Functions

Definition 1. Let \(A\) and \(B\) be sets. A function from \(A\) into \(B\) is a set \(f\subseteq A\times B\) such that \[\forall a\in A \ \exists!b\in B \quad (a,b)\in f.\] We write \(f: A\to B\) to denote that \(f\) is a function from \(A\) into \(B\).

If \(a\in A\), we write \(f(a)\) to denote the unique \(b\in B\) such that \((a,b)\in f\).

Definition 2. A function \(f:A\to B\) is injective if \[\forall x,y\in A \quad f(x)=f(y) \implies x=y\]

Definition 3. A function \(f: A\to B\) is surjective if \[\forall b\in B \ \exists a\in A \quad f(a)=b\] If \(f: A\to B\) is surjective, we say \(f\) is a function from \(A\) onto \(B\).

Definition 4. A function is bijective if it is both injective and surjective.