Question

Let $x=\{n\in N:1\le n\le 50\}$. If $A=\{n\in x:n$ is a multiple of $2$} and $B=\{n\in x:n$ is a multiple of $7$}, then the number of elements in the smallest subset of ‘$x$’ containing both A & B is

Answer 1

Baisc set theory concept:

$A=\{n\in x:n$ is a multiple of $2$}

$A=\{2,4,6,8,10......\}$

$B=\{n\in x:n$ is a multiple of $7$}

$B=\{7,14,21,28,....\}$

$x=\{n\in N:1\le n\le 50\}$

Smallest subset of $x$ which contains elements of both $A\&B$ is a set with multiples of $2$ or $7$ less than $50$.

{$P=\{x:x$ is a multiple of $2$ less than or equal to $50\}$

Therefore, $n\left(P\right)=25$

$Q=\{x:x$ is a multiple of $7$ less than or equal to $50$}

Therefore, $n\left(Q\right)=7$

Also, $\{14,28,42\}$is divisible $2$ & $7$

Therefore, $n(P\cap Q)=3$

$$\begin{gathered} \Rightarrow n(P\cup Q)=n\left(P\right)+n\left(Q\right)-n(P\cap Q) \\ \Rightarrow n(P\cup Q)=25+7-3 \\ \Rightarrow n(P\cup Q)=29 \end{gathered}$$

Hence the smallest subset of $x$ containing both $A\&B$ is $29$