Hi, I've been reading about transfinite cardinal numbers and was wondering if you could answer this question. Supposedly the set of integers has the same cardinality as the set of even integers (both are countably infinite) since there exists a bijection between the two sets. But at the same time, doesn't there also exist a function between the set of even integers and the set of integers that is injective while NOT bijective (g(x) = x), since the image of f does not compose all of the integers (only the even ones)?
To clarify, let f and g be functions from the set of EVEN integers to the set of ALL integers. Let f(x) = 1/2 x, and g(x) = x. Both are injective functions, but f is onto while g is not. So f is a bijection, while g is merely 1-to-1. Why, then, can I not say that the set of even integers and the set of all integers do NOT have the same cardinality since there exists some 1-1 function that is not onto? It seems like I should be able to draw this intuitive conclusion since g is 1-1, so for every element in the first set there exists a corresponding element in the second, but since g is not onto there are more elements in the second than in the first? Thanks!