INTRODUCTION OF SETS

Chapter 4: Sets

Revise
4.1 Intersection of sets

4.2 Union of sets
4.3 Combined operation on sets

Revise : Introduction of sets

Sets

● Set = a collection of objects according to certain characteristics.
● The objects in a set are known as elements.
● Sets are usually denoted by capital letters and notation used for sets is

braces, { }.
● Example:

A= {1, 3, 5, 7, 9}

Elements

Set A

Venn diagram represent sets

● A set can be represented by a Venn diagram using closed geometry

shapes such as circles, rectangles, triangles and etc.

● A dot to the left of an object in a Venn diagram indicates that the object

is an element of the set.

● When a Venn diagram represents the number of elements in a set, no

dot is placed to the left of the number.
● Number of elements represented Venn Diagram:
● Example: A = {2,3,5,7}

n(A) = 4 n(A) = 17

Elements of sets

● In set notation, the symbol ∈ means ‘is an element of’ or ‘belongs to’ and ∉means ‘is not an element of’ or
‘does not belong to’.

● Example :

Given that P= {factors of 15} and Q = {positive perfect squares less than 28}. Complete the following, by
using the symbol ∈ or ∉.

(a) 5 ___ P (b) 20 ___ P (c) 25 ___ Q (d) 8 ___ Q

Solution:

P= {1, 3, 5, 15}, Q = {1, 4, 9, 16, 25}

(a) 5 ∈ P 5 is an element of set P.

(b) 20 ∉ P 20 is not an element of set P.

(c) 25 ∈ Q 25 is an element of set Q.
(d) 8 ∉ Q 8 is not an element of set Q.

Empty set

● A set with no elements is called an empty set or null set. The symbol ø
or empty braces, { }, denotes empty set.

● Example:
● If set A is an empty set, then
● A= { } or A = ø
● n(A) = 0
● If B = {0} or {φ} does not denote that B is an empty set. B = {0} means

that there is an element ‘0’ in set B.
● B= {φ} means that there is an element ‘φ’ in set B.

Revise : SUBSETS

Subsets

● If every element of a set A is also an element of a set B, then

○ set A is subset of set B.

● The symbol ⊂ is used to denote ‘is a subset of’.
● Therefore, set A is a subset of set B. In set notation, it is written as A⊂B.
● Example:
● A = {11, 12, 13} and B = {10, 11, 12, 13, 14}

Every element of set A is an element of set B.

Therefore, A⊂B.

● Venn diagram represent as:

Subsets

● The symbol ⊄is used to denote ‘is not a subset of’.
● An empty set is a subset of any set. For example, ∅⊂A.
● A set is a subset of itself. Example : B ⊂ B.
● The number of subsets for a set with n elements is 2ⁿ.

Example, if A = {3, 7}

● So n = 2, then number of subsets of set A = 2² = 4.
● All the subsets of set A are { }, {3}, {7} and {3, 7}.

Revise : Complement of sets

Complement of set

● The complement of set B is the set of all elements in the universal set, ξ,
which are not elements of set B, and is denoted by B’.

● The Venn diagram below shows the relationshipbetween B, B’ and the
universal set, ξ.

The complement of set B is represented by the green colour shaded region
inside the universal set, ξ, but outside set B.
● Example : If ξ = {17, 18, 19, 20, 21, 22, 23} and B = {17, 20, 21} then,

B’ = {18, 19, 22, 23}

To be continue: 4.1 Intersection of sets