# Number sets || Representing number types in set

In this post we will discuss common number sets used in mathematics and will represent the sets in different set representation.

All the number types will be expressed in three format:

(a) Statement form of set representation
(b) Roster form
(c) Set Builder form

Note that these set of numbers will be used again and again in higher mathematics. So make sure you understand the chapter and remember all the formats explained in this chapter.

## Explaining Number sets

Number sets are basically group of numbers showing similar property.

Number sets are important because instead of writing all the numbers in format, we can express it in the form of formula or set representation.

Given below are important number sets in mathematics:

(01) N = Natural Number

Natural number is a set of number starting from digit 1.

⟹ Notation N is used to represent natural number
⟹ Natural number do not involve fraction or decimal number.

Examples of Natural Numbers are 1, 2, 3, 4, 5, 6 . . . . . etc.

Using set to represent Natural Number

(i) Statement form
X = { Set of all integers starting from 1 }

(ii) Roster form
X = { 1, 2, 3, 4, 5, 6, . . . . . }

(iii) Set builder form
X = { y : y > 0 & y is integer }, or;
X = { y : y is positive integer } , or ;
X = { y : y is counting number starting from 1 }

(02) W = Whole numbers

Whole numbers are counting numbers starting from digit 0.

⟹ Notation W is used to represent whole numbers
⟹ Whole numbers does not includes decimals, fractions and negative numbers.

Example of Whole Numbers are 0, 1, 2, 3, 4, 5, 6, 7 . . .etc.

Note
Whole numbers are similar to natural numbers but it also includes number 0.

Using Set to represent Whole Numbers.

(i) Statement form
X = { Set of all integers starting from 0 }

(ii) Roster form
X = { 0, 1, 2, 3, 4, 5, 6, . . . . . }

(iii) Set builder form
X = { y : y \mathtt{\geqq } 0 & y is integer }, or;
X = { y : y is both positive & zero integer }, or;
X = { y : y is counting number starting from 0 }

(03) Z = Integer

Integers include negative numbers, 0 and positive natural numbers.

⟹ Notation Z is used to represent integer numbers
⟹ Integers do not include decimals and fraction.

Integers are same as whole numbers but they also includes negative numbers.

Using set to represent Integers

(i) Statement form

X = { -ve natural numbers and whole numbers }

(ii) Roster form

X = { . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . . . }

(iii) Set builder form

X = { y : \mathtt{0\geqq y\leqq 0;\ \ y\neq \ decimal} }, or;

X = { y : y is -ve natural number, 0 & positive natural number }

(04) E = Even Natural Number

These are the numbers which are exactly divisible by 2.

Examples of even natural numbers are: 2, 4, 6, 8, 10, 12 . . .

⟹ Annotation E is used to represent even natural numbers

⟹ Just start from number 2 and move forward by skip counting by 2’s, you will get your even natural numbers.

⟹ These do not include decimals or fraction

Representing Even natural numbers using set

(i) Statement Form
X = { natural number divisible by 2 }

(ii) Roster Form
X = { 2, 4, 6, 8, 10, 12, 14 . . .}

(iii) Set Builder form
X = { y : y = 2n & \mathtt{n\ \epsilon \ Natural\ Number} }, or;

X = { y : y is natural number divisible by 2 }

(05) O = Odd Natural Number

The natural numbers not divisible by 2 is known as Odd natural numbers.

Examples are 1, 3, 5, 7, 9, 11 . . . etc.

⟹ Annotation O is used to represent odd natural numbers

⟹ Just start with number 1 and move forward by skip counting 2’s, you will get all odd natural numbers.

⟹ These number doesn’t include fractions or decimals.

Representing Odd Natural numbers through Sets

(i) Statement Form
X = { Natural Numbers not divisible by 2 }

(ii) Roster Form
X = { 1, 3, 5, 7, 9, 11 . . . }

(iii) Set Builder form

X = { y : y = 2n + 1 & \mathtt{n\ \epsilon \ Whole\ Number} }, or;

X = ( y : y is natural number not divisible by 2}

(06) Q = Rational Numbers

The numbers that can be represented in the form of a/b is called rational numbers. (Here b should not be 0)

We know that fractions can be represented in the form of a/b, hence they are part of rational numbers.

Examples of rational numbers are : -3, 2/6, -3.7, 16, 0 . . . etc.

⟹ Annotation Q (i.e. quotient) is used to represent rational numbers.

⟹ All number sets like decimals, fractions, whole numbers and integers are part of rational numbers.

Note:
Rational number are numbers which can be easily converted into fraction form.
For example, 2.5 can be written as 25/10

Representing natural number in a set

(i) Statement form
Q = { All numbers in form of a/b and b cannot be zero}

(ii) Roster form
Cannot be expressed in roster form

(iii) Set Builder form

Q = { y : y = a / b, \mathtt{a\ \&\ b\ \epsilon \ integer,\ b\neq 0} }

(07) P = Irrational Numbers

The decimal numbers which cannot be represented into fractions are called irrational numbers.

Examples of Irrational Numbers are:

(i) 𝜋 = 3.14159 . . . .

We generally use fraction value of 𝜋 = 22/7, but this is just an approximate value.

There is no particular fraction value which can exactly represent the 𝜋 value. That’s why 𝜋 is an irrational number.

(ii) \mathtt{\sqrt{3}} = 1.73205 . . . .

The value of \mathtt{\sqrt{3}} is an unending value in decimals.

There is no fraction which can accurately convert the above decimal. Hence \mathtt{\sqrt{3}} is an irrational number.

Representing irrational number in sets

Let R = Real Numbers
and Q = Rational Numbers

We have already discussed value of Q as;
Q = { y : y = a / b, \mathtt{a\ \&\ b\ \epsilon \ integer,\ b\neq 0} }

Then Set of irrational number will be;
P = R – Q

P= R – { y : y = a / b, \mathtt{a\ \&\ b\ \epsilon \ integer,\ b\neq 0} }