GATE CS 1998 Q1.8 — Sets, Relations and Functions

To find the number of functions from a set $A$ with $m$ elements to a set $B$ with $n$ elements, we analyze the choices available for mapping each element of the domain.

Let the domain set be $A = \{a_1, a_2, \dots, a_m\}$ and the codomain set be $B = \{b_1, b_2, \dots, b_n\}$.

• For the first element $a_1 \in A$, there are $n$ possible choices in $B$ to map to.
• For the second element $a_2 \in A$, there are also $n$ possible choices in $B$.
• Similarly, for each of the $m$ elements in $A$, there are exactly $n$ choices in $B$.

Since the mapping of each element in the domain is independent of the others, the total number of possible functions is the product of the number of choices for each element:

Therefore, the correct option is C.