[puffoflogic]

Not only is the answer "no" like the sibling comment says, but in fact one definition of an infinite set is that it can be put into one-to-one correspondence with a strict subset of itself. In other words, infinite sets are precisely those for which your concept of size doesn't work.

Another sibling comment used the even/odd example, but that's not necessary to dispel this particular misconception. Consider the set of non-negative integers and the set of positive integers. That is, {0,1,2,3,...} and {1,2,3,4,...}. The latter is a strict subset of the former. Maybe I have just done mathematics for too long, but to me these are intuitively, "obviously" the same size. What would it even mean for one of them to be smaller? Which one is the same size as {-1,-2,-3,...}, if either of them? Even doing folk mathematics, if the size of the first is "infinity" then the size of the second is "infinity minus one which is still infinity".