Let $X$ be a set then $X$ is infinite if and only if there is a $1-1$
map $X\to X$ which is not onto.
I don't know how to prove this? I read that,
A set $X$ is infinite if and only if it may be put into one-one
correspondence with a proper subset of itself
but I got a bit confused because when they say it can put into a bijection of its proper subset does that mean the proper subset is also infinite? So the map is a bijection of an infinite set to am infinite set?
Best Answer
I assume you mean "$X$ is infinite if and only if there is a one-to-one map $X\to X$ which is not onto" (since otherwise every set is infinite, as the inclusion map $\Bbb N\to\Bbb R$ is an example of a map that is one-to-one but not onto). This is almost immediately equivalent to the characterization of an infinite set as one that may be put into one-to-one correspondence with a proper subset of itself. (Try writing out the definitions of one-to-one map, one-to-one correspondence, and proper subset to see how.)
In answer to your other questions, yes, the proper subset will be infinite, and that map will be a bijection of an infinite set to an infinite set. Neither of these is particularly relevant to the proof, though.