Thread: Induction proof: Cardinality of power sets

I am trying to prove the following by induction:

The cardinality of the power set of a set with n elements has 2^n elements


Is there a property that says that the cardinality of a power set with x+1 elements is twice the cardinality of a set with x elements? Or how would I prove that this is the case?

Thanks!

Originally Posted by economanc

The cardinality of the power set of a set with n elements has 2^n elements
Is there a property that says that the cardinality of a power set with x+1 elements is twice the cardinality of a set with x elements? Or how would I prove that this is the case?

Well just think about it.
If has elements then
would have the same subsets as the former plus each one of those united with .
That does double the count.

Fair enough, I just wasn't sure if doing it that way would be "rigorous" enough of a proof. It does make sense however.

Thanks again!