>>101
Copypasta:
2.3 Definition If there exists a 1-1 mapping of A onto B, we say that A and B can be put in 1-1 correspondence, or that A and B have the same cardinal number, or, briefly, that A and B are equivalent, and we write A ~ B. This relation clearly has the following properties:
It is reflexive: A ~ A.
It is symmetric: If A ~ B, then B ~ A.
It is transitive: If A ~ B and B ~ C, then A ~ C.
Any relation with these three properties is called an quivalence relation.
2.4 Definition For any positive integer n, let J_n be the set whose elements are the integers 1, 2,..., n; let J be the set consisting of all positive integers. For any set A, we say:
(a) A is finite if A ~ J_n for some n (the empty set is also considered to be finite).
(b) A is infinite if A is not finite.
(c) A is countable if A ~ J.
(d) A is uncountable if A is neither finite nor countable.
(e) A is at most countable if A is finite or countable.
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copied straight from the first book I found, Rudin, Principles of Mathematical Analysis. Honestly, finite vs infinite ain't hard. Just wait till you get to toposes without NNOs. Then it gets hard.