Definition 3.4.1 (Images of sets). If \(f:X\to Y\) is a function from \(X\) to \(Y\), and \(S\) is a set in \(X\), we define \(f(S)\) to be the set $$ f(S):=\{f(x):x\in S\} $$ this set is a subset of \(Y\), and is sometimes called the image of \(S\) under the map \(f\). We sometimes call \(f(S)\) the forward image of \(S\) to distinguish it from the concept of the inverse image \(f^{-1}(S)\) of \(S\), which is defined below.
Note that the set \(f(S)\) is well-defined thanks to the axiom of replacement (Axiom 3.6). One can also define \(f(S)\) using the axiom of specification (Axiom 3.5) instead of replacement, but we leave this as a challenge to the reader.