Let S = {1, 2, 3, ldots, n} and A = {(a, b) m | vedclass

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(C) The set $A = S 	imes S$ contains $n^2$ elements.$A$ subset $B$ of $A$ is a good subset if it contains all elements of the form $(x, x)$ for $x \i

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$2^{n(n-1)}$

Vedclass · Answer

(C) The set $A = S 	imes S$ contains $n^2$ elements.$A$ subset $B$ of $A$ is a good subset if it contains all elements of the form $(x, x)$ for $x \in S$.There are $n$ such elements: $(1, 1), (2, 2), \ldots, (n, n)$.For $B$ to be a good subset,these $n$ elements must be present in $B$.The remaining elements in $A$ are those $(a, b)$ where $a 
eq b$. The number of such elements is $n^2 - n = n(n - 1)$.Each of these $n(n - 1)$ elements can either be in $B$ or not in $B$.Therefore,the number of ways to choose the remaining elements is $2^{n(n - 1)}$.Thus,the total number of good subsets is $2^{n(n - 1)}$.

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

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