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        In the previous session, we showed 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 in Session 1.) Here, we formalize these relationships between sets and whole numbers.

 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:  Let C = {#, $, %, &}. Show n(C) = 4.

              #     $    %    &
               |      |      |      |
              1     2    3     4

 

Hence, C is equivalent to {1, 2, 3, 4} and n(C) = 4 since a 1-1 correspondence can be setup between C and {1, 2, 3, 4}.

 

 

 

Example:  For M = {red, blue, green, yellow, orange}, n(M) = 5.
                  The symbol “n(M) = 5” is read, “The cardinal number of set M is equal to 5.”
                  Take the time to set up a 1-1 correspondence between M and {1, 2, 3, 4, 5}.

 

Example:  For T = {2, 4, 6, 8, 10, 12, 14, 16}, n(T) = 8.
                  On a sheet of paper, set up a 1-1 correspondence between T and {1, 2, 3, 4, 5, 6, 7, 8}.

Example:  In this picture, the circles represent sets A and B.  The dots inside are the elements of the sets. We need to make sure we look at an entire circle, even though the circles overlap.

                             Venn Diagram
                              n(A) = 5        n(B) = 8

Additional Notes:   We will not give precise definitions for the terms finite and infinite. We will consider a finite set to be a set that has a cardinal number that is a whole number and an infinite set as a set that is not finite. Think of a finite set as a set that has a limited number of elements and an infinite set as a set that has an unlimited number of elements.

Side Note. The cardinal number for any set equivalent to the set of all the natural numbers is ℵ0, read as aleph-nought. Aleph is a letter in the Hebrew alphabet.