Final answer:
If sets A and B have equal cardinalities, then the cardinalities of their powersets are also equal because each set's powerset contains 2 to the power of the cardinalality of the original set's elements. Therefore, |P(A)| equals |P(B)| when |A| equals |B|.
Step-by-step explanation:
To show that if A and B are sets such that |A| = |B|, then |P(A)| = |P(B)|, we must understand the concept of powersets and cardinality. The cardinality of a set is the number of elements in the set, and the powerset of a set is the set of all possible subsets of the original set, including the empty set and the set itself.
The cardinality of the powerset, |P(S)|, for any set S is 2|S|. This is because, for each element in the original set, there are two possibilities in a subset: either the element is included, or it is not. Given this, if two sets A and B have the same cardinality, there is a one-to-one correspondence between their elements, which also means there is a one-to-one correspondence between their subsets. Therefore, their power sets will have the same number of subsets.
Since |A| is equal to |B|, applying the formula for the cardinality of a powerset yields |P(A)| = 2|A| = 2|B| = |P(B)|, thereby proving that the cardinalities of the powersets of A and B are equal.