How many non-empty subsets does a set with "n" elements have?

A set with "n" elements has a total of ( 2^n ) subsets. This total includes all possible combinations of the elements within the set, including the empty subset. To find the number of non-empty subsets, one simply excludes the empty subset from this total. Thus, the calculation is:

[

2^n - 1

]

This subtraction accounts for the removal of the single empty set from the overall count of subsets.

To elaborate, if you consider a set with three elements, for example, {a, b, c}, the subsets are:

1. {}

2. {a}

3. {b}

4. {c}

5. {a, b}

6. {a, c}

7. {b, c}

8. {a, b, c}

In this case, there are 8 total subsets (which is ( 2^3 )), and among these, only the empty set is not a non-empty subset. Therefore, the number of non-empty subsets is ( 8 - 1 = 7 ), confirming the formula ( 2^n - 1 ) as correct.

This understanding can be applied universally to any set size "n