Function questions

Question

Consider a function f : {A, B, C} → {1, 2, 3, 4, 5, 6}.

(i) How many different possibilities are there for the range of f?

(ii) If you are also told that f is one-to-one, then how many different possibilities are there for the range of f?

Answer 1

(i). Recall that the range of a function is the set of values in the codomain that elements in the domain take. For the given setup, f maps each of A, B, C to values in {1, 2, 3, 4, 5, 6}. We can break this into cases.

Case 1: The function is constant and maps A, B, C to the same value. In this case, we are choosing 1 value from 6 possible values, for a total of 6 such ranges.

Case 2: The function is not constant, but maps to two values. This means the range is determined by choosing two distinct values from 6 possible values. There are 15 such ways of achieving this, so 15 ranges in this case.

Case 3: The function maps each element to a distinct element, which means the range is a subset of {1,2,3,4,5,6} of size 3. Thus, we are choosing 3 elements from 6 total elements, which yields 20 possible ranges.

Thus, there are 6 + 15 + 20 = 41 possible ranges.

(ii). For a function to be one-to-one, each element must be mapped to its own unique image, different from any other element in the domain. From part (i), we see that this occurs only in Case 3. Hence, there are 20 such ranges for one-to-one functions.