Proving Injectivity and Surjectivity: A Fundamental Concept in Function Theory

In summary, the conversation is about proving the properties of surjective and injective functions. The first question asks for a definition of surjective functions and the second question asks for a comparison between functions from one set to another. The conversation also mentions the rules of Physics Forums for receiving help with homework questions.
  • #1
tlkieu
8
1
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
 
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  • #2
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 [itex]f(x)[/itex] from one set, [itex]A[/itex] to another set, [itex]B[/itex] is surjective.
  2. Now, write down what it means to have a function from [itex]B[/itex] to [itex]A[/itex].
  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.
 
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Likes tlkieu
  • #3
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.
 

1. What is an injective proof?

An injective proof is a type of mathematical proof that shows that a function is one-to-one, meaning that each input has a unique output. This is typically done by showing that if two inputs have the same output, then they must be the same input.

2. How is an injective proof different from a surjective proof?

An injective proof shows that each input has a unique output, while a surjective proof shows that the function covers all possible outputs. In other words, an injective proof focuses on the inputs, while a surjective proof focuses on the outputs.

3. What are some common techniques used in injective proofs?

Some common techniques used in injective proofs include direct proof, proof by contradiction, and proof by contraposition. These techniques involve showing that if two inputs have the same output, then there is a contradiction or inconsistency, which proves that the function is one-to-one.

4. How can I determine if a function is injective or surjective?

To determine if a function is injective, you can use the vertical line test. If each vertical line intersects the function at most once, then the function is injective. To determine if a function is surjective, you can use the horizontal line test. If each horizontal line intersects the function at least once, then the function is surjective.

5. Why are injective and surjective proofs important in mathematics?

Injective and surjective proofs are important in mathematics because they help to prove the properties of functions and understand their behavior. These proofs are also essential in many areas of mathematics, including algebra, calculus, and abstract algebra.

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