Ordering the Whole Numbers

What do we mean by 3 < 5?   We illustrate with an example.

The three apples can only be set up with a 1-1 correspondence with three of the oranges. Two of the oranges cannot be paired with any apples. We say that the set of apples is equivalent to a subset of the set of oranges.

      The above example motivates the following definitions.

Definition. A set A is a subset of a set B, if every element of A is also an element of B. The set A is a proper subset of set B, if A is a subset of B and A is not equal to set B.

      Examples.

            1.   Let A = {1, 3, 5} and B = {1, 2, 3, 4, 5, 6}. Then A is a subset of B and A is a proper subset of B since each element of A is an element of B and 2 is an element of B but 2 is not an element of A.

            2.   Let C = {a, b, c} and D = {b, c, d, e}. Then C is not a subset of D. Note C is not a subset of D since D does not contain the element a of set C.

            3.   A is a subset of A and  Ø is a subset of A.  (A set is a subset of itself, and the empty set is a subset of every set.)

 Definition.  Let a = n(A) and b = n(B) be whole numbers for finite sets A and B. If A is equivalent to a proper subset of B, then a < b or b > a, these are read as a is less that b and b is greater than a.

      Example. Consider our earlier example with the apples and oranges. Since the set of apples is equivalent to a proper subset of the set of oranges, we have that 3 < 5.