Question

If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A =

{0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5}, and
D = {1, 6, 7}, list the elements of the sets corresponding
to the following events:
a) A∪C
b) A∩B
c) C′
d) (C′∩D)∪B
e) (S∩C)′
f) A∩C∩D′

Answer 1

a) = {0, 2, 3, 4, 5, 6, 8}, b = {0} empty set. c = {0, 1, 6, 7, 8, 9} d = {0, 1, 3, 5, 6, 7, 8, 9} e = {0, 1, 5, 6, 7, 8, 9} f = {2,4} .

Explanation:

Given:

S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

A = {0, 2, 4, 6, 8}

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

C = {2, 3, 4, 5}

D = {1, 6, 7}

Solution:

a) A∪C = A + C - A∩C

= {0, 2, 4, 6, 8} + {2, 3, 4, 5} - {2, 4}

= {0, 2, 2, 3, 4, 4, 5, 6, 8} - {2, 4}

= {0, 2, 3, 4, 5, 6, 8}

b) A∩B = ?

A∪B = A + B - A∩B

A∪B + A∩B = A + B

A∩B = A + B - A∪B

= {0, 2, 4, 6, 8} + {1, 3, 5, 7, 9} - {0, 1, 2, 3, 4, 5, 6, 8, 9}

= {0,1, 2, 3, 4, 5, 6,7, 8, 9} - {0, 1, 2, 3, 4, 5, 6, 8, 9}

A∩B = {0} empty set.

c) C′ = S - C

= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5}

= {0, 1, 6, 7, 8, 9}

d) (C′∩D)∪B = ?

(C′∩D) = {0, 1, 6, 7, 8, 9} ∩ {1, 6, 7}

= {1, 6, 7}

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

(C′∩D)∪B = (C′∩D) + B - (C′∩D)∩B

= {0, 1, 6, 7, 8, 9} + {1, 3, 5, 7, 9} - {1, 7, 9,}

= {0, 1, 1, 3, 5, 6, 7, 7, 8, 9, 9} - {1, 7, 9,}

= {0, 1, 3, 5, 6, 7, 8, 9}

e) (S∩C)′ = S - (S∩C)′

(S∩C) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ∩ {2, 3, 4, 5}

= {2, 3, 4, 5}

(S∩C)′ = S - (S∩C)

= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5}

= {0, 1, 5, 6, 7, 8, 9}

f) A∩C∩D′ = ?

D′ = S - D

= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 6, 7}

= {0, 2, 3, 4, 5, 8, 9}

A∩C∩D′ = {0, 2, 4, 6, 8} ∩ {2, 3, 4, 5} ∩ {0, 2, 3, 4, 5, 8, 9}

= {2,4}