Proving Injectivity and Surjectivity: A Fundamental Concept in Function Theory

[tlkieu]

Stumped on a couple of questions, if anyone could help!

In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R.
(a) Show that a function f is surjective if and only if there exists a function g such that f ◦ g = I.
(b) Show that a function f is injective if and only if there exists a function h such that h ◦ f = I.
(c) Suppose f ◦ g = I and h ◦ f = I. Show that g = h

 

[stevendaryl]

Let's take one question at a time.

For question (a), do the following exercise:

1. Write down (in English, or using logical notation, if you know logic) what does it mean to say that a function $f(x)$ from one set, $A$ to another set, $B$ is surjective.
2. Now, write down what it means to have a function from $B$ to $A$.
3. Now, see if you can see any relationship between those two definitions.

Post your answers to 1&2. The rules for Physics Forums is that you have to show your effort in order to get help.

 

[tlkieu]

Ahh thank you for the pointer! First time posting so will keep that in mind. Will post my working out so far in the morning, also will repost this in the homework type question forum as I just read that these types of questions are best directed there.