# infinite equivalent sets query

I want to know what is the criteria for two sets (finite and infinite to be equivalent) .Does the same criteria hold for both type of sets? I read that two sets are equivalent if their no. Of elements( cardinality) is same. Its alright for finite sets. But again in case of infinite sets as in this question cardinality is infinite. Here I read again that one to one correspondence should be there. But in the question above for an infinite set if a subset is proper first of all its cardinality with its superset will never be equal .secondly for one - one correspondence how come a proper subset have one to one correspondence. It goes on till infinity. How can we be sure that there are no term which do not have one to one corresponce. If someone can please make me understand in simple language and convince me I ll be greatful.

The set of positive integers $\mathbb N^+$ does have a one-to-one correspondence with a proper subset.
For example, the set of squares $S=\{1,4,9,\ldots\}$. Just pair $n$ with $n^2$. Since every integer has a unique square, and every square has a unique positive square root, nothing is missed or duplicated.