countability

Question

as i know that two infinite sets are equal in size if there is a correspondence between them
suppose N={1,2,3..........}  --set of natural numbers
              E={2,4,6............}  --set of even natural numbers
             A={0,2,4,...........}   -- set of even numbers >=0
by using cantoor's method wehave a correspondence between E and N so we can say taht N and E are of same size
now  as i m able to think N and A wiil not be of equal size bcoz A={0} U E
so my doubt is that is my approach and answer both are right r not ??

Comments

All N, E & A are of same size.

You claimed that N and E are of same size but clearly, E is a proper subset of N.

The claim was based on the fact that we can define a bijective mapping(or correspondence) from E to N in such a way that an element, say i, is mapped to the element i/2 of N.

For example:  2 in E will be mapped to 1(=2/2) in N, 4 in E will be mapped to 2(=4/2) in N and so on.

Similarly, we can define a bijective mapping(or correspondence) from E to A in such a way that an element of E, say i, is mapped to the element i - 2 of A.

For example:  2 in E will be mapped to 0(=2 - 2) in A, 4 in E will be mapped to 2(=4 -  2) in A and so on.

Thus, although E is a proper subset of A but E & A are of same size.

Similarly, we can define a bijective mapping(or correspondence) from A to N in such a way that an element of A, say i, is mapped to the element (i/2) + 1 of N.

For example:  0 in A will be mapped to 1(= (0/2) + 1) in N, 2 in A will be mapped to 2(=2/2 + 1) in N and so on.

So A & N are also of same size.