When asked to prove that "$\exists$ a ..."; then what you are really doing is actually finding whatever is that you need to prove exists.
For example in your question you must prove that there exists a one-to-one correspondance between $\mathbb{Z}^{+}$ and positive even integers, which I will now denote $\mathbb{Z}_e$ so we should attempt to find a map which takes $\mathbb{Z}^{+} \rightarrow \mathbb{Z}_e$.
So how should we go about finding one? well first lets think what is the formal definition of even? I would say an integer $x$ is even if $x = 2k$ for some $k\in \mathbb{Z}$ so the set of positive even integers is $\mathbb{Z}_e = \{x = 2k : k\in\mathbb{Z}^{+}\}$.
Now once we have actually formalized what a positive even integer is it is not hard to think of a map, for example take:
$f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}_e$ defined by :
$k \mapsto 2k$
Now we've got a map we think we will work, and we just need to check if it is one-to-one.
Suppose $f(r) = f(s)$ Then $2r = 2s$, but this quickly implies that $r = s$ so the map is one-to-one, as desired.
Furthermore the map is also onto, because $\mathbb{Z}_e = \{x = 2k : k\in \mathbb{Z}^{+}\}$ is the set of integers of the form $2k$ by definition.
For infinite sets, we define equinumerous as there being a bijection. For finite sets, you cannot have both a bijection and a non-surjective injection, but as you have shown, that is possible for infinite sets. To prove there is a different number of elements for infinite sets, you have to show that there is no bijection, not just that there is a non-surjective injection (or non-injective sujection).
Best Answer
Here's an example. Consider the set $A=\{\,1,2,3,4,\dots\,\}$ of positive integers and the set $B=\{\,2,4,6,8,\dots\,\}$ of positive even integers. The function $f:A\to B$ given by $f(x)=2x$ is a one-to-one correspondence between these two infinite sets, because 1) it is a function from the one to the other, 2) no two different things in $A$ get mapped to the same thing in $B$, and 3) nothing in $B$ gets left out.
I don't know what you mean by "how would you solve them?" One solves equations, one doesn't solve correspondences.