Let $\mathrm{A}=\{1,2,3, \ldots, 7\}$ and let $\mathrm{P}(\mathrm{A})$ denote the power set of $\mathrm{A}$. If the number of functions $f: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})$ such that $\mathrm{a} \in f(\mathrm{a}), \forall \mathrm{a} \in \mathrm{A}$ 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 _________.
Answer (integer)
44
Solution
<p>$$\begin{aligned}
& f: A \rightarrow P(A) \\
& a \in f(a)
\end{aligned}$$</p>
<p>That means '$a$' will connect with subset which contain element '$a$'.</p>
<p>Total options for 1 will be $2^6$. (Because $2^6$ subsets contains 1)</p>
<p>Similarly, for every other element</p>
<p>Hence, total is $2^6 \times 2^6 \times 2^6 \times 2^6 \times 2^6 \times 2^6 \times 2^6=2^{42}$</p>
<p>Ans. $2+42=44$</p>
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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