# Tagged Questions

2answers
64 views

### is this argument true?

i had a puzzle and used a logical argument to show a point but some says that my argument is wrong but i can't understand the reason they provide ! the puzzles says , Given four cards laid out on a ...
2answers
89 views

### How to prove this with induction

$$(P_0 \lor P_1 \lor P_2\lor\ldots\lor P_n) \rightarrow Q$$ is the same as $$(P_0 \rightarrow Q) \land (P_1 \rightarrow Q) \land (P_2 \rightarrow Q) \land\ldots\land(P_n \rightarrow Q)$$ Do I ...
1answer
92 views

### Prove that a formal system is absolutely inconsistent

I'm using the book An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, and it does not have any solutions and barely any examples. I want to understand how to prove that all ...
4answers
192 views

### Prove equivalence $(P \Rightarrow Q) \land (P \Rightarrow R) \Leftrightarrow P\Rightarrow(Q\land R)$

Prove equivalence $$(P \Rightarrow Q) \land (P \Rightarrow R) \Leftrightarrow P\Rightarrow(Q\land R)$$ What is the step by step for the equivalence of these equations. I can first break down the ...
2answers
138 views

### Providing a counter example for a Logic Statement

How do I give a counter-example of the following logic statement (I think the statement is false): There exists $x$ $\geq$ 0 s.t. (For All real $y$, $x$ = $y$$^2$) Since the statement has a "There ...
2answers
75 views

### Is the set of self-dual connectives incomplete?

A $n$-ary connective $\$$is called self-dual if$f_\$(x_1^*, \ldots , x_n^*) = (f_\$(x_1, \ldots , x_n))^*$where$0^* = 1$and$1^* = 0\$. How to show that the set of such self-dual connectives ...
0answers
156 views

### propositional logic - substitution

Prove: $\varphi_1 =\!\mathrel|\mathrel|\!= \varphi_2 \implies \varphi_1[\psi/p] =\!\mathrel|\mathrel|\!= \varphi_2[\psi/p]$. We've proven that \$\varphi_1 =\!\mathrel|\mathrel|\!= \varphi_2 \implies ...
2answers
509 views

### I want a clear explanation for the Principle of Strong Mathematical Induction

I understood the Principle of Mathematical Induction. I know how to make a recursive definition. But I am stuck with how the "Principle of Strong Mathematical Induction (- the Alternative Form)" ...
2answers
128 views

### Predicate Logic Argument Validity

My question is whether or not the following is a validly structured argument: (P→T)→Q Q → ¬Q ∴ P I'm getting hung up on the Q→¬Q part by itself as a premise, it doesn't seem like that is ...
3answers
123 views

### Logical Equivalance

Determine whether the following pairs of statements are logically equivalent or not. Give a reason. (i) $p \to (q \to r)$ and $(p \to q) \to r$ (ii) $p \to (q \to r)$ and \$q \to (p \to ...
3answers
369 views

### inference rules application (introduction / elimination): two examples

Got stuck while trying out how to apply inference rules (introduction and elimination) for the following examples: From $\lnot(P\land Q)$ and $P$ infer $\lnot Q$ From $P\lor Q$ and $Q$ infer \$\lnot ...
3answers
322 views

### Understanding this proof by contradiction

Let $c$ be a positive integer that is not prime. Show that there is some positive integer $b$ such that $b \mid c$ and $b \leq \sqrt{c}$. I know this can be proved by contradiction, but I'm not ...
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