DETERMINE THE ELEMENTS OF GIVEN SETS FROM VENN DIAGRAM

Problem 1 :

Find the following in the venn diagram.

Solution :

i) set C = {1, 3, 7, 9}

ii) set D = {1, 2, 5}

iii) set U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

iv) set C ∩ D = {1}

v) set C ∪ D = {1, 2, 3, 5, 7, 9}

b) 

i) n(C) = 4

ii) n(D) = 3

iii) n(U) = 9

iv) n(C ∩ D) = 1

v) n(C ∪ D)

n(C ∪ D) = n(C) + n(D) – n(C ∩ D)

= 4 + 3 – 1

= 6

Problem 2 :

Find the following.

Solution :

i) set A = {2, 7}

ii) set B = {1, 4, 6}

iii) set U = {1, 2, 3, 4, 5, 6, 7, 8}

iv) set A ∩ B = { }

v) set A ∪ B = {1, 2, 4, 6, 7}

b Find :

i) n(A) = 2

ii) n(B) = 3

iii) n(U) = 8

iv) n(A ∩ B) = { }

v) n(A ∪ B)

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

= 2 + 3 – {}

= 5

Problem 3 :

a) List the elements of :

i) U       ii) N         iii) M

b) What are n(N) and n(M)?

c) Is M  ⊆ N?

Solution :

i) U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

ii) N = {8, 3}

iii) M = {1, 4, 7}

b) n(N) = 2 and n(M) = 3

c) M ⊆ N = {1, 3, 4, 7, 8}

Problem 4 :

One hundred students were asked whether they currently enrolled in English, History and Spanish. Use the Venn Diagram to answer the questions.

a) How many students are not enrolled in any of these three disciplines?

b) How many are enrolled in English and History?

c) How many are enrolled in English or History?

d) How many are enrolled in Spanish?

Solution :

a) Number of students who enrolled in any of these three discipline

= total number of students - number of students who enroll in any one

= 100 - (23 + 10 + 21 + 7 + 8 + 11 + 12)

= 100 - 92

= 8

b) Number of students who enrolled in English and History = 10 + 7

= 17

c) Number of students who are enrolled in English or History

= 23 + 10 + 7 + 8 + 11 + 21

= 80

d) Number of students who are enrolled in Spanish = 8 + 7 + 11 + 12

= 38

Problem 5 :

The Venn diagram alongside shows the number of people in a sporting club who play tennis (T) and hockey (H). Find the number of people:

(a) in the club

(b) who play hockey 

(c)  who play both sports

(d) who play neither sport

(e) who play at least one sport

(f)  who play tennis but not hockey

Solution :

(a) in the club

= 15 + 27 + 26 + 7

= 75

(b) who play hockey

= 27 + 26

= 53

(c)  who play both sports = 27

(d) who play neither sport = 7

(e) who play at least one sport

= 75 - 7

= 68

(f)  who play tennis but not hockey = 15

Problem 6 :

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

A = multiples of 3

B = multiples of 5

(a) Complete the Venn diagram

One of the numbers is selected at random.

(b) Write down P (A ∩ B)

Solution :

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}

A = {3, 6, 9, 12, 15}

B = {5, 10, 15} 

(a) Complete the Venn diagram

b) A n B = {15}