CAT Practice : Functions

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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

 

  • Correct Answer
    Choice C. 36

Explanatory Answer

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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
Answer Choice (C)

Correct Answer: 36



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