Animated Solution for Mathematics - Sets and Relations: Let $A=\{1,2,3,\dots ,7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f:A\to P(A)$ such that $a\in f(a),\forall a\in A$ is $m^n,m$ and $n\in \mathbb{N}$ and $m$ is least, then $m+n$ is equal to ______

We are tasked with finding the number of functions $f:A\to P(A)$ where $A=\{1,2,3,\dots ,7\}$, subject to the constraint that for every $a\in A$, $a\in f(a)$. Here, $P(A)$ denotes the power set of $A$.

The set $A$ contains $n=7$ elements. The power set $P(A)$ contains $2^n=2^7=128$ distinct subsets.

A Filter of Possibility

Consider a single element $a\in A$. The condition $a\in f(a)$ implies that the subset $f(a)$ must contain the element $a$.

To determine the number of such subsets, we fix $a$ as an element of the subset. For the remaining $n-1=6$ elements of $A$, each element has exactly $2$ choices: it can either be included in the subset or excluded.

Thus, the number of valid subsets $f(a)$ for any specific $a$ is:

Since there are $7$ elements in $A$, and each element $a$ must independently choose a subset $f(a)$ from the $64$ valid options, we apply the fundamental principle of counting. The total number of such functions is the product of the number of choices for each of the $7$ elements.

The total number of functions is given by:

Applying the laws of exponents, we simplify this expression:

We are asked to express this result in the form $m^n$, where $m$ is the least natural number. Given $m^n=2^{42}$, we seek the smallest possible base $m$.

The smallest prime base for this power is $m=2$, which corresponds to $n=42$.

The problem asks for the sum $m+n$: