# 1.10 Number sets

Worksheet
Set builder notation
1

List all the elements of the following sets:

a
\left\{ \text{positive odd numbers less than } 10 \right\}
b

\left\{ \text{positive multiples of } 4 \text{ that are less than } 46 \right\}

c

\left\{ \text{positive even integers that are less than } 16 \right\}

d

\left\{ \text{positive factors of } 42 \right\}

e

\left\{ \text{square numbers less than } 109 \right\}

f

\left\{ \text{numbers that can be rolled on a standard die} \right\}

g

\left\{ \left. x \right\vert x \text{ is a natural number less than } 5 \right\}

h

\left\{ \left. x \right\vert x \text{ is an odd whole number less than } 13 \right\}

i

\left\{ \left. x \right\vert x \text{ is an integer between } - 8 \text{ and } - 3 \text{ (not inclusive)} \right\}

2

Determine whether the following statements are true or false:

a

3 \in \left\{ \left. x \right\vert x \text{ is a rational number} \right\}

b

3 \in \left\{2, 4, 5, 8\right\}

c

1 \in \left\{9, 8, 6, 5, 1\right\}

d

-0.25 \in \left\{\left. x \right\vert x \text{ is a natural number}\right\}

3

Consider the set A = \left\{2, 4, 6, 8\right\}. Construct a set builder notation for A.

Special number sets
4

Determine whether the following statements are true or false:

a

-6 \notin \Z

b
\dfrac{5}{2} \in \Reals
c
\pi \in \N
d
\sqrt{2} \in \Z^+
5

Determine whether the following statements are true or false:

a
\N \subset \Reals
b
\Z^+ \subseteq \Z
c
{0, 1, 5, -9} \nsubseteq \Reals
d
\mathbf{Q} \subseteq \Reals
e
\Reals \subseteq \N
f
\Z \subseteq \N
g
\Z^+ \subset \mathbf{Q}
h
\N \subseteq \N
6

For each of the following numbers sets, state whether they approach positive infinity as the numbers increase. If not, state the set's limit.

a

The real numbers, \Reals.

b

The rational numbers, \mathbf{Q}.

c

The integers, \Z.

d

The positive integers, \Z^+.

e

The natural numbers, \N.

7

For each of the following numbers sets, state whether they approach negative infinity as the numbers decrease. If not, state the set's limit.

a

The real numbers, \Reals.

b

The rational numbers, \mathbf{Q}.

c

The integers, \Z.

d

The positive integers, \Z^+.

e

The natural numbers, \N.

8

For each of the following numbers sets, state whether they are dense.

a

The real numbers, \Reals.

b

The rational numbers, \mathbf{Q}.

c

The integers, \Z.

d

The positive integers, \Z^+.

e

The natural numbers, \N.

9

Complete the following table to indicate whether each number belongs in the number set:

10

Write down the next 3 trianglular numbers in this sequence:

1, 3, 6, 10, 15, ⬚, ⬚, ⬚
11

Beginning with the eighth triangular number, write down the next 3 triangular numbers in this sequence:

36, 45, 55, 66, 78, ⬚, ⬚, ⬚

12

Let A = \left\{ x | \, x \text{ is a triangular number} \lt 60 \right\} and let B = \left\{ x | \, x \text{ is a positive multiple of } 3 \lt 60 \right\}.

a

Write all the terms in set A.

b

Write all the terms in set B.

c

Find A \cap B.

d

e

### Outcomes

#### 9.B1.2

Describe how various subsets of a number system are defined, and describe similarities and differences between these subsets.

#### 9.B1.3

Use patterns and number relationships to explain density, infinity, and limit as they relate to number sets.