XIII OM - III - Task 4

In how many ways can a set of $ n $ items be divided into two sets?
The answer to the posed question will depend on whether - to the considered sets, we include the empty set, i.e., a set that does not have any element (object), or not; in the first case, the sought number is $ 1 $ greater than in the second. We agree to consider only non-empty sets.
Let $ p $ denote the number of all possible partitions of the set $ Z $ having $ n $ elements, into two sets, i.e., into two parts.
Then $ 2p $ denotes the number of all possible parts, i.e., subsets of the set $ Z $, with the condition that we only consider proper subsets, i.e., different from the set $ Z $ itself.
The number $ 2p $ of all subsets of a set with $ n $ elements can be found by adding the numbers of subsets with $ 1, 2, \ldots, (n - 1) $ elements.
The number of subsets with $ k $ elements is as many as there are combinations of $ n $ elements taken $ k $ at a time, i.e., $ \binom{n}{k} $. Therefore,






According to the binomial theorem of Newton,






Subtracting equations (1) and (2) and considering that $ \binom{n}{0} = \binom{n}{n} = 1 $, we obtain






and from this,






