Definition 7.1.1.
A function, \(f:X\rightarrow Y\text{,}\) is a map in which each input is related to one and only one output.
We say \(X\) is the domain and \(Y\) is the codomain of \(f\text{.}\)
Section 7.1 Functions
Functions are familiar mathematical objects from algebra and calculus. We also saw them in Section 1.3.
Example 7.1.2. Exploring the Definition of a Function.
Let \(f: X\rightarrow Y\) be a map as shown in the figure.
A set labeled \(X\) on the left consisting of elements \(x_1, x_2, x_3\text{.}\) A set \(Y\) on the right consisting of elements \(y_1, y_2, y_3\text{.}\) There is an arrow from \(x_1\) to \(y_1\text{,}\) an arrow from \(x_2\) to \(y_1\text{,}\) an arrow from \(x_2\) to \(y_2\text{,}\) and an arrow from \(x_3\) to \(y_3\text{.}\)
Figure 7.1.3 shows a map that is not a function since \(x_2\) maps to two different outputs.
Let \(g: X\rightarrow Y\) be a map as shown in the figure.
A set labeled \(X\) on the left consisting of elements \(x_1, x_2, x_3\text{.}\) A set \(Y\) on the right consisting of elements \(y_1, y_2, y_3\text{.}\) There is an arrow from \(x_2\) to \(y_1\) and an arrow from \(x_3\) to \(y_3\text{.}\)
Figure 7.1.4 shows a map that is not a function since \(x_1\) does not map to any output.
Let \(h: X\rightarrow Y\) be a map as shown in the figure.
A set labeled \(X\) on the left consisting of elements \(x_1, x_2, x_3\text{.}\) A set \(Y\) on the right consisting of elements \(y_1, y_2, y_3\text{.}\) There is an arrow from \(x_1\) to \(y_1\text{,}\) an arrow from \(x_2\) to \(y_1\text{,}\) and an arrow from \(x_3\) to \(y_3\text{.}\)
For a given \(x\in X\text{,}\) \(f(x)\) is
-
the output of \(f\text{,}\)
-
the value of \(f\) at \(x\text{,}\)
-
the image of \(x\) under \(f\text{.}\)
Definition 7.1.6.
The image or range of a set \(X\) under \(f\) is the set of outputs of \(f\) corresponding to inputs from \(X\text{.}\) In notation
\begin{equation*}
\text{Im}(f)=\{y\in Y : y=f(x) \text{ for some } x\in X\}.
\end{equation*}
If \(f(x)=y\) we say \(x\) is a preimage or an inverse image of \(y\text{.}\)
Since \(y\) can have several preimages, we usually care about the set of all of them.
Definition 7.1.7.
Let \(f:X\rightarrow Y\) be a function. The set \(\{x\in X : f(x)=y\}\) is the preimage of \(y\text{.}\)
Activity 7.1.1.
Define \(f:\mathbb{Z}\rightarrow\mathbb{Z}\) by \(f(n)=r\) where \(r\) is the remainder when \(n\) is divided by 3.
(a)
Find \(f(0), f(9), f(5), f(-7), f(10)\text{.}\)
(b)
What is the image of \(\mathbb{Z}\) under \(f\text{?}\)
(c)
What is the set of preimages of 0? In other words, find the preimage of 0.
If we have a map from a finite set to a finite set, we can draw an arrow diagram in which we use arrows to represent the map from \(X\) to \(Y\text{,}\) as in Example 7.1.2.
Example 7.1.8. Arrow Diagram.
Let \(X=\{a, b, c, d\}\) and \(Y=\{0, 1, 2\}\text{.}\) Let \(f: X\rightarrow Y\) be the function given by the following arrow diagram.
A set labeled \(X\) on the left consisting of elements \(a, b, c, d\text{.}\) A set \(Y\) on the right consisting of elements \(0, 1, 2\text{.}\) There is an arrow from \(a\) to \(0\text{,}\) an arrow from \(b\) to \(0\text{,}\) an arrow from \(c\) to \(2\text{,}\) and an arrow from \(d\) to \(2\text{.}\)
Find the domain of \(f\text{.}\)
Answer 1.
\(\{a, b, c, d\}\)
Find the codomain of \(f\text{.}\)
Answer 2.
\(\{0, 1, 2\}\)
Find the range or image of \(f\text{.}\)
Answer 3.
\(\{0, 2\}\)
Find \(f(a)\text{.}\)
Answer 4.
\(0\)
Find the preimage or inverse image of \(0\text{.}\)
Answer 5.
\(\{a, b\}\)
Example 7.1.10. Finding Sets for a Function.
Let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be given by \(f(x)=x^2\text{.}\)
Find the domain of \(f\text{.}\)
Answer 1.
\(\mathbb{R}\)
Find the codomain of \(f\text{.}\)
Answer 2.
\(\mathbb{R}\)
Find the range of \(f\text{.}\)
Answer 3.
\([0, \infty)\)
Find the preimage (or inverse image) of \(1\text{.}\)
Answer 4.
\(\{1, -1\}\)
Let \(f, g\) be functions from \(X\) to \(Y\text{.}\) Then \(f=g\) if \(f(x)=g(x)\) for all \(x\in X\text{.}\)
In this course we want to look at functions to and from sets other than just the real numbers. For example, we may have functions from finite sets to finite sets.
Example 7.1.11. More Functions.
A sequence is a function from \(\mathbb{Z}^+\) to \(\mathbb{R}\text{.}\) For example, \(f(n)=\frac{1}{n}\text{.}\)
We may also have functions involving Cartesian products of sets. For example, \(f:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}\) given by \(f(a, b)=a+b\text{.}\)
Activity 7.1.2.
Define \(f:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}\) by \(f(a, b)=a-b\text{.}\)
(a)
Find \(f(0, 2), f(1, 1), f(2, 0), f(3, 2), f(n, 0)\text{.}\)
(b)
What is the image of \(\mathbb{Z}\times\mathbb{Z}\) under \(f\text{?}\)
(c)
What is the preimage of 0?
Activity 7.1.3.
Define \(f:\mathbb{Z}\rightarrow\mathbb{Z}\times\mathbb{Z}\) by \(f(a)=(a, a)\text{.}\)
(a)
Find \(f(0), f(2), f(-3)\text{.}\)
(b)
What is the image of \(\mathbb{Z}\) under \(f\text{?}\) Is \((1, 1)\) in the image? Is \((-1, 1)\) in the image?
(c)
What is the preimage of \((0, 0)\text{?}\)
Since a function needs to satisfy the property that each \(x\in X\) can only map to one \(y \in Y\text{,}\) we say a function is well-defined if whenever \(a=b\text{,}\) \(f(a)=f(b)\text{.}\) Most of the functions you’ve seen in algebra and calculus are clearly well-defined since when \(a=b\text{,}\) \(f(a)=f(b)\text{.}\) This property is really only interesting when elements of \(X\) have multiple representations. In other words, when two equal elements in \(X\) have two different forms. The most familiar set where this happens is \(\mathbb{Q}\text{.}\) For example, \(\frac{1}{2}=\frac{2}{4}\text{.}\)
Example 7.1.12. A Map That is Not Well-Defined.
Let \(f:\mathbb{Q}\rightarrow \mathbb{Z}\) be given by \(f(p/q)=p+q\text{.}\)
Then \(1/2=2/4\) in \(\mathbb{Q}\text{,}\) but \(f(1/2)=1+2=3\) and \(f(2/4)=2+4=6\text{.}\) Thus, \(1/2=2/4\) but \(f(1/2)\neq f(2/4)\text{.}\)
Thus \(f\) is not well-defined, and hence, \(f\) is not a function.
Activity 7.1.4.
Let \(f:\mathbb{Q}\rightarrow\mathbb{Z}\) be given by \(f(m/n)=m\text{.}\) Show \(f\) is not well-defined by finding two equivalent fractions in \(\mathbb{Q}\) that map to two different integers.
Reading Questions Check Your Understanding
1.
True.
False.
Let \(A=\{1, 2, 3\}, B=\{2, 4, 6, 8\}\text{.}\)
Let \(f:A\rightarrow B\) be given by \(f(1)=4, f(2)=8, f(3)=2\text{.}\)
True or false: \(f\) is a function.
2.
True.
False.
Let \(A=\{1, 2, 3\}, B=\{2, 4, 6, 8\}\text{.}\)
Let \(f:A\rightarrow B\) be given by \(f(1)=2, f(2)=4, f(3)=2\text{.}\)
True or false: \(f\) is a function.
3.
True.
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\(6, 8\) don’t map anywhere.
False.
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\(6, 8\) don’t map anywhere.
Let \(B=\{2, 4, 6, 8\}, C=\{0, 1\}\text{.}\)
Let \(f:B\rightarrow C\) be given by \(f(2)=0, f(4)=1\text{.}\)
True or false: \(f\) is a function.
4.
True.
False.
Let \(B=\{2, 4, 6, 8\}, C=\{0, 1\}\text{.}\)
Let \(f:B\rightarrow C\) be given by \(f(2)=0, f(4)=1, f(6)=1, f(8)=1\text{.}\)
True or false: \(f\) is a function.
5.
True.
-
0 maps to two different values.
False.
-
0 maps to two different values.
Let \(C=\{0, 1\}\text{.}\)
Let \(f:C\rightarrow C\) be given by \(f(0)=0, f(0)=1, f(1)=0, f(1)=1\text{.}\)
True or false: \(f\) is a function.
6.
Let \(A=\{1, 2, 3\}, C=\{0, 1\}\text{.}\)
Let \(f:A\times C\rightarrow C\) be given by \(f(a, c)=c\text{.}\)
Which of the following are in the preimage of \(0\in C\text{.}\)
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(1, 0)
- \((1, 0)\) is in the preimage.
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(2, 0)
- \((2, 0)\) is in the preimage.
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(3, 0)
- \((3, 0)\) is in the preimage.
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(0, 0)
- \((0, 0)\) is not in \(A\times C\text{.}\)
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0
- \(0\) is not in \(A\times C\text{.}\)
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(0, 1)
- \((0, 1)\) is not in \(A\times C\text{,}\) and would not map to 0.
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1
- \(1\) is not in \(A\times C\text{.}\)
Exercises Exercises
1.
Let \(X=\{1, 3, 5\}\) and \(Y=\{a, b, c, d\}\text{.}\) Define the function \(g:X\rightarrow Y\) as the set of ordered pairs
\begin{equation*}
\{(1, b), (3, b), (5, b)\}.
\end{equation*}
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Write the domain and codomain of \(g\text{.}\)
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What is the range of \(g\text{?}\)
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Is 3 an inverse image of \(a\text{?}\) Is 1 an inverse image of \(b\text{?}\)
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What is the inverse image of \(b\text{?}\) What is the inverse image of \(c\text{?}\)
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Represent \(g\) as an arrow diagram.
2.
Determine if the following are true or false. Justify your answer.
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If two elements in the domain of a function are equal, then their images in the codomain are equal.
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If two elements in the codomain of a function are equal, then their preimages in the domain are equal.
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A function can have the same output for more than one input.
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A function can have the same input for more than one output.
3.
Let \(A=\{1, 2, 3, 4, 5\}\) and define a function \(F:{\cal P}(A)\rightarrow \mathbb{Z}\) where for all sets \(X\) in \({\cal P}(A)\text{,}\)
\begin{equation*}
F(X)=
\begin{cases}
0 &\text{if $X$ has an even number of elements}\\
1 &\text{if $X$ has an odd number of elements.}
\end{cases}
\end{equation*}
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Find \(F(\{1, 3, 4\})\text{.}\)
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Find \(F(\emptyset)\text{.}\)
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Find \(F(\{2, 3\})\text{.}\)
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\(F(\{2, 3, 4, 5\})\text{.}\)
4.
Let \(D\) be the set of all finite subsets of positive integers. Let \(T:\mathbb{Z}^+\rightarrow D\) be the function where for each positive integer \(n\text{,}\) \(T(n)\) is the set of positive divisors of \(n\text{.}\)
-
Find \(T(1)\text{.}\)
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Find \(T(15)\text{.}\)
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Find \(T(17)\text{.}\)
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Find \(T(5)\text{.}\)
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Find \(T(18)\text{.}\)
-
Find \(T(21)\text{.}\)
5.
Define the function \(F:\mathbb{Z}\times\mathbb{Z}\rightarrow \mathbb{Z}\times\mathbb{Z}\) where for all ordered pairs \((a, b)\) of integers, \(F(a, b)=(2a+1, 3b-2)\text{.}\)
-
Find \(F(4, 4)\text{.}\)
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Find \(F(2, 1)\text{.}\)
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Find \(F(3, 2)\text{.}\)
-
Find \(F(1, 5)\text{.}\)
6.
Let \(X=\{1, 2, 3, 4\}\) and \(Y=\{a, b, c, d, e\}\text{.}\) Define \(g:X\rightarrow Y\) by
\begin{equation*}
g(1)=a, g(2)=a, g(3)=a, g(4)=d\text{.}
\end{equation*}
-
Draw an arrow diagram for \(g\text{.}\)
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Let \(A=\{2, 3\}, C=\{a\}\text{,}\) and \(D=\{b, c\}\text{.}\) Find \(g(A), g(X), g^{-1}(C), g^{-1}(D)\text{,}\) and \(g^{-1}(Y)\text{.}\)
7.
Show that each of the following maps is not a function by showing it is not well-defined.
-
Define \(g:\mathbb{Q}\rightarrow\mathbb{Z}\) by the rule\begin{equation*} g\Big(\frac{m}{n}\Big)=m-n \end{equation*}for all integers \(m\) and \(n\) with \(n\neq 0\text{.}\)
-
Define \(h:\mathbb{Q}\rightarrow\mathbb{Q}\) by the rule\begin{equation*} h\Big(\frac{m}{n}\Big)=\frac{m^2}{n} \end{equation*}for all integers \(m\) and \(n\) with \(n\neq 0\text{.}\)
