Total Number of Functions from Set A to Set B
Question
Let \(A\) and \(B\) be any two sets such that
\[ n(A)=p \]and
\[ n(B)=q \]Then the total number of functions from \(A\) to \(B\) is equal to ?
Solution
Let
\[ A=\{a_1,a_2,a_3,\ldots,a_p\} \]and
\[ B=\{b_1,b_2,b_3,\ldots,b_q\} \]A function from \(A\) to \(B\) assigns exactly one element of \(B\) to each element of \(A\).
For each element of set \(A\), there are \(q\) possible choices in set \(B\).
Since \(A\) contains \(p\) elements, the total number of possible functions is:
\[
q \times q \times q \times \cdots \times q
\]
(\(p\) times)
\[ =q^p \]Final Answer
\[ \boxed{q^p} \]Therefore, the total number of functions from \(A\) to \(B\) is
\[ q^p \]