Even N and (even + odd) N have the same size because there is a 1 to 1 correspondence between the sets. For every even number x in N I can pair it with x/2 which is also in N and that gives me all of N so N must be at least as large as even N. Likewise for x in N I can pair x with 2x and that gives me all the even N, so even N must be at least as large as N. So N and even N must have the same size.
However, while I can pair every rational number up with a real number, if I try to go the other way I find there is no rational number to pair with some (actually infinitely many) real numbers. I picked one, e, to show this. So the real numbers must be strictly larger than the rationals. Because the rationals are a strict subset of the reals it’s simpler than the Even/odd natural numbers because you don’t need the correspondence- every x in Q is in R but there exists at least one x in R not in Q, and no amount of correspondence shenanigans can get around that so |R|>|Q|.
However, while I can pair every rational number up with a real number, if I try to go the other way I find there is no rational number to pair with some (actually infinitely many) real numbers. I picked one, e, to show this. So the real numbers must be strictly larger than the rationals. Because the rationals are a strict subset of the reals it’s simpler than the Even/odd natural numbers because you don’t need the correspondence- every x in Q is in R but there exists at least one x in R not in Q, and no amount of correspondence shenanigans can get around that so |R|>|Q|.