Please Correct if I am wrong at any point since at some points I must go wrong in my basic understanding of the question and data.
My understanding of this is that set A={1,2,3..,n} refers to a set, A, having n number of natural numbers in order for example if n=5 A={1,2,3,4,5} and if n=10 A={1,2,3,4,5,6,7,8,9,10}. The Cardinality of a set is the number of elements in the set for example if n=5 set A has a cardinality of 5 and etc.
For the question itself my understanding is that they've defined a set {a,S} such that, a, is a member of, S, and, S, is a member of the power set of, A, and I'm supposed to find the number of elements in this set. My current understanding is: The Cardinality of P(A) is 2^n. S, is a member of P(A) so therefore should have the same cardinality and, a, is a member of, S, and so should have the same cardinality making the overall set have a cardinality of 2^2n.
However this answer is completely wrong, the answer says that there a n choices for a and 2^(n-1) choices for S since we can choose the set of S \ {a} in 2^(n-1) ways giving us the answer of n2^(n-1).
Best Answer
Hint:
Consider instead $\{(a,T)~:~a\in A,~a\notin T,~T\in\mathcal{P}(A)\}$
See if you can come up with a bijection from that set to yours
Now, as for counting the size of this set, break apart via multiplication principle. First pick the element $a$. Then, pick a subset not containing $a$.