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.

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

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.