# Calculus

## Set Theory # Functions as Relations

### Definition of a Function

Recall that a binary relation $$R$$ from set $$A$$ to set $$B$$ is defined as a subset of the Cartesian product $$A \times B,$$ which is the set of all possible ordered pairs $$\left( {a,b} \right),$$ where $$a \in A$$ and $$b \in B.$$

If $$R \subseteq A \times B$$ is a binary relation and $$\left( {a,b} \right) \in R,$$ we say that $$a$$ is related to $$b$$ by $$R.$$ It is denoted as $$aRb.$$

A function, denoted by $$f,$$ is a special type of binary relation. A function from set $$A$$ to set $$B$$ is a relation $$f \subseteq A \times B$$ that satisfies the following two properties:

• Each element $$a \in A$$ is mapped to some element $$b \in B.$$
• Each element $$a \in A$$ is mapped to exactly one element $$b \in B.$$

As a counterexample, consider a relation $$R$$ that contains pairs $$\left( {1,1} \right),\left( {1,2} \right).$$ The relation $$R$$ is not a function, because the element $$1$$ is mapped to two elements, which violates the second requirement.

In the next example, the second relation (on the right) is also not a function since both conditions are not met. The input element $$11$$ has no output value, and the element $$3$$ has two values – $$6$$ and $$7.$$

If $$f$$ is a function from set $$A$$ to set $$B,$$ we write $$f : A \to B.$$ The fact that a function $$f$$ maps an element $$a \in A$$ to an element $$b \in B$$ is usually written as $$f\left( a \right) = b.$$

### Domain, Codomain, Range, Image, Preimage

We will introduce some more important notions. Consider a function $$f : A \to B.$$

The set $$A$$ is called the domain of the function $$f,$$ and the set $$B$$ is the codomain. The domain and codomain of $$f$$ are denoted, respectively, $$\text{dom}\left({f}\right)$$ and $$\text{codom}\left({f}\right)$$.

If $$f\left( a \right) = b,$$ the element $$b$$ is the image of $$a$$ under $$f.$$ Respectively, the element $$a$$ is the preimage of $$b$$ under $$f.$$ The element $$a$$ is also often called the argument or input of the function $$f,$$ and the element $$b$$ is called the value of the function $$f$$ or its output.

The set of all images of elements of $$A$$ is briefly referred to as the image of $$A.$$ It is also known as the range of the function $$f,$$ although this term may have different meanings. The range of $$f$$ is denoted $$\text{rng}\left({f}\right)$$. It follows from the definition that the range is a subset of the codomain.

## Solved Problems

Click or tap a problem to see the solution.

### Example 1

Let $$A = \left\{ {a,b,c,d} \right\}$$ and $$B = \left\{ {1,2,3} \right\}.$$ Determine which of the following relations from $$A$$ to $$B$$ are functions:
1. $$\left\{ {\left( {a,1} \right),\left( {b,2} \right),\left( {c,1} \right),\left( {d,3} \right)} \right\}$$
2. $$\left\{ {\left( {a,1} \right),\left( {b,1} \right),\left( {b,2} \right),\left( {c,3} \right)} \right\}$$
3. $$\left\{ {\left( {a,2} \right),\left( {b,2} \right),\left( {c,2} \right),\left( {d,2} \right)} \right\}$$
4. $$\left\{ {\left( {a,1} \right),\left( {b,2} \right),\left( {c,3} \right),\left( {d,1} \right)} \right\}$$
5. $$\left\{ {\left( {b,3} \right),\left( {c,2} \right),\left( {d,1} \right),\left( {c,3} \right)} \right\}$$

### Example 2

Find the domain and range of the following functions:
1. The function $$f_1$$ assigns to each natural number the product of its digits.
2. The function $$f_2$$ assigns to each integer its last digit squared.
3. The function $$f_3$$ assigns to each pair of natural numbers the average value of these numbers.
4. The function $$f_4$$ is a Boolean function in three variables.

### Example 3

List all functions $$f:\left\{ {a,b,c} \right\} \to \left\{ {0,1} \right\}.$$

### Example 4

List all functions $$g:\left\{ {0,1} \right\} \to \left\{ {a,b,c} \right\}.$$

### Example 5

The function $$f : \mathbb{R} \to \mathbb{R}$$ is defined by $$f\left( x \right) = 5{x^2} – {x^4}.$$ Find all preimages of $$6.$$

### Example 6

The function $$g : \mathbb{Z} \to \mathbb{Z}$$ is defined by $$g\left( x \right) = 3{x^2} – {x^4} + 5.$$ Find all preimages of $$7.$$

### Example 7

The function $$f : \mathcal{P}\left(\left\{{a,b,c}\right\}\right) \to \left\{{0,1,2,3}\right\}$$ is given by the formula $$f\left( x \right) = \left| x \right|,$$ where $$\left| x \right|$$ is the cardinality of the set $$x.$$ Calculate the average value of the function $$f.$$

### Example 8

The function $$g : \mathcal{P}\left({\mathcal{P}\left(\left\{{a,b,c}\right\}\right)}\right) \to \mathbb{N}\cup 0$$ is given by the formula $$g\left( x \right) = \left| x \right|,$$ where $$\left| x \right|$$ is the cardinality of the set $$x.$$ Calculate the maximum value of the function $$g.$$

### Example 1.

Let $$A = \left\{ {a,b,c,d} \right\}$$ and $$B = \left\{ {1,2,3} \right\}.$$ Determine which of the following relations from $$A$$ to $$B$$ are functions:
1. $$\left\{ {\left( {a,1} \right),\left( {b,2} \right),\left( {c,1} \right),\left( {d,3} \right)} \right\}$$
2. $$\left\{ {\left( {a,1} \right),\left( {b,1} \right),\left( {b,2} \right),\left( {c,3} \right)} \right\}$$
3. $$\left\{ {\left( {a,2} \right),\left( {b,2} \right),\left( {c,2} \right),\left( {d,2} \right)} \right\}$$
4. $$\left\{ {\left( {a,1} \right),\left( {b,2} \right),\left( {c,3} \right),\left( {d,1} \right)} \right\}$$
5. $$\left\{ {\left( {b,3} \right),\left( {c,2} \right),\left( {d,1} \right),\left( {c,3} \right)} \right\}$$

Solution.

1. A function.
2. Not a function since the element $$b$$ is related to two output values, $$1$$ and $$2$$, and the element $$d$$ has no output value.
3. A function.
4. A function.
5. Not a function because the element $$c$$ has two output values, $$2$$ and $$3$$, and the element $$a$$ has no output value.

### Example 2.

Find the domain and range of the following functions:
1. The function $$f_1$$ assigns to each natural number the product of its digits.
2. The function $$f_2$$ assigns to each integer its last digit squared.
3. The function $$f_3$$ assigns to each pair of natural numbers the average value of these numbers.
4. The function $$f_4$$ is a Boolean function in three variables.

Solution.

1. The domain of the function $$f_1$$ is the set of natural numbers. The range of $$f_1$$ is the set of nonnegative integers. We can write this in the form ${\text{dom}\left( {{f_1}} \right) = \mathbb{N},\;\;}\kern0pt{\text{rng}\left( {{f_1}} \right) = \mathbb{N} \cup \left\{ 0 \right\}.}$ For example, ${{f_1}\left( {135} \right) = 1 \times 3 \times 5 = 15;\;\;}\kern0pt{{f_1}\left( {140} \right) = 1 \times 4 \times 0 = 0.}$
2. The domain of the function $$f_2$$ is the set of integers, and the range is the set of nonnegative integers: ${\text{dom}\left( {{f_2}} \right) = \mathbb{Z},\;\;}\kern0pt{\text{rng}\left( {{f_2}} \right) = \mathbb{N} \cup \left\{ 0 \right\}.}$ Examples: ${{f_2}\left( {-125} \right) = 5^2 = 25;\;\;}\kern0pt{{f_2}\left( {-150} \right) = 0^2 = 0.}$
3. The domain of the function $$f_3$$ is the Cartesian Product $$\mathbb{N} \times \mathbb{N}.$$ The range of $$f_3$$ is the set of rational numbers $$\mathbb{Q}:$$ ${\text{dom}\left( {{f_3}} \right) = \mathbb{N} \times \mathbb{N},\;\;}\kern0pt{\text{rng}\left( {{f_3}} \right) = \mathbb{Q}.}$ Example: ${{f_3}\left( {10,11} \right) = \frac{10+11}{2} = 10.5}$
4. The domain of the Boolean function $$f_4$$ is $${\left\{ {0,1} \right\}^3},$$ and the range is the two-element set $$\left\{ {0,1} \right\}.$$ So ${\text{dom}\left( {{f_4}} \right) = {\left\{ {0,1} \right\}^3},\;\;}\kern0pt{\text{rng}\left( {{f_4}} \right) = \left\{ {0,1} \right\}.}$ Examples: ${{f_4}\left( {0,1,1} \right) = 1,\;\;}\kern0pt{{f_4}\left( {1,1,1} \right) = 0.}$

### Example 3.

List all functions $$f:\left\{ {a,b,c} \right\} \to \left\{ {0,1} \right\}.$$

Solution.

Total there are $$2^3 = 8$$ different functions that are listed below:

${{f_1} = \left\{ {\left( {a,0} \right),\left( {b,0} \right),\left( {c,0} \right)} \right\},\;\;}\kern0pt{{f_2} = \left\{ {\left( {a,0} \right),\left( {b,0} \right),\left( {c,1} \right)} \right\},\;\;}\kern0pt{{f_3} = \left\{ {\left( {a,0} \right),\left( {b,1} \right),\left( {c,0} \right)} \right\},\;\;}\kern0pt{{f_4} = \left\{ {\left( {a,0} \right),\left( {b,1} \right),\left( {c,1} \right)} \right\},\;\;}\kern0pt{{f_5} = \left\{ {\left( {a,1} \right),\left( {b,0} \right),\left( {c,0} \right)} \right\},\;\;}\kern0pt{{f_6} = \left\{ {\left( {a,1} \right),\left( {b,0} \right),\left( {c,1} \right)} \right\},\;\;}\kern0pt{{f_7} = \left\{ {\left( {a,1} \right),\left( {b,1} \right),\left( {c,0} \right)} \right\},\;\;}\kern0pt{{f_8} = \left\{ {\left( {a,1} \right),\left( {b,1} \right),\left( {c,1} \right)} \right\}.}$

### Example 4.

List all functions $$g:\left\{ {0,1} \right\} \to \left\{ {a,b,c} \right\}.$$

Solution.

The mapping $$g$$ between the sets $$\left\{ {0,1} \right\}$$ and $$\left\{ {a,b,c} \right\}$$ contains $$3^2 = 9$$ different functions:

${{g_1} = \left\{ {\left( {0,a} \right),\left( {1,a} \right)} \right\},\;\;}\kern0pt{{g_2} = \left\{ {\left( {0,a} \right),\left( {1,b} \right)} \right\},\;\;}\kern0pt{{g_3} = \left\{ {\left( {0,a} \right),\left( {1,c} \right)} \right\},\;\;}\kern0pt{{g_4} = \left\{ {\left( {0,b} \right),\left( {1,a} \right)} \right\},\;\;}\kern0pt{{g_5} = \left\{ {\left( {0,b} \right),\left( {1,b} \right)} \right\},\;\;}\kern0pt{{g_6} = \left\{ {\left( {0,b} \right),\left( {1,c} \right)} \right\},\;\;}\kern0pt{{g_7} = \left\{ {\left( {0,c} \right),\left( {1,a} \right)} \right\},\;\;}\kern0pt{{g_8} = \left\{ {\left( {0,c} \right),\left( {1,b} \right)} \right\},\;\;}\kern0pt{{g_9} = \left\{ {\left( {0,c} \right),\left( {1,c} \right)} \right\}.}$

### Example 5.

The function $$f : \mathbb{R} \to \mathbb{R}$$ is defined by $$f\left( x \right) = 5{x^2} – {x^4}.$$ Find all preimages of $$6.$$

Solution.

To find the preimages of $$6$$ we need to solve the equation

$f\left( x \right) = 5{x^2} – {x^4} = 6.$

Let $$u = x^2.$$ Then we can write the original equation in terms of $$u:$$

${u^2} – 5u + 6 = 0.$

The roots of the quadratic equation are $${u_1} = 2,$$ $${u_2} = 3.$$ So we see that either $${x^2} = 2$$ or $${x^2} = 3.$$ Undo the $$x^2$$ by taking the square root:

${{x^2} = {u_1} = 2,}\;\; \Rightarrow {{x_{1,2}} = \pm\sqrt 2;}$

${{x^2} = {u_2} = 3,}\;\; \Rightarrow {{x_{3,4}} = \pm\sqrt 3 .}$

Hence, the set of all preimages of the number $$6$$ is given by

$x \in \left\{ { – \sqrt 3 , – \sqrt 2 ,\sqrt 2 ,\sqrt 3 } \right\}.$

### Example 6.

The function $$g : \mathbb{Z} \to \mathbb{Z}$$ is defined by $$g\left( x \right) = 3{x^2} – {x^4} + 5.$$ Find all preimages of $$7.$$

Solution.

The value of the function is equal to $$7.$$ Hence, the preimages are defined by the biquadratic equation

$g\left( x \right) = 3{x^2} – {x^4} + 5 = 7.$

We substitute $$x^2 = t.$$ This produces the quadratic equation

${t^2} – 3t + 2 = 0,$

which has solutions $${t_1}= 1$$ and $${t_2} = 2.$$

Now, since $$t = x^2,$$ we obtain:

${{x_1} = – 1,\;\;}\kern0pt{{x_2} = 1,\;\;}\kern0pt{{x_3} = – \sqrt 2 ,\;\;}\kern0pt{{x_4} = \sqrt 2 .}$

We see that only $${x_1}, {x_2} \in \mathbb{Z}.$$ Hence, the set of preimages of $$7$$ is given by

$x \in \left\{ { – 1,1} \right\}.$

### Example 7.

The function $$f : \mathcal{P}\left(\left\{{a,b,c}\right\}\right) \to \left\{{0,1,2,3}\right\}$$ is given by the formula $$f\left( x \right) = \left| x \right|,$$ where $$\left| x \right|$$ is the cardinality of the set $$x.$$ Calculate the average value of the function $$f.$$

Solution.

The power set of $$\left\{ {a,b,c} \right\}$$ includes $$8$$ subsets, which are listed along with their cardinality in the table below.

$\require{AMSsymbols}{\begin{array}{c|c} \text{Subset} & \text{Cardinality} \\ \hline \varnothing & 0 \\ \left\{ a \right\} & 1 \\ \left\{ b \right\} & 1 \\ \left\{ c \right\} & 1 \\ \left\{ {a,b} \right\} & 2 \\ \left\{ {a,c} \right\} & 2 \\ \left\{ {b,c} \right\} & 2 \\ \left\{ {a,b,c} \right\} & 3 \end{array}}$

The average value of the function $$f$$ is given by

${\overline f = \frac{{0 + 1 \cdot 3 + 2 \cdot 3 + 3}}{8} }={ \frac{{12}}{8} }={ 1.5}$

### Example 8.

The function $$g : \mathcal{P}\left({\mathcal{P}\left(\left\{{a,b,c}\right\}\right)}\right) \to \mathbb{N}\cup 0$$ is given by the formula $$g\left( x \right) = \left| x \right|,$$ where $$\left| x \right|$$ is the cardinality of the set $$x.$$ Calculate the maximum value of the function $$g.$$

Solution.

The power set $$\mathcal{P}\left( {\left\{ {a,b,c} \right\}} \right)$$ consists of $$2^3 = 8$$ subsets.

The power set of a set of $$8$$ elements has $$2^8$$ subsets. The subset with the greatest cardinality is the set itself. Hence, in our case the subset with the greatest cardinality is the power set $$\mathcal{P}\left( {\mathcal{P}\left( {\left\{ {a,b,c} \right\}} \right)} \right).$$

Thus, the maximum value of the function $$g$$ is given by

${{g_{\max }} = \left| {\mathcal{P}\left( {\mathcal{P}\left( {\left\{ {a,b,c} \right\}} \right)} \right)} \right| }={ {2^{{2^3}}} }={ {2^8} }={ 256.}$