We have shown a close relationship between the concept of one-to-one correspondence and the idea of the number of elements in a set, called the cardinality of a set. (See the counting of the insects above.) This exploration has led us to the following definitions relating the sets of natural and whole numbers to sets. Further, we note that this relationship is closely related to how small children learn to count.

Sets of Numbers:  The set of natural numbers (or counting numbers) is the set N = {1, 2, 3, …}.
                             The set of whole numbers is the set W = {0, 1, 2, 3, …}.

Cardinal Number of a Set:  The number of elements in a set is the cardinal number of that set.

       Notation:  If a set A is equivalent to the set {1, 2, 3, …, N}, we write n(A) = N and say “The cardinal number of set A is N.”
                        Also, n(Ø) = 0. The cardinal number for an empty set is zero.

       Example:  When we counted the insects in the above example, we have shown a one-to-one correspondence between the set {1, 2, 3, …, 12} and the set of insects, i.e., we showed the set of insects and the set {1, 2, 3, ..., 12} are equivalent. We showed the two sets are equivalent. This means that after we counted the insects and said there were twelve insects, we were saying that the cardinal number for the set of insects is 12.

       More examples for the cardinal numbers for sets will be given in the next session.

       You probably learned the cardinal number zero, 0, much later in life, well after you learned how to count. This is also true in the history of humans. The cardinal number zero was invented much later than any of the natural numbers.