Proof involving surjective/onto and image/preimage of sets

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In summary, the conversation discusses proving that f(f^-1(B))=B when f is surjective and B is a subset of Y. The first step is to show that f(f^-1(B)) is a subset of B and then prove that B is a subset of f(f^-1(B)). The concept of surjectivity is then introduced as a key factor in proving that B is a subset of f(f^-1(B)).
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mathmajor2013
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EXERCISE: Suppose f is surjective, and B is a subset of Y. Prove that f(f^-1(B))=B.

SOLUTION: We must show that f(f^-1(B)) is a subset of B and that B is a subset of f(f^-1(B)). I have already proven that f(f^-1(B)) is a subset of B. Now I must prove that B is a subset of f(f^-1(B)) when f is surjective. Fix x is an element of B.

After this I am lost. Help please! I know that the surjectivity must come in handy at some point since B is not a subset of f(f^-1(B)) for all f.
 
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Well

[tex] f^{-1} : B \rightarrow Y [/tex]

So let x be in B as you said. Now [tex] f^{-1}(x) [/tex] is an element of Y provided that there is some y in Y such that [tex] f^{-1}(x) = y [/tex]. This is where you need to use surjectivity. Hope this helps.
 

1. What is a surjective/onto function?

A surjective/onto function is a type of function in mathematics where every element in the output (codomain) has at least one corresponding input (domain). In other words, every element in the output is mapped from an element in the input.

2. How do you prove that a function is surjective/onto?

To prove that a function is surjective/onto, you must show that for every element in the output, there exists at least one element in the input that maps to it. This can be shown using a direct proof or by using the contrapositive method.

3. What is the image of a set?

The image of a set is the set of all outputs produced by a function when the input is taken from the given set. In other words, it is the set of all elements that the function maps to when the input is taken from the given set.

4. How do you find the image of a set?

To find the image of a set, you must apply the function to each element in the given set. The resulting set of outputs is the image of the set.

5. What is the preimage of a set?

The preimage of a set is the set of all inputs that produce a given output when the function is applied. In other words, it is the set of all elements from the domain that map to a specific element in the codomain.

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