Question

Let $S=\left\{1,2,3,.......,n\right\}$ and $A=\left\{\left(a,b|\right)1\le a,b\le n\right\}=S\times S$ A subset $B$ of $A$ is said to be a good subset if $\left(x,x\right)\in B$ for every $x\in S$ Then, the number of good subsets of $A$ is

• A. $1$
• B. $2^n$
• C. $2^{n\left(n-1\right)}$
• D. $2^{n^2}$

Answer 1

Answer: (B)

We have,
$S=\left\{1,2,3,4,\dots ,n\right\}$
$A=\left\{\left(a,b\right):1\le a,b\le n\right\}=S\times S$
$B=\left\{\left(x,x\right):x\in S\right\}$
$\therefore B=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\dots ,\left(n,n\right)\right\}$
Number of elements in $B=n$
Total number of subset of $B$ is $2^n$
$\therefore$ Total number of good subset of $A$ is $2^n$