Question

The number of non-surjective mappings that can be defined from A={1,4,9,16} to B={2,8,16,32,64} is
a) 1024
b) 20
c) 505
d) 625

Answer 1

Hint: In this question we are given two sets from A={1,4,9,16} and B={2,8,16,32,64} and we have to find the number of non-surjective mappings that can be defined from A to B. Therefore, we should first understand the definition of a surjective mapping and then find out the number of ways in which the elements of A can be mapped to B with a non-surjective mapping.
Complete step by step solution:
A mapping from a set A to a set B is a function, which associates each element of the first set to exactly one element of the second set. A surjective mapping from a set A to a set B is a mapping such that for every element in B, there is at least one element of A which maps to that element………………………(1.1)
As we asked to find non-surjective mappings, we should have every element of A mapped to exactly one element of B. However, two or more elements in A can map to the same element of B. Therefore, as there are four elements in A, the number of elements they map to can be 1, 2, 3 or 4……………………….(1.2)
We thus see that as the total number of elements of A is less than that of B, in any mapping all the elements of B cannot be chosen because of the reasoning in (1.2). Therefore, all maps from A to B will be non-surjective………………………(1.3)
Therefore, we should find out the mapping by first selecting the element to which the first element of A will map to and then selecting the element of B to which second element of A will map to and so on. However, as there is no restriction on the number of times an element in can be mapped by some elements of B, the choice of mapping of each element will be equal to the total number of elements of B which is B. For example, here each element of A can map to 2, 18, 16, 32 or 62 regardless of where the other elements have been mapped…………………..(1.4)
We know that
No. of ways of total mapping = No. of ways the first element of A can be mapped $\times$ No. of ways the second element of A can be mapped $\times$ No. of ways the third element of A can be mapped $\times$ No. of ways the fourth element of A can be mapped …………………………….(1.5)
And using (1.4) in (1.5), we see that
Total number of ways of the mapping =$5\times 5\times 5\times 5=625$
Now, as from (1.3), we know that all the maps from A to B are non-surjective, 625 should be the correct answer which matches option (d). Hence, option (d) is the correct answer.

Note: We should note that even though we are finding the number of mappings by calculating the number of ways in which elements of B can be mapped by number of elements in A, we should not use the concepts of permutations and combinations because in them each element can be chosen only once but here each element in B can be mapped more than one time by different elements of A.