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(C) Given the set $S = \{ 1, 2, 3 \}$.The number of elements in $S$ is $n(S) = 3$.$P(S)$ is the power set of $S$,which is the set of all subsets of $S

Vedclass · Accepted Answer

$336$

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(C) Given the set $S = \{ 1, 2, 3 \}$.The number of elements in $S$ is $n(S) = 3$.$P(S)$ is the power set of $S$,which is the set of all subsets of $S$.The number of elements in the power set is $n(P(S)) = 2^{n(S)} = 2^3 = 8$.$A$ function $f: S 	o P(S)$ is one-to-one (injective) if each element in $S$ maps to a unique element in $P(S)$.The number of one-to-one functions from a set with $m$ elements to a set with $n$ elements is given by the permutation formula $^nP_m = \frac{n!}{(n-m)!}$.Here,$m = n(S) = 3$ and $n = n(P(S)) = 8$.Therefore,the number of one-to-one functions is $^8P_3 = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 	imes 7 	imes 6 = 336$.

$A$. $f: R \rightarrow R$ defined by $f(x) = \cos(112x - 37)$	$I$. Injection but not surjection
$B$. $f: A \rightarrow B$ defined by $f(x) = x\|x\|$ when $A = [-2, 2]$ and $B = [-4, 4]$	$II$. Surjection but not injection
$C$. $f: R \rightarrow R$ defined by $f(x) = (x-2)(x-3)(x-5)$	$III$. Bijection
$D$. $f: N \rightarrow N$ defined by $f(n) = n+1$	$IV$. Neither injection nor surjection
	$V$. Composite function

If $P(S)$ denotes the set of all subsets of a given set $S$,then the number of one-to-one functions from the set $S = \{ 1, 2, 3 \}$ to the set $P(S)$ is

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