Question

Given the following sets:

U = {2, 7, 10, 15, 22, 27, 31, 37, 45, 55}
A = {10, 22, 27, 37, 45, 55}
B = {2, 15, 31, 37}
C = {7, 10, 15, 37}
Give the set Ac U (B ∩ C).
a) {2, 7, 10, 31, 37}
b) {2, 7, 15, 31, 37}
c) {2, 10, 15, 31, 37}
d) {2, 7, 15, 27, 37}
e) ∅
f) None of the above.

Answer 1

Option b) is correct ie.,
`A^(c)\bigcup (B \bigcap C)={\{2, 7, 15, 31, 37\}}`

Explanation:

Given sets are

`U ={\{2, 7, 10, 15, 22, 27, 31, 37, 45, 55\}}`

`A = {\{10, 22, 27, 37, 45, 55\}}`

`B = {\{2, 15, 31, 37\}}`

`C = {\{7, 10, 15, 37\}}`

To find
`A^(c)\bigcup (B \bigcap C)`

First to find
`A^(c)`

`A^(c)={\{2,7,15,31\}}`

to find
`B\cap C`

`B\cap C={\{2, 15, 31, 37\}}\cap {\{7, 10, 15, 37\}}`

`B\cap C={\{37,15\}}`

`A^(c)\bigcup (B \bigcap C)={\{2,7,15,31\}}\cup {\{37,15\}}`

`A^(c)\bigcup (B \bigcap C)={\{2,7,15,31,37\}}`

Therefore option b) is correct

Therefore
`A^(c)\bigcup (B \bigcap C)={\{2,7,15,31,37\}}`

Answer 2

b) {2, 7, 15, 31, 37}

Explanation:

Ac is the complement of A, that is, the elements that are in the U(universe) but not in A.

Ac - {2,7,15,31}

`B \cap C` are the elements that are in both B and C. So

(B ∩ C) = {15,37}

Ac U (B ∩ C) are the elements that are in at least one of Ac or (B ∩ C).

Ac U (B ∩ C) = {2,7,15,31,37}

So the correct answer is:

b) {2, 7, 15, 31, 37}