Question

Let A and B be two sets such that : $n\left(A\right)=20,n\left(A\cup B\right)=42\mathrm{and}n\left(A\cap B\right)=4$. Find
(i) $n\left(B\right)$
(ii) $n\left(A-B\right)$
(iii) $n\left(B-A\right)$

Answer 1

Given:
$n\left(A\right)=20,n\left(A\cup B\right)=42\mathrm{and}n\left(A\cap B\right)=4$

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{We}\mathrm{know}: \\ n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B) \\ \Rightarrow 42=20+n\left(B\right)-4 \\ \Rightarrow n\left(B\right)=26 \\ \left(ii\right)n(A-B)=n\left(A\right)-n(A\cap B) \\ \Rightarrow n(A-B)=20-4=16 \\ \left(\mathrm{iii}\right)\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{sets}\mathrm{follow}\mathrm{the}\mathrm{commutative}\mathrm{property}. \\ \therefore n(A\cap B)=n(B\cap A) \\ n(B-A)=n\left(B\right)-n(B\cap A) \\ \Rightarrow n(B-A)=26-4=22 \end{gathered}$$