# How Can Injectivity Prove f^(-1)(f(A)) = A?

• m_kosak
In summary, the conversation discusses a proof involving injective functions and subsets. It is suggested to show two inclusions and then use the definition of injective functions to prove the statement.
m_kosak
who can help me?
ı want to prove this
If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A.
but how can I do this :(

So being injective means that whenever f(a) = f(b) in Y, then a = b in X.

The usual proof for such statements is to show two inclusions. Let's start with $f^{-1}(f(A)) \subseteq A$.
Let x be an element of the set on the left hand side. So x is an element in X, for which $x \in f^{-1}(f(A))$. Can you show that x should in fact be an element of A?

## 1. What is an injective function?

An injective function is a type of mathematical function that maps each element of the input set to a unique element in the output set. This means that no two elements in the input set will have the same output value.

## 2. Why is it important to prove that a function is injective?

Proving that a function is injective is important because it ensures that each input has a unique output, making it easier to analyze and understand the behavior of the function. It also allows us to use the inverse function to solve equations and perform other mathematical operations.

## 3. What are the steps to prove that a function is injective?

The steps to prove that a function is injective are as follows:
1. Assume that the function is injective.
2. Take two different elements from the input set and set them equal to each other.
3. Use the properties of the function to show that the two elements must be equal to each other.
4. This contradiction proves that the function is injective.

## 4. Can you give an example of a function that is injective?

Yes, the function f(x) = x^2 is injective because for every input x, there is a unique output x^2. No two inputs will have the same output value.

## 5. What are some common mistakes to avoid when proving a function is injective?

Some common mistakes to avoid when proving a function is injective are:
- Assuming the function is injective without proof
- Using circular reasoning
- Not considering all elements in the input set
- Making incorrect algebraic manipulations
It is important to carefully follow each step in the proof and check for any errors.

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