### CAT Functions, Onto functions

**Question**

How many onto functions can be defined from the set A = {1, 2, 3, 4} to
{a, b, c}?

- 81
- 79
- 36
- 45

**Answer: Choice (C)**

**Explanatory Answer:**

First let us think of the number of potential functions possible. Each
element in A has three options in the co-domain. So, the number of possible
functions = 3

^{4}= 81.
Now, within these, let us think about functions that are not onto. These
can be under two scenarios

**Scenario 1:**Elements in A being mapped on to exactly two of the elements in B (There will be one element in the co-domain without a pre-image).

Ø Let us assume that elements are mapped into A and B. Number of ways in
which this can be done = 2

^{4}– 2 = 14
o
2

^{4}because the number of options for each element is 2. Each can be mapped on to either A or B
o
-2 because these 2

^{4}selections would include the possibility that all elements are mapped on to A or all elements being mapped on to B. These two need to be deducted
Ø The elements could be mapped on B & C only or C & A only. So,
total number of possible outcomes = 14 * 3 = 42.

**Scenario 2:**Elements in A being mapped to exactly one of the elements in B. (Two elements in B without pre-image). There are three possible functions under this scenario. All elements mapped to a, or all elements mapped to b or all elements mapped to c.

Total number of onto functions = Total number of functions – Number of
functions where one element from the co-doamin remains without a pre-image -
Number of functions where 2 elements from the co-doamin remain without a
pre-image

ð Total number of onto functions = 81 – 42 – 3 = 81 – 45 = 36

**Answer Choice (C)**

Labels: 2IIM, 2iim Chennai, CAT Chennai, CAT Coaching classes Chennai, CAT Functions, Onto functions, Rajesh Balasubramanian

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