Readings for Lecture/Lab 1 – Sets and Whole Number

 

Sets and Natural & Whole Numbers

 

      The first number set we learned as children was the set of natural or counting numbers. This is also true in the history of humans; early humans first counted the objects around them.

 Example. How many insects are illustrated below?

 

 A small child would point at each insect and count them.

 

 

         The child would say there are 12 insects since the child has set up a 1-1 correspondence with the set {1, 2, 3, …, 12} and the set of insects. Now we define these connections between sets and counting numbers.

Definition. If each element of a set A can be paired with exactly one element of a set B and if each element of B can be paired with exactly one element of A, then there is a one-to-one correspondence between A and B. The sets A and B are said to be equivalent, denoted as A ~ B.

       Example.
            Show set A = {a, b, c} and set B = {●,♦, ♥} are equivalent, i.e., show A ~ B.

a   b   c  |    |    | ●   ♦   ♥

The two sets are equivalent since a one-to-one correspondence can be made between the two sets. Note that A ~ B, but A ≠ B.

 We illustrate that there are other 1-to-1 correspondences that could have been made to show the sets are equivalent.

a   b   c  |    |    | ●  ♥  ♦

a   b   c  |    |    | ♦  ●   ♥

a   b   c  |    |    | ♦   ♥   ●

How many other distinct one-to-one correspondences could be made where a, b, c are kept in the same order? What are they? That is, how many different one-to-one correspondences could be made?

Important Note.  Equal sets are equivalent, but equivalent sets may not be equal. This was illustrated in the above example where A ~ B, but A ≠ B. Two sets are equal when they have exactly the same elements, and sets are equivalent when a one-to-one correspondence can be set up between the two sets.  

      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 finite sets. Further, we note that this relationship is closely related to how small children learn to count. 

Definitions. The set of natural numbers (or counting numbers) is the set {1, 2, 3, …}.
                       The set of whole numbers is the set {0, 1, 2, 3, …}.

 Definition. Let A be a finite set. If set A is equivalent to the subset {1, 2, 3, …, N} of the natural numbers, we say that the cardinal number for set A is N.  Notation:  n(A) = N. The cardinal number for the empty set, Ø, is 0, i.e., n(Ø ) = 0.  

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

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

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

 

 

 

      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 the natural numbers.

 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 natural 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 אo  read as aleph-nought. Aleph is a letter in the Hebrew alphabet.   

 

    

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