UNION AND INTERSECTION OF TWO SETS

Problem 1 :

If A is the set of all factors of 36 and B is the set of all factors of 63, find : 

a)  A ∩ B   

b)  A U B

Solution :

Given, A is the set of all factors of 36

So, set A = {1, 2, 3, 4, 6, 9, 12, 18, 36}

B is the set of all factors of 63.

So, set B = {1, 3, 7, 9, 21, 63}

a)  

A ∩ B = {1, 3, 9}

b)

{1, 2, 3, 4, 6, 7, 9, 12, 18, 21, 36, 63}

Problem 2 :

If X = {A, B, D, M, N, P, R, T, Z} and Y = {B, C, M, T, W, Z}, find :

a)  X ∩ Y

b)  X U Y

Solution :

Given, X = {A, B, D, M, N, P, R, T, Z} and Y = {B, C, M, T, W, Z}

a)  X ∩ Y = {B, M, T, Z}

b)  X U Y = {A, B, C, D, M, N, P, R, T, W, Z}

Problem 3 :

If U = {x │ x ≤ 30, x  Z⁺},

A = {factors of 30} and B = {prime numbers ≤ 30}

a)  Find :

b)  Use a to verify that n(A U B) = n(A) + n(B) - n(A ∩ B)

Solution :

Given, A = {1, 2, 3, 5, 6, 10, 15, 30}

B = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}

a)

(b) n(A U B) = n(A) + n(B) - n(A ∩ B)

15 = 8 + 10 - 3

15 = 18 - 3

15 = 15

Hence, it is verified.

Problem 4 :

(a)  Use the Venn diagram given to show that :

n(A U B) = n(A) + n(B) - n(A ∩ B)

Note :

(a)  Means that there are a elements in this region,

so n(A) = a + b.

(b)  Suppose A and B are disjoint events.

Explain why n(A U B) = n(A) + n(B).

Solution :

a)  From Venn diagram,

n(A) = a + b

n(B) = b + c

n(A ∩ B) = b

n(A U B) = a + b + c ---- > (1)

n(A U B) = n(A) + n(B) - n(A ∩ B)

n(A U B) = (a + b) + (b + c) - b

n(A U B) = a + b + c ---- > (2)

(1)  = (2)

Hence, it is verified.

(b)  A and B are disjoint sets.

n(A ∩ B) = Ø

n(A U B) = n(A) + n(B) - n(A ∩ B)

n(A U B) = n(A) + n(B) - Ø

n(A U B) = n(A) + n(B)

Problem 5 :

Simplify :

a)  X ∩ Y for X = {1, 3, 5, 7} and Y = {2, 4, 6, 8}

b)  A U A’ for any set A Є U.

c)  A ∩ A’ for any set A Є U.

Solution :

a)  Given X = {1, 3, 5, 7} and Y = {2, 4, 6, 8}

X ∩ Y = Ø

b)  Let U = {1, 2, 3, 4, 5}

A = {1, 2, 3}

A’ = {4, 5}

A U A’ = {1, 2, 3, 4, 5}

c)  Let U = {1, 2, 3, 4, 5}

A = {1, 2, 3}

A’ = {4, 5}

A ∩ A’ = Ø

Problem 6 :

Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(AUB) = 36, find n(A n B).

Solution :

n(A U B) = n(A) + n(B) - n(A ∩ B)

n(A) = 20, n(B) = 28 and n(AUB) = 36

Applying these known values, we get

36 = 20 + 28 - n(A ∩ B)

n(A ∩ B) = 48 - 36

= 12

So, the value of n(A ∩ B) is 12.

Problem 7 :

In a group of 100 persons, 72 people speak English and 43 can speak French. How many can speak English only ? How many can speak French only and how many can speak both English and French ?

Solution :

Let A and B be number of people who are speaking English and French respectively.

n(A) = 72, n(B) = 43 and n(AUB) = 100

Number of people who speak only English = n(A) - n(AnB)

Number of people who speak only French = n(B) - n(AnB)

Number of people who speak both English and French = n(AnB)

n(A U B) = n(A) + n(B) - n(A ∩ B)

100 = 72 + 43 - n(A ∩ B)

n(A ∩ B) = 72 + 43 - 100

= 115 - 100

= 15

Number of people who speak only English = 72 - 15

= 57

Number of people who speak only French = 43 - 15

= 28

Number of people who speak both English and French = 15