Question

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8, 10}, and C = {1, 2, 4, 5, 8, 9}. List the elements of each set. (Enter your answers using roster notation. Enter EMPTY or ∅ for the empty set.)

(a) Ac ∩ (B ∩ C c)

(b) (A ∪ Bc) ∪ (B ∩ C c)

(c) (A ∪ B)c ∩ C c

Answer 1

(a) {6,10}

(b) {1, 3, 5, 6, 7, 9, 10}

(c) ∅

Explanation:

Given,

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

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

B = {2, 4, 6, 8, 10},

C = {1, 2, 4, 5, 8, 9},

(a) Since,

`A^c =U -A= \{ 2, 4, 6, 8, 10\}` ,

`C^c = U - C= \{ 3, 6, 7, 10\}`,

`B\cap C^c = \{6, 10\}`,

`\implies A^c\cap (B\cap C^c)=\{6,10\}`

(b)

`B^c= U-B=\{1, 3, 5, 7, 9\}`

`A\cup B^c= \{1, 3, 5, 7, 9\}`

`\implies (A\cup B^c)\cup (B\cap C^c)=\{1,3,5,6,7,9, 10\}`

(c)

`A\cup B=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}`

`(A\cup B)^c=U-(A\cup B) = \phi`