Question

Without writing them all out, what is the number of subsets of set A ={king, queen, knight, prince, princess, duke, earl}?

Answer 1

When there is 0 items, the number of subset is 1=2^0 {}=φ.

When there is one item, the number of subsets is 2=2^1, namely {}, {A}.

Assume that when there is n (different) items, there are 2^n subsets.
Then when there are 2^(n+1) items, we have the 2^n subsets for the first n items. By adding the (n+1)th item to each of the 2^n subsets, we end up with twice 2^n subsets, or 2^(n+1) subsets.

So by the principle of mathematical induction, the number of subsets of n different items is 2^n.

How many different items are there in set A?