Question

If a set has 31 different proper subsets, then what is the number of elements in the set?

Answer 1

There are \(5\) elements in the set.

Answer 2

\[\sum_{n=1}^{5}\binom{5}{n}=31\]

Answer 3

The number of subsets of a set with n elements is 2^n. An element of the given set is either in the set or not; hence, the 2 ways to deal with a given element. 2^n includes the set itself as well as the empty set. The set itself is not considered to be a proper subset of itself.

If your set has 31 proper subsets, then add 1 to the 31 to get 32 subsets.

2^n = 32 = 2^5 which gives n = 5. There are five elements in your given set.

Note: The empty set is a proper subset of every set except itself.