee. Yeet the bee, Dee!

The bee is mad.

Can a bee yeet Dee?

No.

A seed.

Can a bee yeet the seed?

Yes!

Feet.

The bee has six feet.

Can you see the bee's feet?

Dee.

Dee has two feet.

Can you see Dee's feet?

Can you yeet your feet?

No.

title: Wikijunior:Yeet


text: Category:Book:Wikijunior:Yeet

title: Category:Book:Wikijunior:Yeet


text:
Wikijunior:Loan me a boat

This is an early phonics reader to teach the digraph 'oa'. Loan me a boat

Loan me a boat

For the moat.

To get the toad

and the goat.

Be a bud!

Loan me the soap.

There is mud,

On the goat.

title: Wikijunior:Loan me a boat


text:
Wikijunior:The Cow that said Bow Wow

This is an early reader phonics book to teach the digraph "ow".

The Cow that said "Bow Wow"

"Bow Wow" said the Cow.

"Bow Wow" said the Owl.

Now YOU say Bow Wow!

"Down", I said to the dog.

"Moo", said the dog.

Now YOU say moo, too!

title: Wikijunior:The Cow that said Bow Wow


text:
Understanding Air Safety in the Jet Age

Civilian airliners rarely crash, but when they do it is big news and leads to serious questions about the safety of flying. Is flying really the safest form of travel? It depends how "safety" is measured. To understand the risk, and make decisions, it's helpful to understand how incidents and accidents occur, and the factors that add to your chances of avoiding or surviving an incident.

This book examines the history of civilian air safety since the beginning of the jet age. The first commercial civilian jet airliner was the de Haviland Comet. It first flew in prototype form in 1949 with the first production aircraft flying on 9 January 1951. The first flight with paying passengers, Comet registration G-ALYS, took off on 2 May 1952. The first accident happened less than a year later on 3 March 1953. A Comet 1A, registration CF-CUN, failed to take off on a delivery flight and crashed killing all 11 people on board. It was the beginning of a grim toll of incidents and accidents that continues to this day. However, all is not "doom and gloom". By making sensible choices, you affect the odds of being involved in an incident, and your chances of surviving it. Also, knowing what to expect in an accident and planning what to do if it happens will help.

Part 1: Introduction

 * /How Safe is Flying?/
 * /The Dawn of the Jet Age/

Part 2: Mechanical Failure

 * /Metal Fatigue/
 * /Terror at 22,000 ft - United Airlines Flight 811/
 * /Impossible Landing - United Airlines Flight 232/

 * /Structural Failure/
 * /Fire/
 * /Fire Down Below - ValuJet Flight 592/

 * /Getting the Basics Right/
 * /Hanging on by the Fingertips - British Airways Flight 5390/

Part 3: Design Failure

 * /Falling from the Sky - The DC10 Panic/
 * /Disaster Foreshadowed - American Airlines Flight 96/
 * /Explosive Decompression - Turkish Airlines Flight 981/
 * /The Final Straw - American Airlines Flight 191/

 * /Falling from the Sky Again - The 737 Max Panic/
 * /Bad Design, Bad Maintenance - TWA 800/

Part 4: Human Factors

 * /Pull Up! Pull Up!/
 * /Terrorism and Pilot Suicide/
 * /Mass Murder by Pilot - Germanwings Flight 9525/

 * /Taking Shortcuts/
 * /Air Florida Flight 90/

Part 5: Acts of God?

 * /Wildlife Encounters/
 * Bird Strikes
 * A Dip in the Hudson - Cactus 1549

 * /Ice and Rain/

Part 6: Surviving Disaster

 * /Common Themes/
 * /Being a Survivor/

Information for Contributors

 * /Style Guide and Authors/

This book is dual licensed under the GFDL and Creative Commons Attribution-ShareAlike 3.0 Unported License.

title: Understanding Air Safety in the Jet Age


text:
Understanding Air Safety in the Jet Age/Style Guide and Authors

Please follow the style guide. Changes to the guide should be proposed on the Talk page here for agreement of consensus between contributors.

Style Guide

 * This book is written in British English. Don't worry if you write in American English - it will be changed to British English by the next editor.
 * This book is designed to be fully self contained. It is not an encyclopedic article and there are no links to Wikipedia. Where a term needs describing it is linked to the Glossary within the book.
 * In line references are not used. Where a reference is quoted, please include it in the References chapter, not within the individual chapters. This also applies to bare URLs and external links - put them in the references chapter.


text: Category:Book:Understanding Air Safety in the Jet Age


text:
Understanding Air Safety in the Jet Age/The Dawn of the Jet Age

The British de Havilland Comet was the first jet airliner to fly (1949), the first in service (1952), and the first to offer a regular jet-powered transatlantic service (1958). One hundred and fourteen of all versions were built but the Comet 1 had serious design problems, and out of nine original aircraft, four crashed (one at takeoff and three broke up in flight), which grounded the entire fleet. The Comet 4 solved these problems but the program was overtaken by the Boeing 707 on the trans-Atlantic run. The Comet 4 was developed into the Hawker Siddeley Nimrod which retired in June 2011.

Following the grounding of the Comet 1, the Tu-104 became the first jet airliner to provide a sustained and reliable service, its introduction having been delayed pending the outcome of investigations into the Comet crashes. It was the world's only jet airliner in operation between 1956 and 1958 (after which the Comet 4 and Boeing 707 entered service). The plane was operated by Aeroflot (from 1956) and Czech Airlines ČSA (from 1957). ČSA became the first airline in the world to fly jet-only routes, using the Tu-104A variant.

The first western jet airliner with significant commercial success was the Boeing 707. It began service on the New York City|New York to London route in 1958, the first year that more trans-Atlantic passengers traveled by air than by ship. Comparable long-range airliner designs were the DC-8, VC10 and Il-62. The Boeing 747, the "Jumbo jet", was the first widebody aircraft that reduced the cost of flying and further accelerated the Jet Age.

One exception to the domination by turbofan engines was the turboprop-powered Tupolev Tu-114 (first flight 1957). This airliner was able to match or even exceed the performance of contemporary jets, however the use of such powerplants in large airframes was restricted to the military after 1976.

Jet airliners are able to fly much higher, faster, and further than piston-powered propliners, making transcontinental and intercontinental travel considerably faster and easier than in the past. Aircraft making long transcontinental and trans-oceanic flights could now fly to their destinations non-stop, making much of the world accessible within a single day's travel for the first time. As demand grew, airliners became larger, further reducing the cost of air travel. People from a greater range of social classes could afford to travel outside of their own countries.

General aviation
The use of mass-production techniques similar to those of the motor industry lowered the cost of private aircraft, with types such as the Cessna 172 and Beechcraft Bonanza seeing widespread use, the 172 eclipsing even wartime production levels.

Aircraft came to be used increasingly in specialist roles such as crop spraying, policing, fire fighting, air ambulances and many others.

As helicopter technology developed, they also came into widespread use, dominated by Sikorsky's approach of a single main rotor plus tail counter-torque rotor.

Sport flying also developed, with both powered aeroplanes and gliders becoming more sophisticated. The introduction of glass fibre construction allowed sailplanes to achieve new levels of performance. In the 1960s the re-introduction of the hang-glider, now using the flexible Rogallo wing, ushered in a new era of ultralight aircraft.

The development of safe gas burners led to the re-introduction of hot air ballooning, and it became a popular sport.

Supersonic transport
The introduction of the Concorde supersonic transport (SST) airliner to regular service in 1976 was expected to bring similar social changes, but the aircraft never found commercial success. After several years of service, the fatal crash of Air France Flight 4590 near Paris in July 2000 and other factors eventually caused Concorde flights to be discontinued in 2003. This was the only loss of an SST in civilian service. Only one other SST design was used in a civilian capacity, the Soviet era Tu-144, but it was soon withdrawn due to high maintenance and other issues. McDonnell Douglas, Lockheed and Boeing were three U.S. manufacturers that had originally planned to develop various SST designs since the 1960s, but these projects were eventually abandoned for various developmental, cost, and other practical reasons.

Manufacturing
The fabrication of riveted stressed-skin aluminium airframes was widespread by the end of the Second World War, although the use of wood for private aviation continued. The pursuit of greater strength for less weight led to the introduction of advanced, and often expensive, manufacturing techniques. Key developments during the 1960s and 70s included; milling a complex part from a solid billet rather than building it up from smaller parts, the use of synthetic resin adhesives in place of rivets to avoid stress concentrations and fatigue around the rivet holes, and electron beam welding.

The development of composite materials such as fibreglass and, later, carbon fibre, freed up designers to make more fluid, aerodynamic shapes. However the unknown properties of these novel materials meant that introduction has been slow and methodical.

Airports
Many military aerodromes became civilian airports after the war, while pre-war airports reverted to their former role. The rapid growth in air travel ushered in by the jet age required an equally rapid enlargement of airport facilities worldwide.

As jet airliners grew larger and passenger numbers per flight increased, larger and more sophisticated equipment was developed for handling the aircraft, passengers and baggage.

Radar systems became commonplace, with Air traffic control facilities needed to manage the large number of aircraft in the sky at any one time.

Runways were made longer and smoother to accommodate new, larger and faster aircraft, while safety considerations and night flying led to much improved runway lighting.

Major airports became such vast and busy places that their environmental impact became substantial and the siting of any new airport, or even the expansion of an existing one, became a major social and political affair.


text: MediaWiki:Abusefilter-warning-spam-willblock


text:
Wikijunior talk:Yeet

deletion nomination
This is an early reader to assist with learning the digraph "ee."

They do tend to resemble nonsense but this is in fact how we teach kids to read. See a similar work by Dr Seuss. Cheers. Mvolz (discuss • contribs) 21:24, 4 February 2021 (UTC)

title: Wikijunior talk:Yeet


text: Talk:Exercise as it relates to Disease/The energy expenditure from combat sports and martial arts training and how it can help reach health recommendations


text: Chinese (Mandarin)/Buildings

title: Chinese (Mandarin)/Buildings


text: Chinese (Mandarin)/Engineering

title: Chinese (Mandarin)/Engineering


text:
User talk:Absolute Reality

Unblocked
I apologise for your being blocked, it was an automated filter that I set up to prevent abuse. Leaderboard (discuss • contribs) 08:19, 5 February 2021 (UTC)

title: User talk:Absolute Reality


text:
Wikijunior:Early phonics readers

Early phonics readers are intended to teach children how to read in a systematic way using common sounds. These phonics readers are designed around the UK National Curriculum approach, and are suitable for other curricula as well. They should be done in order.

Phase 2 phonics books

 * g (tag, gag, sag, gas, pig) -
 * ss (less, hiss, mass, mess, boss) -

Phase 3 phonics books

 * j - Johnny and Jill
 * v - The Violet Van
 * w (we swim wide swirling waves) -
 * x  -
 * y  -
 * z  -
 * bl & br - Boy Blue Boldly Brags
 * zz (a puzzled buzzard guzzles fizzy water) -
 * qu (quiet quick! quit quacking) -
 * ch (a bunch of crunchy french fries) -
 * cl (claws clenched on a clump of clothes -
 * sh - Rush to the Mart
 * th - The Cat is Thin
 * ng (King Cole Sings a Song) -
 * ai (tail in the rain) -
 * ee - Yeet the Bee
 * igh (a light in the night) -
 * oa - Loan me a Boat
 * oo - My Spoon is Too Small
 * oo - This is a Book
 * ar (march in the dark) -
 * or (born with a horn) -
 * ur (curl the fur) -
 * ow - The Cow that said Bow Wow
 * oi - The Soup Pot
 * ear (get near to hear) -
 * air (chair in the air) -
 * ure (picture a natural mixture of pure manure) -
 * er (a bigger digger) -

Additional phonics books

 * The Rat and the Cat Can Run

title: Wikijunior:Early phonics readers


text:
Wikijunior:The cat is thin

This is an early reader phonics book to teach the sound 'th'.

The cat is thin.

This cat is thin.

That cat is thick.

The cat with the spots is too thin.

Get them food!

title: Wikijunior:The cat is thin


text:
User talk:E p hajder

Welcome
Welcome to Wikibooks, E p hajder!

 First steps tutorial Wikibooks is for collaborative development of free textbooks.

You do not need technical skills to contribute. You can easily change most books. Please introduce yourself, and let us know what interests you.

If you already contribute at other Wikimedia projects, our Wikimedia Orientation should quickly get you started. (Would you like to provide feedback on this message?)

  Getting help   Made a mistake?   Goodies, tips and tricks
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--Pi zero (discuss • contribs) 13:51, 5 February 2021 (UTC)

title: User talk:E p hajder


text:
User talk:Splibubay

Welcome!
Welcome, Splibubay!

If you have any questions, you can ask in the assistance reading room or possibly contact me personally. —Atcovi (Talk - Contribs) 14:51, 5 February 2021 (UTC)

title: User talk:Splibubay


text:
Understanding Air Safety in the Jet Age/Wildlife Encounters/Bird Strikes/Cactus 1549

On January 15, 2009, US Airways Flight 1549, an Airbus A320 on a flight from New York City's LaGuardia Airport to Charlotte, North Carolina, struck a flock of birds shortly after take-off, losing all engine power. Unable to reach any airport for an emergency landing, pilots Chesley Sullenberger and Jeffrey Skiles glided the plane to a ditching in the Hudson River off Midtown Manhattan. All 155 people on board were rescued by nearby boats, with a few serious injuries.

This water landing of a powerless jetliner became known as the "Miracle on the Hudson", and a National Transportation Safety Board official described it as "the most successful ditching in aviation history". The Board rejected the notion that the pilot could have avoided ditching by returning to LaGuardia or diverting to nearby Teterboro Airport.

The pilots and flight attendants were awarded the Master's Medal of the Guild of Air Pilots and Air Navigators in recognition of their "heroic and unique aviation achievement".

Background

On January 15, 2009, US Airways Flight 1549 with call sign 'Cactus 1549' was scheduled to fly from New York City's LaGuardia Airport (LGA) to Charlotte Douglas (CLT), with direct onward service to Seattle–Tacoma International Airport. The aircraft was an Airbus A320-214 powered by two GE Aviation/Snecma-designed CFM56-5B4/P turbofan engines.

The captain and pilot in command was 57-year-old Chesley B. Sullenberger, a former fighter pilot who had been an airline pilot since leaving the United States Air Force in 1980. At the time, he had logged 19,663 total flight hours, including 4,765 in an A320; he was also a glider pilot and expert on aviation safety. First officer Jeffrey Skiles, 49, had accrued 20,727 career flight hours with 37 in an A320, but this was his first A320 assignment as pilot flying. There were 150 passengers and three flight attendants on board.

Takeoff and bird strike
The flight was cleared for takeoff to the northeast from LaGuardia's Runway 4 at 3:24:56 pm Eastern Standard Time (20:24:56 UTC). With Skiles in control, the crew made its first report after becoming airborne at 3:25:51 as being at 700 ft and climbing.

The weather at 2:51 p.m. was 10 mi visibility with broken clouds at 3,700 ft, wind 8 knots from 290°; an hour later it was few clouds at 4,200 ft, wind 9 knots from 310°. At 3:26:37 Sullenberger remarked to Skiles: "What a view of the Hudson today."

At 3:27:11 during climbout, the plane struck a flock of Canada geese at an altitude of 2818 ft about 4.5 miles north-northwest of LaGuardia. The pilots' view was filled with the large birds; passengers and crew heard very loud bangs and saw flames from the engines, followed by silence and an odor of fuel. Realizing that both engines had shut down, Sullenberger took control while Skiles worked the checklist for engine restart. The aircraft slowed but continued to climb for a further 19 seconds, reaching about 3,060 ft at an airspeed of about 185 kn, then began a glide descent, accelerating to 210 kn at 3:28:10 as it descended through 1,650 ft.

At 3:27:33, Sullenberger radioed a mayday call to New York Terminal Radar Approach Control (TRACON): "... this is Cactus 1539 [sic – correct call sign was Cactus 1549], hit birds. We've lost thrust on both engines. We're turning back towards LaGuardia". Air traffic controller Patrick Harten told LaGuardia's tower to hold all departures, and directed Sullenberger back to Runway 31. Sullenberger responded, "Unable".

Sullenberger asked controllers for landing options in New Jersey, mentioning Teterboro Airport. Permission was given for Teterboro's Runway 1, Sullenberger initially responded "Yes", but then: "We can't do it ... We're gonna be in the Hudson". The aircraft passed less than 900 ft above the George Washington Bridge. Sullenberger commanded over the cabin address system, "Brace for impact", and the flight attendants relayed the command to passengers. Meanwhile, air traffic controllers asked the Coast Guard to caution vessels in the Hudson and ask them to prepare to assist with rescue.

Ditching and evacuation

About ninety seconds later, at 3:31 pm, the plane made an unpowered ditching, descending southwards at about 125 kn into the middle of the North River section of the Hudson tidal estuary, at 	40.76944°N, -74.00444°W on the New York side of the state line, roughly opposite West 50th Street (near the Intrepid Sea, Air & Space Museum) in Manhattan and Port Imperial in Weehawken, New Jersey. Flight attendants compared the ditching to a "hard landing" with "one impact, no bounce, then a gradual deceleration." The ebb tide then began to take the plane southward.

Sullenberger opened the cockpit door and gave the order to evacuate. The crew began evacuating the passengers through the four overwing window exits and into an inflatable slide/raft deployed from the front right passenger door (the front left slide failed to operate, so the manual inflation handle was pulled). The evacuation was made more difficult by the fact that someone opened the rear left door, allowing more water to enter the plane; whether this was a flight attendant or a passenger is disputed. Water was also entering through a hole in the fuselage and through cargo doors that had come open, so as the water rose the attendant urged passengers to move forward by climbing over seats. One passenger was in a wheelchair. Finally, Sullenberger walked the cabin twice to confirm it was empty.

The air and water temperatures were about -7 C and 5 C, respectively. Some evacuees waited for rescue knee-deep in water on the partially submerged slides, some wearing life vests. Others stood on the wings or, fearing an explosion, swam away from the plane. One passenger, after helping with the evacuation, found the wing so crowded that he jumped into the river and swam to a boat.

Rescue
Sullenberger had ditched near boats, which facilitated rescue. Two NY Waterway ferries arrived within minutes and began taking people aboard using a Jason's cradle; numerous other boats, including from the US Coast Guard, were quickly on scene as well. Sullenberger advised the ferry crews to rescue those on the wings first, as they were in more jeopardy than those on the slides, which detached to become life rafts. The last person was taken from the plane at 3:55 pm.

About 140 New York City firefighters responded to nearby docks, as did police, helicopters, and various vessels and divers. Other agencies provided medical help on the Weehawken side of the river, where most passengers were taken.

Aftermath

Among the people on board there were 95 minor injuries and 5 serious, including a deep laceration in flight attendant Doreen Welsh's leg. Seventy-eight people were treated, mostly for minor injuries and hypothermia; twenty-four passengers and two rescuers were treated at hospitals, with two passengers kept overnight. One passenger now wears glasses because of eye damage from jet fuel. No pets were being carried on the flight.

Each passenger later received a letter of apology, $5,000 in compensation for lost baggage (and $5,000 more if they could demonstrate larger losses), and refund of the ticket price. In May 2009, they received any belongings that had been recovered. In addition, they reported offers of $10,000 each in return for agreeing not to sue US Airways.

Many passengers and rescuers later experienced post-traumatic stress symptoms such as sleeplessness, flashbacks, and panic attacks; some began an email support group. Patrick Harten, the controller who had worked the flight, said that "the hardest, most traumatic part of the entire event was when it was over", and that he was "gripped by raw moments of shock and grief".

In an effort to prevent similar accidents, officials captured and gassed 1,235 Canada geese at 17 locations across New York City in mid-2009 and coated 1,739 goose eggs with oil to smother the developing goslings. To date (2017) 70,000 birds have been intentionally slaughtered in NYC as a result of the Hudson ditching.

Investigation

The partially submerged plane was moored to a pier near the World Financial Center in Lower Manhattan, roughly 4 mi downstream from the ditching location. The left engine, detached by the ditching, was recovered from the riverbed. On January 17 the aircraft was barged to New Jersey.

The initial National Transportation Safety Board (NTSB) evaluation that the plane had lost thrust after a bird strike was confirmed by analysis of the cockpit voice and flight data recorders.

Two days earlier the plane had experienced a less severe compressor stall, but the affected engine was restarted. A faulty temperature sensor was replaced, and inspection verified the engine had not been damaged in that incident.

On January 21, the NTSB found evidence of soft-body damage in the right engine along with organic debris including a feather. The left engine also evidenced soft body impact, with "dents on both the spinner and inlet lip of the engine cowling. Five booster inlet guide vanes are fractured and eight outlet guide vanes are missing." Both engines, missing large portions of their housings, were sent to the manufacturer for examination. On January 31, the plane was moved to Kearny, New Jersey. The bird remains were later identified by DNA testing to be Canada geese, which typically weigh more than engines are designed to withstand ingesting.

Because the plane was assembled in France, the European Aviation Safety Agency (EASA; the European counterpart of the FAA) and the Bureau d'Enquêtes et d'Analyses pour la Sécurité de l'Aviation Civile (BEA; the French counterpart of the NTSB) joined the investigation, with technical assistance from Airbus and GE Aviation/Snecma, respectively the manufacturers of the airframe and the engines.

The NTSB used flight simulators to test the possibility that the flight could have returned safely to LaGuardia or diverted to Teterboro; only seven of the thirteen simulated returns to La Guardia succeeded, and only one of the two to Teterboro. Furthermore, the NTSB report called these simulations unrealistic: "The immediate turn made by the pilots during the simulations did not reflect or account for real-world considerations, such as the time delay required to recognize the bird strike and decide on a course of action." A further simulation, in which a 35-second delay was inserted to allow for those, crashed. In testimony before the NTSB, Sullenberger maintained that there had been no time to bring the plane to any airport and that attempting to do so would likely have killed those onboard and more on the ground.

The Board ultimately ruled that Sullenberger had made the correct decision, reasoning that the checklist for dual-engine failure is designed for higher altitudes when pilots have more time to deal with the situation, and that while simulations showed that the plane might have just barely made it back to LaGuardia, those scenarios assumed an instant decision to do so, with no time allowed for assessing the situation.

On May 4, 2010, the NTSB issued its final report, which identified the probable cause as "the ingestion of large birds into each engine, which resulted in an almost total loss of thrust in both engines." The final report credited the outcome to four factors: good decision-making and teamwork by the cockpit crew (including decisions to immediately turn on the APU and to ditch in the Hudson); the fact that the A320 is certified for extended overwater operation (and hence carried life vests and additional raft/slides) even though not required for that route; the performance of the flight crew during the evacuation; and the proximity of working vessels to the ditching site. Contributing factors were good visibility and fast response times from the ferry operators and emergency responders. The report made 34 recommendations, including that engines be tested for resistance to bird strikes at low speeds; development of checklists for dual-engine failures at low altitude, and changes to checklist design in general "to minimize the risk of flight crewmembers becoming stuck in an inappropriate checklist or portion of a checklist"; improved pilot training for water landings; provision of life vests on all flights regardless of route, and changes to the locations of vests and other emergency equipment; research into improved wildlife management, and technical innovations on aircraft, to reduce bird strikes; research into possible changes in passenger brace positions; and research into "methods of overcoming passengers' inattention" during preflight safety briefings.

Author and pilot William Langewiesche asserted that insufficient credit was given to the A320's fly-by-wire design, by which the pilot uses a side-stick to make control inputs to the flight control computers. The computers then impose adjustments and limits of their own to keep the plane stable, which the pilot cannot override even in an emergency. This design allowed the pilots of Flight 1549 to concentrate on engine restart and deciding the course, without the burden of manually adjusting the glidepath to reduce the plane's rate of descent. However, Sullenberger said that these computer-imposed limits also prevented him from achieving the optimum landing flare for the ditching, which would have softened the impact.

Crew awards and honors
An NTSB board member called the ditching "the most successful... in aviation history. These people knew what they were supposed to do and they did it and as a result, no lives were lost." New York State Governor David Paterson called the incident "a Miracle on the Hudson." U.S. President George W. Bush said he was "inspired by the skill and heroism of the flight crew," and praised the emergency responders and volunteers. President-elect Barack Obama said that everyone was proud of Sullenberger's "heroic and graceful job in landing the damaged aircraft." He thanked the crew, whom he invited to his inauguration five days later.

The Guild of Air Pilots and Air Navigators awarded the crew the rarely bestowed Master's Medal on January 22, 2009 for outstanding aviation achievement, at the discretion of the Master of the Guild. New York City Mayor Michael Bloomberg presented the crew with the Keys to the City, and Sullenberger with a replacement copy of a library book lost on the flight, Sidney Dekker's Just Culture: Balancing Safety and Accountability. Rescuers received Certificates of Honor.

The crew received a standing ovation at the Super Bowl XLIII on February 1, 2009, and Sullenberger threw the ceremonial first pitch of the 2009 Major League Baseball season for the San Francisco Giants. His Giants jersey was inscribed with the name "Sully" and the number 155 – the count of people aboard the plane.

On July 28, passengers Dave Sanderson and Barry Leonard organized a thank you luncheon for emergency responders from Hudson County, New Jersey, on the shores of Palisades Medical Center in North Bergen, New Jersey, where 57 passengers had been brought following their rescue. Present were members of the U.S. Coast Guard, North Hudson Regional Fire and Rescue, NY Waterway Ferries, the American Red Cross, Weehawken Volunteer First Aid, the Weehawken Police Department, West New York E.M.S., North Bergen E.M.S., the Hudson County Office of Emergency Management, the New Jersey E.M.S. Task Force, the Guttenberg Police Department, McCabe Ambulance, the Harrison Police Department, and doctors and nurses who treated survivors.

Sullenberger was named Grand Marshal for the 2010 Tournament of Roses Parade in Pasadena, California.

In August 2010, aeronautical chart publisher Jeppesen issued a humorous approach plate titled "Hudson Miracle APCH," dedicated to the five crew of Flight 1549 and annotated "Presented with Pride and Gratitude from your friends at Jeppesen."

Sullenberger retired on March 3, 2010, after thirty years with US Airways and its predecessor, Pacific Southwest Airlines. At the end of his final flight he was reunited with Skiles and a number of the passengers from Flight 1549.

N106US, the accident aircraft, was moved to a salvage yard in New Jersey and put up for auction a week after the accident, but remained without takers for over two years. In 2011, it was purchased by the Carolinas Aviation Museum in Charlotte, North Carolina, and reassembled, minus the engines, in the museum's main hangar, where it is currently on display.

In 2013, the entire crew was inducted into the International Air & Space Hall of Fame at the San Diego Air & Space Museum.

Media
The ditching was recorded by several closed-circuit television cameras. Television reports and documentaries produced soon afterward contained extensive video of the ditching and rescue, and recorded interviews with the aircrew, passengers, rescuers, and other key participants.

Sullenberger's 2009 memoir, Highest Duty: My Search for What Really Matters was adapted into a feature film Sully, directed by Clint Eastwood. It starred Tom Hanks as Sullenberger and Aaron Eckhart as co-pilot Jeff Skiles. It was released by Warner Bros. on September 9, 2016.