Bijective

A mapping is bijective if it is a Bijection.

A Bijection is a mapping between two sets of equal numbers of elements such that each element of one set maps to a unique element in the other set. Because there are equal numbers of elements and every mapping is to a unique element, every element in either set is a part of a mapping.

Appendix A

A relation might be expressed, as shown below. Let’s have a function we call “Doubler” and two sets A={1,2,3} and B={2,4,6}.

• 1 R 2
• 2 R 4
• 3 R 6

Appendix B

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Mapping

The word mapping might seem a bit confusing. Mapping is more general than Function, just like Animal is more general than Mammal. To make this clear, we need to provide an example of something that qualifies as a mapping but it fails to be a Function. Let’s go to a restaurant.

The restaurant ticket maps each guest at a table to several selections from the menu.

• f R person=Coke
• f R person=Cheeseburger
• f R person=Large fries

It is legal at a restaurant to order more than one thing, and they hope you do… But the moment that person ordered the second item, the relation was no longer a function because that person no longer mapped to a unique element in the set of things sold by the restaurant.

Appendix C

We also have a way of making a restaurant story into a Bijection.

Three people go to a restaurant and on the counter are three plates: Beef, Chicken, Fish.

The first person has two choices, the second person has two choices and the third person has to take what’s left. At the end three people map to three dishes, or alternatively, three dishes map to three people, and all mappings are unique.