Topic: Discrete Mathematics
If A is a set with m elements and n is a positive integer, the number of different elements in $A^{n}$ is:
- A. $m\times n$
- B. $m\times m$
- C. $n^{m}$
- D. $m^{n}$
Answer: D. $m^{n}$
Explanation: The notation $A^n$ denotes the n-ary Cartesian product of the set A with itself, i.e., $A \times A \times \dots \times A$ ($n$ times). By the rule of product in combinatorics, the cardinality of the Cartesian product of sets is the product of their cardinalities. Thus, $|A^n| = |A| \times |A| \times \dots \times |A|$ ($n$ times) $= m^n$.