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1.Set Theory
normal
Let $S=\{1,2,3, \ldots \ldots, n\}$ 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
A
$1$
B
$2^n$
C
$2^{n(n-1)}$
D
$2^{n^2}$
(KVPY-2012)
Solution
(b)
We have,
$S=\{1,2,3,4, \ldots, n\}$
$A=\{(a, b): 1 \leq a, b \leq n\}=S \times S$
$B=\{(x, x): x \in S\}$
$\therefore \quad B=\{(1,1),(2,2),(3,3), \ldots,(n, n)\}$
Number of elements in $B=n$
Total number of subset of $B$ is $2^n$.
$\therefore$ Total number of good subset of $A$ is $2^n$.
Standard 11
Mathematics
