# Name for a set in which some of the elements are also contained in other set elements?

Is there a name for when a set contains individual elements as well as elements that are combinations of those elements?

For example let's take the set of all permutations (not requiring every element to be used) of 5 distinct instructions, like 1 = do laundry, 2 = mop floor, etc. Let's just refer to them as 1 2 3 4 5. So obviously in the set would be 1, 2, 3, 4, 5, 12, 13, 14, etc, all the way up to all 5 digit permutations.

What I am confused about, is that the set only has five truly unique elements. The rest of the elements are just combinations of those unique 5. So is the set redundant or ill formed in some way? Like, performing set element 12 would be the same as individually performing element 1 and element 2, but still it's different in a way because it is the element representative of their combination.Or perhaps there is a specific name describing a set that has this characteristic? Or is it considered no different than any other set?

• The set $\mathcal{P}$ of all subsets of a given set $X$ is called the "power set of $X$" (in your case, you describe the powerset of $X=\{1,2,3,4,5\}$ – mm8511 Apr 17 '18 at 19:07

You may be looking for the power set of $\{1,2,3,4,5\}$, which is the set of subsets of $\{1,2,3,4,5\}$. It has $2^5=32$ members. It does not care about the order of elements in the subsets, so considers $\{1,2\}$ the same as $\{2,1\}$ and only contains one of them. You can define the set of tuples with no repeated elements from $\{1,2,3,4,5\}$, which would distinguish $(1,2)$ from $(2,1)$ and contain both of them. It would contain $120\ 5-$ tuples, $20\ 2-$tuples, and a number of the other sizes.