First of all big thanks to 
The question was: Which set of numbers is bigger, P or E.
P= all natural numbers 1, 2, 3, 4, 5, 6, 7, ... to infinity
E= all even numbers 2, 4, 6, 8, 10, 12, 14, ... to infinity
So Alice and Betty were arguing. Alice said, P is bigger. Due to the fact, that it contains numbers, which are not in P. Betty's Opinion was, that both are of the same size, because every number of P can be multiplied by 2 and become a number of E.
To answer the question: Both set's are equaly big.
Why?
Simply because both are infinit. P contains infinit natural numbers. You could start to count. But you would never finish counting. There is an infinit amount of numbers in this set. Just like in E. The only difference is, that those are not just natural numbers, but even numbers. But also an infinit amount of them. You could not count them all either.
If the set's would not go to infinity, that would be a huge difference.
For example if we would say the limit is 100. There would be 100 natural numbers. But only 50 even ones.
With an even number as limit, P will always be twice as big as E.
But infinity is not a number!
Therefore they are of the same size.