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