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Question: Find number of surjection from A to B where . \[A = \{ 1,2,3,4\},B = \{ a,b\}\]...

Find number of surjection from A to B where .

A={1,2,3,4},B={a,b}A = \{ 1,2,3,4\},B = \{ a,b\}

A

13

B

14

C

15

D

16

Answer

14

Explanation

Solution

Number of surjection from A to B =∑r=12(−1)2−r2Cr(r)4= \sum_{r = 1}^{2}{( - 1)^{2 - r}}{_{}^{2}C}_{r}(r)^{4}

=(−1)2−12C1(1)4+(−1)2−22C2(2)4=−2+16=14= ( - 1)^{2 - 1}{_{}^{2}C}_{1}(1)^{4} + ( - 1)^{2 - 2}{_{}^{2}C}_{2}(2)^{4} = - 2 + 16 = 14

Therefore, number of surjection from A to B = 14.

Trick : Total number of functions from A to B is 242^{4} of which two function f(x)=af(x) = a for all x∈Ax \in A and g(x)=bg(x) = b for all x∈Ax \in A are not surjective. Thus, total number of surjection from A to B =24−2=14.= 2^{4} - 2 = 14.