I'm currently working through set theory at the moment and have stumbled across a question that I'm not entirely sure how to go about answering
For the set of A = {1,2,3,...,n) how many subsets, B, have the property that B ∪ {1,2} = A?
I know that the number of subsets B of A have the property that B ∩ {1, 2} = ∅ is equal to 2^n - 2^n-2 as the number of subsets of A that contain 1 and 2 is 2^n-2. However I'm unsure how to go about answering this other question.
For $B\cup\{1,2\}$ to equal $\{1,2\ldots,n\}$, $B$ needs to have all of $3$, $4\ldots,n$ as elements. How many subsets of $\{1,2,\ldots,n\}$ contain $\{3,4,\ldots,n\}$?