Let A = {1, 2, 3, 4} and B = {1, 4, 9, 16}. T | vedclass

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(B) The total number of functions from set $A$ to set $B$ is $|B|^{|A|} = 4^4 = 256$. The number of one-one functions is $4! = 24$. The number of many

Vedclass · Accepted Answer

$151$

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(B) The total number of functions from set $A$ to set $B$ is $|B|^{|A|} = 4^4 = 256$. The number of one-one functions is $4! = 24$. The number of many-one functions is $	ext{Total} - 	ext{One-one} = 256 - 24 = 232$. Now,we need to find the number of many-one functions such that $1 \in f(A)$. This is equal to $(	ext{Total many-one functions}) - (	ext{Many-one functions where } 1 
otin f(A))$. If $1 
otin f(A)$,then the range of $f$ is a subset of $\{4, 9, 16\}$. The total number of functions from $A$ to $\{4, 9, 16\}$ is $3^4 = 81$. Among these $81$ functions,the number of one-one functions is $0$ (since $|A| > |\{4, 9, 16\}|$). Thus,all $81$ functions are many-one. Therefore,the number of many-one functions such that $1 \in f(A)$ is $232 - 81 = 151$.

Let $A = \{1, 2, 3, 4\}$ and $B = \{1, 4, 9, 16\}$. Then the number of many-one functions $f: A \rightarrow B$ such that $1 \in f(A)$ is equal to:

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