Question

Digital Logic - I don't quite understand the definition of a power set and how many elements it contains Digital Logic - I don't quite understand the definition of a power set and how many elements it contains @Mathematics

Answer 1

I know that if given a set A then it's all the subsets of set A. But I don't quite understand what subsets are

Answer 2

like if given P({1,2,{1,2}})

Answer 3

Suppose A = {1,2}, then the power set of A, P(A) is the set of all subsets. So first of all what are all the subsets? They are all the sets whose elements are all members of A.
Well, one is A itself (both 1 and 2) and another is the empty set {} (neither 1 or 2) and the sets with one element: {1}, {2}.

Hence

P(A) = { {}, {1}, {2}, {1,2} }

Notice that the power set of A has 2^2 = 4 members.

Another example. B = {a, b, ,c}, then the subsets are
no members: {}
1 member: {a}, {b}, {c}
2 members: {a,b}, {b,c}, {a,c}
3 members: {a,b,c}

Hence
P(B)= { {}, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c} }

Now the power set has 2^3 = 8 members. The intuition for this power relationship 2^3, is that each member of B is either in or out of a subset. So there are 2x2x2 possibilities for subsets; i.e., 2^3 possibilities.

Answer 4

Hence in general, given a set X with a finite number of elements n, the power set of X, P(X), has \( 2^n \) members.