Set A = {2, 3, 5, 6, 7}, Set B = {a, b, c}. How many onto functions can be defined from Set B to Set A?

Explanatory Answer :

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 two cases

Case 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).

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

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

b) -2 because these 2^4selections 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

ii. The elements could be mapped on B & C only or C & A only. So, total number of possible outcomes = 14 * 3 = 42.

case 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)