Question

Using complete sentences, explain what an infinite set is. In your answer give at least two examples of infinite sets.

Answer 1

Here, 'infinite' will mean 'Dedekind-infinite'. A set S which is infinite has at least one proper subset P (P only contains elements of S, but not all elements of S) such that there is a bijective mapping between S and P, i.e. a one-to-one pairing off of elements of S and P that associates exactly one element of S to each element of P, and vice versa.
This means that S, which contains all elements of P and some elements which are not found in P, has the same "number" of elements as P.
For example, the set of non-negative integers {0, 1, 2, 3, ...} includes all elements of the set of all positive integers {1, 2, 3, 4, ...}, and an additional element, namely 0, not found in {1, 2, 3, 4, ...}. You might therefore think that {0, 1, 2, 3, ...} contains a greater number of elements than {1, 2, 3, 4, ...}, but this is not the case because you can completely pair off the elements of the two sets: (0, 1), (1, 2), (2, 3), ... . Therefore, {0, 1, 2, 3, ...} is infinite.
Other examples of infinite sets are the rationals Q, consisting of all fractions with integer numerators and denominators<>0, and the reals IR.
On the other hand, {1, 2} is finite because there is no bijective mapping between it and any of its proper subsets {1}, {2} and {}.