# Thread:

1. ## Proving bijective functions

I have 2 I am struggling with:

1. Let f:A to B. Prove that f is a bijection iff for all b that is an element of B, there exists exactly one a that is an element of A f(a)=f(b).

2. Prove: If f:A to B is a bijection and g:B to C is a bijection, then the composition of g of f is a bijection.

2. Originally Posted by IIuvsnshine
I have 2 I am struggling with:

1. Let f:A to B. Prove that f is a bijection iff for all b that is an element of B, there exists exactly one a that is an element of A f(a)=f(b).

2. Prove: If f:A to B is a bijection and g:B to C is a bijection, then the composition of g of f is a bijection.
You appear to have a typo. Shouldn't it read:
"Let f:A to B. Prove that f is a bijection iff for all b that is an element of B, there exists exactly one a that is an element of A f(a)=b."?

I'm not sure how much of a help this will be as I don't know what problems you are having with these, but in general to show a function is bijective you need to show that it is both injective and surjective. My advice is to simply apply the definition of each.

For example, the function in problem 1 is clearly surjective, since for every element b of B there exists (at least) one element of A such that f(a) = b.

-Dan

3. Yes, I did have a typo.

My concern is the proof (lol).

Do I need to approach this by cases?

For the 1st part I have:

Suppose f is a bijection. Then f is injective and surjective.
Case 1: Suppose f is injective. Then for all a,b that is an element of A, f(a)=f(b).
Case 2: Suppose f is surjective. Then for all of b that is an element of B, there exists an a such that f(b) = a.

I'm not sure I'm headed down the correct path.

4. A bijection is surjective injection.

A function is a surjection if and only if each element of the final set, in this case B, is the image of some element in the initial set A.

A function is an injection if and only if no two of its pairs have the same second term.

From the given does the function f have those two properties?

5. I understand the definitions...I'm struggling with wording for the proof.

6. Its an $\Leftrightarrow$ statement so you have 2 directions to go in.

7. Originally Posted by IIuvsnshine
I understand the definitions
Well that is not at all clear to me that you do understand how to use them.

You are given that each b is the image of some a therefore f is onto.

You are given that each b is the image of exactly one a therefore f is one-to-one.

8. It is sometimes difficult to state exactly what the meaning of words typed into a thread.
I do understand the definitions. I apologize if that is not clear to you. I also understand that i'm given a b that is an image of a such that f is onto, and that i'm given a b that is the image of exactly one a such that f is one-to-one.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.962291955947876, "perplexity_flag": "head"}