Image and preimage

Key defintions

Definition: Let \(f:A\to B\) be a function.

  1. If \(X\subseteq A\), then the image of \(X\) is the set \(f(X)=\{f(x):x\in X\}\subseteq B\).
  2. If \(Y\subseteq B\), then the preimage of \(Y\) is the set \(f^{-1}(Y)=\{x\in A: f(x)\in Y\}\subseteq A\).

Note: \(f^{-1}(Y)\) is defined even when \(f^{-1}\) is not a function, i.e. even when \(f\) is not bijective.

Example 12.13

Problem 12.6.7

Problem: Prove that, if \(f:A\to B\) is a function, and \(W\) and \(X\) are subsets of \(A\), then \[f(W\cap X)\subseteq f(W)\cap f(X)\]

Problem 12.6.9

Problem: Prove that, if \(f:A\to B\) is a function, and \(W\) and \(X\) are subsets of \(A\) then \[f(W\cup X) = f(W)\cup f(X)\]