Functions

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.

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Example 7.1.11. More Functions.

A sequence is a function from

Z^{+}

R .

For example,

f (n) = \frac{1}{n} .

We may also have functions involving Cartesian products of sets. For example,

f : Z \times Z \to Z

given by

f (a, b) = a + b .

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Activity 7.1.2.

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Define

f : Z \times Z \to Z

f (a, b) = a - b .

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(a)

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Find

f (0, 2), f (1, 1), f (2, 0), f (3, 2), f (n, 0) .

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(b)

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What is the image of

Z \times Z

under

f ?

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(c)

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What is the preimage of 0?

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Activity 7.1.3.

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Define

f : Z \to Z \times Z

f (a) = (a, a) .

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(a)

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Find

f (0), f (2), f (- 3) .

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(b)

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What is the image of

Z

under

f ?

(1, 1)

in the image? Is

(- 1, 1)

in the image?

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(c)

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What is the preimage of

(0, 0) ?

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Since a function needs to satisfy the property that each

x \in X

can only map to one

y \in Y,

we say a function is well-defined if whenever

a = b,

f (a) = f (b) .

Most of the functions you've seen in algebra and calculus are clearly well-defined since when

a = b,

f (a) = f (b) .

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

Q .

For example,

\frac{1}{2} = \frac{2}{4} .

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Example 7.1.12. A Map That is Not Well-Defined.

Let

f : Q \to Z

be given by

f (p / q) = p + q .

Then

1 / 2 = 2 / 4

Q,

but

f (1 / 2) = 1 + 2 = 3

and

f (2 / 4) = 2 + 4 = 6 .

Thus,

1 / 2 = 2 / 4

but

f (1 / 2) \neq f (2 / 4) .

Thus

f

is not well-defined, and hence,

f

is not a function.

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Activity 7.1.4.

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Let

f : Q \to Z

be given by

f (m / n) = m .

Show

f

is not well-defined by finding two equivalent fractions in

Q

that map to two different integers.

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Reading Questions Check Your Understanding

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1.

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Let

A = {1, 2, 3}, B = {2, 4, 6, 8} .

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Let

f : A \to B

be given by

f (1) = 4, f (2) = 8, f (3) = 2 .

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True or false:

f

is a function.

True.
False.

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2.

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Let

A = {1, 2, 3}, B = {2, 4, 6, 8} .

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Let

f : A \to B

be given by

f (1) = 2, f (2) = 4, f (3) = 2 .

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True or false:

f

is a function.

True.
False.

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3.

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Let

B = {2, 4, 6, 8}, C = {0, 1} .

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Let

f : B \to C

be given by

f (2) = 0, f (4) = 1 .

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True or false:

f

is a function.

True.
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$6, 8$ don't map anywhere.
False.
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$6, 8$ don't map anywhere.

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4.

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Let

B = {2, 4, 6, 8}, C = {0, 1} .

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Let

f : B \to C

be given by

f (2) = 0, f (4) = 1, f (6) = 1, f (8) = 1 .

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True or false:

f

is a function.

True.
False.

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5.

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Let

C = {0, 1} .

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Let

f : C \to C

be given by

f (0) = 0, f (0) = 1, f (1) = 0, f (1) = 1 .

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True or false:

f

is a function.

True.
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0 maps to two different values.
False.
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0 maps to two different values.

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6.

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Let

A = {1, 2, 3}, C = {0, 1} .

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Let

f : A \times C \to C

be given by

f (a, c) = c .

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Which of the following are in the preimage of

0 \in C .

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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 $A \times C .$
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0
$0$ is not in $A \times C .$
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(0, 1)
$(0, 1)$ is not in $A \times C,$ and would not map to 0.
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1
$1$ is not in $A \times C .$

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Exercises Exercises

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1.

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Let

X = {1, 3, 5}

and

Y = {a, b, c, d} .

Define the function

g : X \to Y

as the set of ordered pairs

{(1, b), (3, b), (5, b)} .

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Write the domain and codomain of $g .$
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What is the range of $g ?$
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Is 3 an inverse image of $a ?$ Is 1 an inverse image of $b ?$
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What is the inverse image of $b ?$ What is the inverse image of $c ?$
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Represent $g$ as an arrow diagram.

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2.

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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.

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3.

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Let

A = {1, 2, 3, 4, 5}

and define a function

F : P (A) \to Z

where for all sets

X

P (A),

F (X) = {\begin{cases} 0 & if X has an even number of elements \\ 1 & if X has an odd number of elements. \end{cases}

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Find $F ({1, 3, 4}) .$
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Find $F (\emptyset) .$
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Find $F ({2, 3}) .$
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$F ({2, 3, 4, 5}) .$

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4.

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Let

D

be the set of all finite subsets of positive integers. Let

T : Z^{+} \to D

be the function where for each positive integer

n,

T (n)

is the set of positive divisors of

n .

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Find $T (1) .$
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Find $T (15) .$
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Find $T (17) .$
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Find $T (5) .$
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Find $T (18) .$
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Find $T (21) .$

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5.

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Define the function

F : Z \times Z \to Z \times Z

where for all ordered pairs

(a, b)

of integers,

F (a, b) = (2 a + 1, 3 b - 2) .

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Find $F (4, 4) .$
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Find $F (2, 1) .$
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Find $F (3, 2) .$
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Find $F (1, 5) .$

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6.

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Let

X = {1, 2, 3, 4}

and

Y = {a, b, c, d, e} .

Define

g : X \to Y

g (1) = a, g (2) = a, g (3) = a

and

g (4) = d .

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Draw an arrow diagram for $g .$
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Let $A = {2, 3}, C = {a},$ and $D = {b, c} .$ Find $g (A), g (X), g^{- 1} (C), g^{- 1} (D),$ and $g^{- 1} (Y) .$

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7.

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Show that each of the following maps is not a function by showing it is not well-defined.

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Define $g : Q \to Z$ by the rule

$g (\frac{m}{n}) = m - n$

for all integers $m$ and $n$ with $n \neq 0 .$
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Define $h : Q \to Q$ by the rule

$h (\frac{m}{n}) = \frac{m^{2}}{n}$

for all integers $m$ and $n$ with $n \neq 0 .$

Discrete Mathematics: An Active Approach to Mathematical Reasoning

Search Results:

Section 7.1 Functions

Definition 7.1.1.

Example 7.1.2. Exploring the Definition of a Function.

Definition 7.1.6.

Definition 7.1.7.

Activity 7.1.1.

(a)

(b)

(c)

Example 7.1.8. Arrow Diagram.

Example 7.1.10. Finding Sets for a Function.

Example 7.1.11. More Functions.

Activity 7.1.2.

(a)

(b)

(c)

Activity 7.1.3.

(a)

(b)

(c)

Example 7.1.12. A Map That is Not Well-Defined.

Activity 7.1.4.

Reading Questions Check Your Understanding

1.

2.

3.

4.

5.

6.

Exercises Exercises

1.

2.

3.

4.

5.

6.

7.