The document discusses the number of onto functions that can be defined from the set A = {1, 2, 3, 4} to the set B = {a, b, c}. It explains that there are 81 possible functions overall, with 45 non-onto functions, resulting in 36 onto functions. Two types of non-onto functions are identified: those mapped to exactly two elements and those mapped to exactly one element.

1. Functions

2. Functions
How many onto functions can be defined from the set A = {1, 2, 3, 4}
to {a, b, c}?
(a) 81 (b) 79
(c) 36 (d) 45

3. Functions - Onto defintion
There is no element in the co-domain that does not have a pre-
image.
1
2
3
A
B
Example of an Onto
function
1
2
3
A
B
C
>
> X
Example of a function
that is not onto

4. Functions
First, let us think about how
many functions are possible
from Set A to Set B. We can then
deduct functions that are not
onto.
How many functions can be
defined from A to B?
How many onto functions can be defined from the set A = {1, 2, 3, 4}
to {a, b, c}?
1
2
3
4
a
b
c
Function
A B

5. Functions
How many functions can be
defined from A to B?
34 = 81
Now, non onto functions can be
of two types.
i. Functions mapped to exactly
two elements
ii. Functions mapped to exactly
one element.
How many onto functions can be defined from the set A = {1, 2, 3, 4}
to {a, b, c}?
1
2
3
4
a
b
c
Function
A B

6. Functions
i. Functions mapped to exactly
two elements
How many functions can be
mapped to just {a, b}?
How many onto functions can be defined from the set A = {1, 2, 3, 4}
to {a, b, c}?
1
2
3
4
a
b
c
Function
A B

7. Functions
i. Functions mapped to exactly
two elements
How many functions can be
mapped to just {a, b}?
24 - 2 = 14
Total number of functions that
can be mapped to just two
elements = 14 * 3 = 42.
How many onto functions can be defined from the set A = {1, 2, 3, 4}
to {a, b, c}?
1
2
3
4
a
b
c
Function
A B

8. Functions
ii. Functions mapped to exactly
one element
There are three such functions
possible. Everything mapped to
a, b or c.
So, totally we have 42 + 3 = 45
non onto functions.
Number of onto functions =
81 - 45 = 36.
How many onto functions can be defined from the set A = {1, 2, 3, 4}
to {a, b, c}?
1
2
3
4
a
b
c
Function
A B

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