The question is about Onto Functions. We need to find out number of functions from one set to another set. Functions is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. A range of CAT questions can be asked based on this simple concept of Functions questions in the CAT Exam. Make sure you practice a lot of questions in CAT Functions Algebra to push your CAT preparation in the right direction. Functions as an idea is a fabulous idea if one can wrap their head around it the right way, and it has been one of the staple ideas tested by the CAT exam historically.

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

- 81
- 79
- 36
- 45

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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).

Let us assume that elements are mapped into A and B. Number of ways in which this can be done = 2^{4} – 2 = 14

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

-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.

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

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

Choice C is the correct answer.

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