# Proof about Functions

Let f: X-->Y be a function from one set X to another set Y, let S be a subset of X, and let U be a subset of Y. What in general can one say about f^-1(f(S)) and S? What about f(f^-1(U)) and U?

© BrainMass Inc. brainmass.com October 10, 2019, 4:22 am ad1c9bdddfhttps://brainmass.com/math/computing-values-of-functions/proof-functions-457741

#### Solution Preview

Dear Student

Please find the attachment for the solutions.

Let f: X-->Y be a function from one set X to another set Y, let S be a subset of X, and let U be a subset of Y. What in general can one say about and f^-1(f(S)) and S? What about f(f^-1(U)) and U?

We know that f(S) denotes the set of the images of all elements of S under the function f.

So, f(s) = {b element of Y/ f(a) = b, for some a element of S}

Similarly f^-1(b) = {a element of S/f(a) = b, b element of Y} indicates the set of all elements of S that are mapped to some element b in Y through f.

Let us see two examples ...

#### Solution Summary

This solution explains how to provide proof for the given functions.