1. Union of subsets

Let be a set with elements. In how many different ways can one select two not necessarily distinct or disjoint subsets of so that the union of the two subsets is ? The order of selection does not matter. For example, the pair of subsets represents the same selection as the pair

Firstly what does "two not necessarily distinct or disjoint subsets" mean?

And I've tried experimenting a bit. I wrote down the 16 different subsets for . There seems to be some pattern but I don't think brute force like this is the way to go...

So can someone explain an easier method? (Please don't leave out any steps cause I'm a beginner at combinatorics )

2. Originally Posted by usagi_killer
Let be a set with elements. In how many different ways can one select two not necessarily distinct or disjoint subsets of so that the union of the two subsets is ? The order of selection does not matter. For example, the pair of subsets represents the same selection as the pair

Firstly what does "two not necessarily distinct or disjoint subsets" mean?
Once again, I must ask: Don't you have a textbook or lecture notes?

3. Two sets A and B are not necessarily disjoint if their intersection does not necessarily equal to the empty set. And a group of integers in a set S do not necessarily have distinct subset sums if the set $\{ \sum x \in X, x: x \subset S \}$ does not necessarily have $2^{|S|}$ distinct elements.

4. Originally Posted by Plato
Once again, I must ask: Don't you have a textbook or lecture notes?
No, I'm not in university yet, I'm just self-learning because I am on holidays. The book doesn't really go into detail.