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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.

                          Insects

                             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 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.