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Question:
The number of surjective functions from $A$ to $B$ where $A = \{1, 2, 3, 4 \}$ and $B = \{a, b\}$ is
The number of surjective functions from $A$ to $B$ where $A = \{1, 2, 3, 4 \}$ and $B = \{a, b\}$ is
Updated On: Sep 24, 2024
- 14
- 12
- 2
- 15
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The Correct Option is A
Solution and Explanation
If A and B are two sets having m and n elements such that
$1 \le n \le m = \sum^{n}_{r=1} (-1)^{n-r} {^nC_{r}} r^{m}$
$= \sum^{n}_{r=1}(-1)^{2-r} {^{2}C_{r}} (r)^{4}$
$ = (-1)^{2-1} {^{2}C_{1}} (1)^{4} + (-1)^{2-2} {^{2}C_{2}} (2)^{4} = -2 + 16$
$ = 14 $
$1 \le n \le m = \sum^{n}_{r=1} (-1)^{n-r} {^nC_{r}} r^{m}$
$= \sum^{n}_{r=1}(-1)^{2-r} {^{2}C_{r}} (r)^{4}$
$ = (-1)^{2-1} {^{2}C_{1}} (1)^{4} + (-1)^{2-2} {^{2}C_{2}} (2)^{4} = -2 + 16$
$ = 14 $
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Concepts Used:
Types of Functions
Types of Functions
One to One Function
A function is said to be one to one function when f: A → B is One to One if for each element of A there is a distinct element of B.
Many to One Function
A function which maps two or more elements of A to the same element of set B is said to be many to one function. Two or more elements of A have the same image in B.
Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function.
One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function.
Read More: Types of Functions
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