Functions Question 9
Question 9 - 2024 (30 Jan Shift 1)
Let $A={1,2,3, \ldots 7}$ and let $P(1)$ denote the power set of $A$. If the number of functions $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})$ such that $\mathrm{a} \in \mathrm{f}(\mathrm{a}), \forall \mathrm{a} \in \mathrm{A}$ is $\mathrm{m}^{\mathrm{n}}, \mathrm{m}$ and $\mathrm{n} \in \mathrm{N}$ and $\mathrm{m}$ is least, then $\mathrm{m}+\mathrm{n}$ is equal to
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Answer (44)
Solution
$\mathrm{f}: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})$
$a \in f(a)$
That means ‘a’ will connect with subset which contain element ’ $a$ '
Total options for 1 will be $2^{6}$. (Because $2^{6}$ subsets contains 1 )
Similarly, for every other element
Hence, total is $2^{6} \times 2^{6} \times 2^{6} \times 2^{6} \times 2^{6} \times 2^{6} \times 2^{6}=2^{42}$
Ans. $2+42=44$
