# The Universal \xi = {multiples of 3 less than 30} A = {odd numbers} B = {numbers greater than 15} A and B are subsets of the universal set. Find the value of: n(A \cap B')

The Universal $$\displaystyle\xi=$$ {multiples of 3 less than 30}
$$\displaystyle{A}=$$ {odd numbers}
$$\displaystyle{B}=$$ {numbers greater than 15}
A and B are subsets of the universal set.
(a). List the members of: (i). $$\displaystyle\xi$$
(ii). A
(iii) B
(b). List the members of:
(i). $$\displaystyle{A}\cup{B}$$
(ii). $$\displaystyle{A}\cap{B}$$
(iii). $$\displaystyle{A}'\cap{B}$$
(c). List the members of: $$\displaystyle{\left({A}\cup{B}'\right)}\cap{A}'$$
(d). Find the value of: $$\displaystyle{n}{\left({A}\cap{B}'\right)}$$

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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) $$\displaystyle{A}\cap{B}'=\{{3},{9},{15}\}$$
$$\displaystyle{n}{\left({A}\cap{B}'\right)}={3}$$

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