19.02 Special number sets

Recall that a set is a list of objects, each of which is called an element. Let's review two types of notations we can use to refer to a subset of the real numbers.

Set builder notation

Set builder notation follows a common convention.

Many subsets of the real numbers have abbreviations which are commonly used.

Set	Letter Abbreviation
Real numbers	$\mathbb{R}$ℝ
Rational numbers	$\mathbf{Q}$Q
Irrational numbers	$\mathbf{Q}'$Q′
Integers	$\mathbb{Z}$ℤ
Positive integers	$\mathbb{Z}^+$ℤ+
Natural numbers	$\mathbb{N}$ℕ
Empty set	$\varnothing$∅

Worked examples

Example 1

Describe the set of even numbers using set builder notation.

Think: The set of even numbers are all natural numbers that are multiples of $2$2. If $x$x is an even number, then $x=2n$x=2n, where $n$n is a natural number. The symbol $\in$∈ means 'is an element of'.

Do: Translate to set builder notation.

$\lbrace x\mid x=2n,n\in\mathbb{N}\rbrace${x∣x=2n,n∈ℕ}

"the set of all x such that $x=2n$x=2n, where $n$n is an element of the natural numbers"

 

example 2

Let set  $A=\lbrace x|\quad x\text{ is a triangle number }\rbrace$A={x| x is a triangle number }.

a) Write down the first ten elements in set $A$A.

Think: Triangle numbers are numbers that can be arranged to form a triangle. Some examples are in the table below: 

Diagram	Triangle number	Amount being added from previous number
	$1$1
	$3$3	$2$2
	$6$6	$3$3
	$10$10	$4$4

We can see in the table above that the number being added increases by one each time:

Do: So the first ten triangle numbers are: 
 

Triangle number	$1$1	$3$3	$6$6	$10$10	$15$15	$21$21	$28$28	$36$36	$45$45	$55$55
Add to get next number	$+2$+2	$+3$+3	$+4$+4	$+5$+5	$+6$+6	$+7$+7	$+8$+8	$+9$+9	$+10$+10

b) Is $A\subset\mathbb{Z}^+$A⊂ℤ+?

Do: Yes, all the values in set $A$A are positive integers.

 

 

Practice questions

Question 1

Question 2