CHAPTER 4 OPERATIONS ON SETS(4.1)

Chapter 4:
Operations on Sets

4.1 Intersection of sets

4.1 INTERSECTION OF SETS

4.1.1 Determine and describe the intersection of sets using various
representations.

Determine and describe the intersection of sets
using various representations.

A = { x : x is the odd number less than 10 } B = { 3, 6, 9, 12, 15 }

All the elements in set A and set B respectively.
There are elements common to Set A and set B, which are {3, 9}.

Determine and describe the intersection of sets
using various representations.

Moreover, the area in both set A and set B, where the two sets
overlap with the elements common to set A and set B, is called

intersection of A and B.

Determine and describe the intersection of sets
using various representations.

The symbol of
intersection of
sets is ⋂ .

A ⋂ B = { 3, 9 }

The grey colour region is
the Intersection of two set.

Determine Intersection of two sets and represented
by using Venn diagram

● The intersection of set P and set Q, denoted by P∩Q is the set consisting
of all elements common to set P and set Q.

● The overlapping region is the intersection of two sets which
represented by venn diagram as below :

P∩Q Q ⊂ P, then P ∩ Q = Q

Determine Intersection of more than two sets and
represented by using Venn diagram

● The intersection of set P, set Q and set R, denoted by P∩Q∩R is the set
consisting of all elements common to set P, set Q and set R.

● Venn diagram represents intersection of sets :

P∩Q∩R

Intersection of sets

● P ∩ Q = ø, There is no intersection between set P and set Q .
● Therefore P ∩ Q is empty set.
● Venn diagram represents as :

Example : Intersection of sets :

Given that A= {3, 4, 5, 6, 7}, B = {4, 5, 7, 8, 9, 12} and C = {3, 5, 7, 8, 9, 10}.

(a) Find A ∩ B ∩ C.

(b) Draw a Venn diagram to represent A ∩ B ∩ C.

Solution :

(a) A ∩ B ∩ C = { 5, 7 } (b)

More example:

Complement of intersection of sets

4.1.2 Determine the complement of the intersection of sets.

Complement of the intersection of two sets

● The complement of the intersection of two sets, P and Q, represented
by (P ∩ Q)’, is a set that consists of all the elements of the universal set, ξ,
but not the elements of P ∩ Q.

● (P ∩ Q)’ is represented by the shaded region as shown in the Venn
diagram.

Other complement of intersection of sets in venn
diagram:

More example:

Given the universal set, ξ = {x : x is an integer, 1 ≤ x ≤ 8}, set A = {1, 2, 3, 4, 5,
6}, set B = {2, 4, 6} and set C = {1, 2, 3, 4}, list all the elements and state the
number of elements of the following sets.

(a) (A ∩ B)′

Solution:
ξ = {1, 2, 3, 4, 5, 6, 7, 8}

(a) A ∩ B = {2, 4, 6} A = {1, 2, 3, 4, 5, 6} B = { 2, 4, 6 }

(A ∩ B)′ = {1, 3, 5, 7, 8}

n(A ∩ B)′ = 5

More example:

Given the universal set, ξ = {x : x is an integer, 1 ≤ x ≤ 8}, set A = {1, 2, 3, 4, 5,
6}, set B = {2, 4, 6} and set C = {1, 2, 3, 4}, list all the elements and state the
number of elements of the following sets.
(b) (A ∩ B ∩ C)′
Solution:
ξ = {1, 2, 3, 4, 5, 6, 7, 8}
(b) A ∩ B ∩ C = { 2, 4 } A = {1, 2, 3, 4, 5, 6} B = {2, 4, 6} C = {1, 2, 3, 4}

( A ∩ B ∩ C )′ = { 1, 3, 5, 6, 7, 8 }

n( A ∩ B ∩ C )′ = 6

To be continued : 4.2 UNION OF SETS