Lets say I have a set of $\{a, b, c, d\}$, is there a means by which I can generate a set that contains all permutations of sets, such as... $$\{ ab, ac, ad, bc, bd, cd, abc, abd, acd, bcd, abcd \}$$
I'm unsure what the name for this computation is, or a means to do so that are not a manual process. I'm able to write computer code that does this using nested loops, but that seems to be an inelegant solution to the problem. Perhaps there's a matrix multiplication I can use?