Alex V. answered • 09/29/21
PhD student with 5+ years of teaching experience
Something is a subset if it is a member of, but not equal to, the larger set. So when we compare a proper subset, call it B, to the larger set its a member of, call it A, A will always have at least one member that B lacks. Basically, if we have a set {P, Q, R}, call it C, then {P, Q, R} is a subset of C, but not a proper subset. It's the same as the difference between less than and less than or equal to. Subsets can be equal to, but proper subsets cannot.
Ok, so that means there are a lot of subsets of the set {H, I, J, K, L, M, N, O, P}, which we'll call S for set. Remembering that we can always include the null set, there are 10 ways to form a proper subset with 9 members (8 of the members of S and the null set), 45 ways to form a proper subset with 8 members (7 from S + null), 120 ways to form a proper subset with 7 members (6 from S + null), and so on...
To get the answer, you'll need to find the number of combinations for proper subsets of each size (down to 1) and then add that to the number of combinations of every other sized proper subset. So far, it would be 10 + 45 + 120... So we definitely know it's going to be a big number!
Hope that helps get you started!
