Step 1

Here a universal set and two subsets are given.

We have to find the following members in the each relation.

Step 2

a) The universal \(\displaystyle\xi=\) {multiples of 3 less than 30}

\(\xi = \{3,6,9,12,15,18,21,24,27\} \)

A and B are subset of the universal set \(\displaystyle\xi\)

\(\displaystyle{A}=\) {add numbers}

\(A = \left\{3,9,15,21,27\right\}\)

\(\displaystyle{B}=\) {numbers greater than 15}

\(\displaystyle{B}=\{{18},{21},{24},{27}\}\)

i) \(\displaystyle\xi=\{{3},{6},{9},{12},{15},{18},{21},{24},{27}\}\)

ii) \(\displaystyle{A}=\{{3},{9},{15},{21},{27}\}\)

iii) \(\displaystyle{B}=\{{18},{21},{24},{27}\}\)

Step 3

b) i) \(\displaystyle{A}\cup{B}\{{3},{9},{15},{18},{21},{24},{27}\}\)

ii) \(\displaystyle{A}\cap{B}=\{{21},{27}\}\)

\(\displaystyle{A}'=\{{6},{12},{18},{24}\}\)

iii) \(\displaystyle{\left({A}'\cap{B}\right)}=\{{18},{24}\}={A}'\cap{B}\)

c) \(\displaystyle{B}'=\{{3},{6},{9},{12},{15}\}\)

\(\displaystyle{A}\cup{B}'=\{{3},{6},{9},{12},{15},{21},{27}\}\)

\(\displaystyle{\left({A}\cup{B}'\right)}\cap{A}'=\{{6},{12}\}\)

d) \(\displaystyle{A}\cap{B}'=\{{3},{9},{15}\}\)

\(\displaystyle{n}{\left({A}\cap{B}'\right)}={3}\)