# CAT Practice : Functions

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Surjective, Injective, Bijective, etc. If you have heard these terms but do not exactly know what these mean, this is the question for you. If you have not even heard these terms, then start now, hit wikipedia.

## Onto Functions

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

Choice C. 36

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## Detailed Solution

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 = 34 = 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).

• Let us assume that elements are mapped into A and B. Number of ways in which this can be done = 24 – 2 = 14

• 24 because the number of options for each element is 2. Each can be mapped on to either A or B
• -2 because these 24 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.

• 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.

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

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Functions.