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Onto Function

If A = 1, 2, ...

Question

If $A = {1, 2, 3, 4}$ and $B = {5, 7, 9}$ , then the number of onto function from $A$ to $B$ is

A

36

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B

45

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C

81

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D

64

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Solution

The correct option is A $36$
If the set $A$ has $n$ elements and set $B$ has $r$ elements, then the number of onto function from set $A$ to set $B$ is equivalent to distributing $n$ distinct objects into $r$ distinct boxes such that each box gets at least one object.

Total number of onto functions $= r^{n} -^{r} C_{1} (r - 1)^{n} +^{r} C_{2} (r - 2)^{n} - . . . . . .$

Here, $n = 4 & r = 3$
$= 3^{4} -^{3} C_{1} (3 - 1)^{4} +^{3} C_{2} (3 - 2)^{4} = 81 - 48 + 3 = 36$

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Similar questions

A = {1, 2, 3, 4}

B = {5, 7, 9}

, then the number of onto function from

A

B

Q. Let f = { (2, 7), (3, 4), (7, 9), (-1, 6), (0, 2), (5, 3) } be a function from A = { -1, 0, 2, 3, 5, 7 } to B = { 2, 3, 4, 6, 7, 9 }. Is this (i) an one-one function (ii) an onto function (iii) both one-one and onto function?

A = {1, 2, 3}

B = {4, 5}

then the number of function

f : A \to B

which is not onto is ______

A = {1, 3, 5, 7}

B = {1, 2, 3, 4, 5, 6, 7, 8}

then the number of one-to-one functions from

A

B

Q. Assertion :Let

A = {x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}

B = {y_{1}, y_{2}, y_{3}}

. The number of functions from

A

B

which are not onto is

45

Reason: The number of onto functions from

A

B

is equal to the coefficient of

x^{5}

5! (e^{x} - 1)^{3}

.

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