1.10 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	$\varnothing$∅

 

Infinity and Limits

The elements in the number set Integers, approach infinity as the terms get bigger:

e.g. $1,2,3,...,50,51,...,1000,1001,...20560,20560,...$1,2,3,...,50,51,...,1000,1001,...20560,20560,...

There is no maximum number, so we say that as the numbers increase they approach positive infinity. This also means that the set has no limit. The number of values in the set cannot be counted because there are an infinite number. 

As the terms get smaller, the elements in the number set Integers approach negative infinity:

e.g.  $4,3,2,0,-1,-2,...-24,-25,...,-560,-561,...-12000,-12100,...$4,3,2,0,−1,−2,...−24,−25,...,−560,−561,...−12000,−12100,...

There is no minimum number, so we say that as the numbers decrease they approach negative infinity. This also means that there is no negative limit, because the numbers keep going forever in the negative direction.

The elements in the natural number set also approach infinity as the numbers increase:

e.g. $1,2,3,...,67,68,69,...,4000,4001,...$1,2,3,...,67,68,69,...,4000,4001,...

However, as the numbers decrease there is a smallest value: $6,5,4,3,2,1.$6,5,4,3,2,1.

$1$1 is the smallest natural number. Therefore, the limit when the natural numbers decrease is $1$1.

 

 

Density

The set of real numbers is considered to be dense because between any two real numbers, there are an infinite number of real numbers. For example between the numbers $56$56 and $57$57 there are real numbers such as: $56.5,56.032,56.9814,56.00000000001,56\frac{3}{8}.$56.5,56.032,56.9814,56.00000000001,5638​. We can always find more numbers by adding more decimal places. 

However, the set of integers is not dense. Because between two consecutive integers, there are no other integers.  For example any number that is between the numbers $56$56  and $57$57 would be a fraction or a decimal. So there are no integers between two consecutive integers.

 

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.

 

c) Does the set of triangle numbers have a limit as the numbers increase?

Do: No. The triangle numbers approach infinity as they increase. There is no maximum value.

d) Does the set of triangle numbers have a limit as the numbers decrease?

Do: Yes. The smallest triangle number is $1$1, so the limit as the numbers decrease is $1$1. 

e) Is the set of triangle numbers dense?

Do: No. There are no triangle numbers between two consecutive triangle numbers.

 

Practice questions

Question 1

Question 2