Bijection - Examples

Examples

As a concrete example of a bijection, consider the batting line-up of a baseball team (or any list of all the players of any sports team). The set X will be the nine players on the team and the set Y will be the nine positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list.

In a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:

1. Every student was in a seat (there was no one standing),
2. No student was in more than one seat,
3. Every seat had someone sitting there (there were no empty seats), and
4. No seat had more than one student in it.

The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.

As another example, consider the relationship between the set of all adults in the U.S. and the set of all social security numbers (SSNs) in current use. (For non-Americans, think of any sort of government-assigned identification number, e.g. the national identification numbers of many countries.) Ideally there should exist a bijection, i.e. one-to-one mapping, between the two: Every adult has an SSN, and every SSN should correspond to exactly one adult. In such a case, the SSN can be used as a unique identifier of a given individual. The four properties would mean:

1. Every person has a social security number. (Not true in practice: Some people who have never paid taxes don't have them.)
2. No person has two or more social security numbers. (Also not true. Some people have multiple SSNs, e.g. due to errors in the database or deliberately, in order to commit fraud.)
3. Every social security number corresponds to at least one person. (True by assumption, since we are considering only SSNs in actual use.)
4. No social security number corresponds to multiple people. (Again, not true. Some SSNs do correspond to multiple people, again either due to database errors or for the purposes of committing fraud.)