MATHS QUESTIONS

Simple Maths Questions

Simple Maths Questions

Question 1

Bijective

In mathematics, a bijection (or bijective function or one-to-one correspondence) is a function giving an exact pairing of the elements of two sets. Every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In formal mathematical terms, a bijective function f: X ? Y is a one to one and onto mapping of a set X to a set Y.

A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets the picture is more complex, leading to the concept of cardinal number, a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a permutation. Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map.

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:

Every student was in a seat (there was no one standing),

No student was in more than one seat,

Every seat had someone sitting there (there were no empty seats), and

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, ...