elementary set theory - Equinumerosity of infinite sets

Key issue: For infinite sets, does the existence of a bijection mean they have the same number of elements?

For example, does the set of natural numbers N = {1,2,3,4...} have the same number of elements as the set of positive even integers E = {2,4,6,8...}? There is a one-to-one correspondence between these sets:

1 -> 2
2 -> 4
3 -> 6
...

However, depending on how you look at it, there is also an injective function (one-to-one but not onto) between E and N:

  -> 1
2 -> 2

  -> 3
4 -> 4
  -> 5
6 -> 6
...

In this way, all of the elements in E are in N, but not all of the elements in N are in E. Does this "prove", in some way, that N and E are not equinumerous? If not, does this mean that while E is a proper subset of N, it still has the same number of members as N - something that is true for infinite sets but not true of finite sets?

A second example:

What about the following sets:

A: {a1,a2,a3,…}
B: {b1,b2,b3,…}

A has one-to-one correspondence (bijection) with B:

a1 -> b1

a2 -> b2
a3 -> b3
…

If this, bijection from A to B, shows that A and B have the same number of elements, what happens if we shift A up so that there is an injective function (one-to-one but not onto) between A and B, e.g.

?  -> b1
a1 -> b2
a2 -> b3

a3 -> b4
…

Does this show that A and B do not necessarily have the same number of elements, after all - despite having bijection between them in one arrangement - since we can arrange them in such a way that there is a one-to-one function, but not onto?