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CAT Questions | Algebra | Functions - Onto Functions

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}?

  1. 81
  2. 79
  3. 36
  4. 45

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Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Functions: 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.

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

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

Hence the answer is "36"

Choice C is the correct answer.


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