Important ideas that we discussed in this section include the following.
A function \(f\) from a nonempty set \(A\) to a set \(B\) is a collection of ordered pairs \((a,b)\) so that for each \(a \in A\) there is a pair \((a,b)\) in \(f\text{,}\) and if \((a,b)\) and \((a,b')\) are in \(f\text{,}\) then \(b=b'\text{.}\) If \(f\) is a function we use the notation \(f(a) = b\) to indicate that \((a,b) \in f\text{.}\)
If \(f\) is a function from \(A\) to \(B\text{,}\) the set \(A\) is the domain of the function.
If \(f\) is a function from \(A\) to \(B\text{,}\) the set \(B\) is the codomain of the function. The set
\begin{equation*}
\{f(a) \mid a \in A\}
\end{equation*}
is the range of the function. So the range of a function is a subset of the codomain.
A function \(f\) from a set \(A\) to a set \(B\) is an injection if, whenever \(f(a) = f(a')\) for \(a\text{,}\)\(a' \in A\text{,}\) then \(a = a'\text{.}\) The function \(f\) is a surjection if, whenever \(b \in B\text{,}\) then there is an \(a \in A\) so that \(f(a)=b\text{.}\)
If \(f\) is a function from a set \(A\) to a set \(B\) and if \(g\) is a function from \(B\) to a set \(C\text{,}\) then the composite \(g \circ f\) is a function from \(A\) to \(C\) defined by \((g \circ f)(a) = g(f(a))\) for every \(a \in A\text{.}\)
A function \(f\) from a set \(A\) to a set \(B\) is a bijection if \(f\) is both a surjection and injection. When \(f\) is a bijection from \(A\) to \(B\text{,}\) then \(f\) has an inverse \(f^{-1}\) defined by \(f^{-1}(b) = a\) when \(f(a) = b\text{.}\)
If \(f\) is a function from a set \(A\) to a set \(B\text{,}\) and if \(C\) is a subset of \(A\text{,}\) then image of \(C\) under \(f\) is the set
\begin{equation*}
f(C) = \{f(c) \mid c \in C\}\text{,}
\end{equation*}
and if \(D\) is a subset of \(Y\text{,}\) the inverse image of \(D\) is the set
Important properties that relate images and inverse images of sets and set unions are the following. If \(f\) is a function from a set \(X\) to a set \(Y\text{,}\) and if \(\{A_{\alpha}\}\) is a collection of subsets of \(X\) for \(\alpha\) in some indexing set \(I\text{,}\) and \(\{B_{\beta}\}\) be a collection of subsets of \(Y\) for \(\beta\) in some indexing set \(J\text{,}\) then
\(f\left(\bigcup_{\alpha \in I} A_{\alpha}\right) = \bigcup_{\alpha \in I} f(A_{\alpha})\) and
