Let A = {1, 2, 3, ..., 7} and let P(A) denote the power set of A. If the number of functions f : A ? P ( A ) such that a ? f ( a ) , ? a ? A is m n , m and n ? N and m is least, then m + n is equal to _____. [2024]

Question

Let A = {1, 2, 3, ..., 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 $n \in N$ and $m$ is least, then $m + n$ is equal to _____. [2024]

Smart Achievers · Accepted Answer

(44)

Given, $f : A \to P (A) \Rightarrow a \in f (a)$

It means $' a'$ will connect with subset which contain element $a .$

Total options for 1 will be $2^{6}$ ( $∵$ $2^{6}$ subsets contains 1)

Similarly, for every other element

Now, number of functions from A to P(A) = ${(2^{6})}^{7} = 2^{42}$

i.e., $m + n = 2 + 42 = 44$

Solution

Q.
Let A = {1, 2, 3, ..., 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 $n \in N$ and $m$ is least, then $m + n$ is equal to _____. [2024]

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Solution

Q. Let A = {1, 2, 3, ..., 7} and let P(A) denote the power set of A. If the number of functions f:A→P(A) such that a∈f(a), ∀ a∈A is mn, m and n∈N and m is least, then m+n is equal to _____. [2024]

Ans. (44) Given, f:A→P(A)⇒a∈f(a) It means 'a' will connect with subset which contain element a. Total options for 1 will be 26(∵ 26 subsets contains 1) Similarly, for every other element Now, number of functions from A to P(A) = (26)7=242 i.e., m+n=2+42=44

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Q.
Let A = {1, 2, 3, ..., 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 $n \in N$ and $m$ is least, then $m + n$ is equal to _____. [2024]