Let A = {1, 2, 3, ldots, 7} and let P(A) deno | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The set $A$ has $7$ elements,so the power set $P(A)$ has $2^7 = 128$ elements.For each element $a \in A$,we must choose a subset $f(a) \subseteq A

Vedclass · Accepted Answer

$44$

Vedclass · Answer

(D) The set $A$ has $7$ elements,so the power set $P(A)$ has $2^7 = 128$ elements.For each element $a \in A$,we must choose a subset $f(a) \subseteq A$ such that $a \in f(a)$.The number of subsets of $A$ that contain a specific element $a$ is $2^{7-1} = 2^6 = 64$.Since there are $7$ elements in $A$,and for each element $a$,there are $2^6$ choices for $f(a)$,the total number of such functions is $(2^6) 	imes (2^6) 	imes (2^6) 	imes (2^6) 	imes (2^6) 	imes (2^6) 	imes (2^6) = (2^6)^7 = 2^{42}$.We are given that the number of functions is $m^n$ where $m$ is the least value.Since $2^{42} = (2^2)^{21} = 4^{21}$ and $2^{42} = (2^1)^{42} = 2^{42}$,the least base $m$ is $2$ with $n = 42$.Thus,$m + n = 2 + 42 = 44$.

Let $A = \{1, 2, 3, \ldots, 7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f: A \rightarrow P(A)$ such that $a \in f(a)$ for all $a \in A$ is $m^n$,where $m, n \in N$ and $m$ is the least possible value,then $m + n$ is equal to . . . . . . .

Similar Questions

$f: X \rightarrow R$, where $X = \{x \mid 0 < x < 1\}$, is defined as $f(x) = \frac{2x-1}{1-|2x-1|}$. Then:

If the set $x$ contains $7$ elements and set $y$ contains $8$ elements,then the number of bijections from $x$ to $y$ is

If $(x, y) \in R$ and $x, y \neq 0$,and the function $f(x, y) = \frac{x}{y}$,then this function is a/an:

The function $f: N \rightarrow N$ defined by $f(x) = \begin{cases} x+1, & x \text{ is odd} \\ x-1, & x \text{ is even} \end{cases}$ is . . . . . . .

Let $f:[0,1] \rightarrow [-1,1]$ and $g:[-1,1] \rightarrow [0,2]$ be two functions such that $g$ is injective and $g \circ f: [0,1] \rightarrow [0,2]$ is surjective. Then,

Vedclass Test Series

Exam Paper Generator

Online Exam Module