Example 1:The Venn diagram below shows the number of elements in the universal set, ξ, set P, set Q and R.

Given n(Q) = n (P υ R)’, find n(ξ).

Solution:

n(Q) = n (P υ R)’2x + 6 + 1 + 5 = 2x + 2x2x + 12 = 4x2x = 12x = 6

n(ξ) = 2x + 2x + x + 7 + 6 + 1 + 5        = 5x + 19        = 5(6) + 19        = 30 + 19        = 49

Example 2:Diagram below is a Venn diagram showing the universal set, ξ = {Form 3 students}, set A = {Students who play piano} and set B = {Students who play violin}.

Given n(ξ) = 60, n(A) = 25, n(B) = 12 and n(A ∩ B) = 8, find the number of students who do not play the two instruments.

Solution:The students who do not play the two instruments are represented by the

shaded region, (A υ B)’.

Number of students who do not play the two instruments

= n (A υ B)’= 60 – 17 – 8 – 4 = 31

Example 1: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)

4. 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.

5. The complement of set (P ∩ Q)’ is represented by the shaded region     as shown in the Venn diagram.