Let A = {a, b, c} and B = {1, 2, 3, 4}. Then | vedclass

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(B) The set $C$ consists of all functions $f: A \rightarrow B$ such that $2 \in f(A)$ and $f$ is not one-one.Total functions from $A$ to $B$ where $2

Vedclass · Accepted Answer

$19$

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(B) The set $C$ consists of all functions $f: A ightarrow B$ such that $2 \in f(A)$ and $f$ is not one-one.Total functions from $A$ to $B$ where $2 \in f(A)$ is calculated as: (Total functions) - (Functions where $2 
otin f(A)$).Total functions $= 4^3 = 64$.Functions where $2 
otin f(A)$ (i.e.,$f: A ightarrow \{1, 3, 4\}$) $= 3^3 = 27$.So,functions where $2 \in f(A) = 64 - 27 = 37$.Now,we subtract the one-one functions where $2 \in f(A)$ from this total.Number of one-one functions from $A$ to $B$ is $P(4, 3) = 4 	imes 3 	imes 2 = 24$.In these $24$ one-one functions,how many contain $2$ in their range? If $2$ is not in the range,the function maps $A$ to ${1, 3, 4}$. Number of one-one functions $= P(3, 3) = 3 	imes 2 	imes 1 = 6$.So,one-one functions where $2 \in f(A) = 24 - 6 = 18$.Therefore,the number of functions that are not one-one and $2 \in f(A) = 37 - 18 = 19$.

Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$. Then the number of elements in the set $C = \{f : A \rightarrow B \mid 2 \in f(A) \text{ and } f \text{ is not one-one}\}$ is

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