Question

Consider two sets A = {2, 3, 5, 7, 11, 13} and B = {1, 8, 27}. Let f be a function from A to B such that for every element b in B, there is at least one element a in A such that f(a) = b. Then, the total number of such functions f is

• A. 537
• B. 540
• C. 667
• D. 665

Answer 1

Answer: (B)

540

Answer 2

Each element of set $B$ must be mapped to at least one element of set $A$, and we need to count how many such functions are possible.

We have 6 elements in set $A$ and 3 elements in set $B$. The condition is that each element in $B$ must have at least one pre-image in $A$, so we are looking for surjections (onto functions).

The total number of surjections from a set of size 6 to a set of size 3 can be calculated using the inclusion-exclusion principle.

The number of surjections from a set of size 6 to a set of size 3 is given by:

$3^6 - \binom{3}{1} 2^6 + \binom{3}{2} 1^6 = 729 - 192 + 3 = 540$

Thus, the total number of such functions is 540.