﻿ Sets and 1-1 Correspondence

Readings for Session 1 – (Continued)

Sets and One-to-One Correspondence

Set
:  In mathematics, we call collections of objects sets.

In the next session, a more carefully formed definition will be given for a collection of objects to be a set. Also, the proper notation used for sets will be given. This proper notation will be used in this course and is used in other mathematics courses. Though, for this introduction, this informal definition is good enough.

Example: The collection of insects on the previous page is a set of insects.

Example: The collection of numerals that represents the first twelve counting numbers is the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}.

One-to-One Correspondence and Equivalence of Sets:  If the elements of two sets can be paired so that each element is paired with exactly one element from the other set, then there is a one-to-one correspondence between the two sets. The two sets are said to be equivalent.

Notation:  The sets A and B are equivalent and is denoted as A ~ B.

Example: From the earlier example, 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.  Example:  Show set A = {a, b, c} and set B = {C, D, H} are equivalent, i.e., show A ~ B.
 a    b    c  |    |    |  C    D   H 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.

Note:  Here equal and equivalent mean two different things. Equal sets are equivalent, but equivalent sets may not be equal. This was illustrated in the above example where A ~ B, but AB. 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 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. 