Note 10.3.1.
An identity function is always a bijection.
Section 10.3 Important examples
- identity function (on a set \(A\))
- the function \(A \to A\) defined by \(a \mapsto a\)
- \(\funcdef{\id_A}{A}{A}\)
- the identity function on on set \(A\)
- inclusion function (on subset \(A \subseteq X\))
- the function \(A \to X\) defined by \(a \mapsto a\)
- \(\funcdef{\inclfunc{A}{X}}{A}{X}\)
- the inclusion function on subset \(A \subseteq X\)
Note 10.3.2.
An inclusion function is always an injection.
- projection functions (on a Cartesian product \(A \cartprod B\))
- the functions \(A \cartprod B \to A\) and \(A \cartprod B \to B\) defined by \((a,b) \mapsto a\) and \((a,b) \mapsto b\)
- \(\funcdef{\projfunc{A}}{A \cartprod B}{A}\)
- the projection function onto the first factor \(A\) in the Cartesian product \(A \cartprod B\)
- \(\funcdef{\projfunc{B}}{A \cartprod B}{B}\)
- the projection function onto the second factor \(B\) in the Cartesian product \(A \cartprod B\)
Example 10.3.3. Projection images.
Consider \((\frac{1}{2},\pi) \in \Q \cartprod \R\text{.}\) Then
\begin{align*}
p_{\Q} \left( \frac{1}{2}, \pi \right) \amp = \frac{1}{2}, \amp p_{\R} \left( \frac{1}{2}, \pi \right) \amp = \pi \text{.}
\end{align*}
Extend.
We may of course similarly define a projection function on a Cartesian product with any number of factors. Write
\begin{equation*}
\funcdef{\projfunc{i}}{A_1 \cartprod A_2 \cartprod \dotsb \cartprod A_n}{A_i}
\end{equation*}
to mean the projection function onto the \(\nth[i]\) factor \(A_i\) in the Cartesian product
\begin{equation*}
A_1 \cartprod A_2 \cartprod \dotsb \cartprod A_n\text{.}
\end{equation*}
Alternatively, we may write
\begin{equation*}
\funcdef{\proj_i}{A_1 \cartprod A_2 \cartprod \dotsb \cartprod A_n}{A_i}
\end{equation*}
for this function.
Note 10.3.4.
A projection is always surjective (except possibly when one or more of the factors in the Cartesian product is the empty set).
- restricting the domain
- the “induced” function \(A \to Y\) created from function \(\funcdef{f}{X}{Y}\) and subset \(A \subseteq X\) by “forgetting” about all elements of \(X\) that do not lie in \(A\)
- \(\funcres{f}{A}\)
- restriction of function \(\funcdef{f}{X}{Y}\) to subset \(A \subseteq X\)
- \(\altfuncres{f}{A}\)
- alternative domain restriction notation
- \(\res_A^X f\)
- alternative domain restriction notation
Example 10.3.6. Domain restriction.
For \(\funcdef{f}{\Z}{\N}\text{,}\) \(f(m) = \abs{m}\text{,}\) we have \(\funcres{f}{\N} = \id_{\N}\text{.}\)
Checkpoint 10.3.7. Properties of restrictions.
Consider function \(\funcdef{f}{X}{Y}\) and subset \(A \subseteq X\text{.}\)
- If \(f\) is injective, is \(\funcres{f}{A}\) injective?
- If \(\funcres{f}{A}\) is injective, must \(f\) be injective?
- Answer the previous two questions replacing “injective” with “surjective”.
Remark 10.3.8.
The concept of restricting the domain makes our previously defined concept image of a function on a subset unnecessary: for function \(\funcdef{f}{X}{Y}\) and subset \(A \subseteq X\text{,}\) the image of \(f\) on \(A\) is the same as the image of the restriction \(\funcres{f}{A}\text{.}\)
- restricting the codomain
- the “induced” function \(X \to B\) created from function \(\funcdef{f}{X}{Y}\) and subset \(B \subseteq Y\) by “forgetting” about all elements of \(Y\) that do not lie in \(B\text{,}\) where \(B\) must contain the image of \(f\)
Example 10.3.10. Codomain restriction.
Consider \(\funcdef{f}{\R}{\R}\text{,}\) \(f(x) = x^2\text{.}\) It would be more precise to write \(\funcdef{f}{\R}{\nnegset{\R}}\text{,}\) since \(x^2 \ge 0\) for all \(x \in \R\text{.}\)
Note 10.3.11.
If we restrict the codomain all the way down to the image set \(f(X)\text{,}\) the resulting map \(\funcdef{f}{X}{f(X)}\) is always surjective. In particular, if \(\ifuncdef{f}{X}{Y}\) is injective, then by restricting the codomain we can obtain a bijection \(\funcdef{f}{X}{f(X)}\text{.}\)
- extension of a function
- relative to function \(\funcdef{f}{A}{B}\) and superset \(X \supseteq A\text{,}\) a function \(\funcdef{g}{X}{B}\) so that \(g(a) = f(a)\) for all \(a \in A\)
Note 10.3.13.
The condition defining the concept extension function can be more succinctly stated as requiring function \(\funcdef{g}{X}{B}\) with \(A \subseteq X\) satisfy \(\funcres{g}{A} = f\text{.}\)
Example 10.3.14. Floor function.
Write \(\funcdef{\flr}{\R}{\Z}\) to mean the floor function: for real input \(x\text{,}\) the output \(\flr(x)\) is defined to be the greatest integer that is less than or equal to \(x\text{.}\) Usually we write
\begin{equation*}
\flr(x) = \floor{x} \text{.}
\end{equation*}
As every integer is less than or equal to itself, we have \(\flr(z) = z\) for every \(z \in \Z\text{.}\) This says that the floor function is an extension of the identity function \(\id_{\Z}\text{.}\)
One of the most common ways to extend a function to a larger domain is to pick an appropriate constant value in the codomain to assign to all “new” inputs in the enlarged domain.
- extenstion by zero
- relative to function \(\funcdef{f}{A}{Z}\) and superset \(X \supseteq A\text{,}\) where \(Z\) is a set of “numbers” containing a zero element, the extension function \(\funcdef{g}{X}{Z}\) defined by\begin{equation*} g(x) = \begin{cases} f(x)\text{,} \amp x \in X \text{,} \\ 0\text{,} \amp \text{otherwise.} \end{cases} \end{equation*}
Example 10.3.15. Extending the identity function by zero.
Define \(\funcdef{{\widetilde{\id}}_{\Z}}{\R}{\Z}\) by
\begin{equation*}
{\widetilde{\id}}_{\Z}(x) =
\begin{cases}
x\text{,} \amp x \in \Z \text{,} \\
0\text{,} \amp \text{otherwise.}
\end{cases}
\end{equation*}
Then \({\widetilde{\id}}_{\Z}\) is the extension by zero of the identity function \(\id_{\Z}\text{.}\)
