What is a bijection?

A bijection or bijective mapping from one set A to another B is one that is both injective and surjective. As a result, it sets up a correspondence in which each element of A can be paired with exactly one element of B and vice versa. In particular, we can define the inverse mapping from B to A that is also a bijection. In many ways, the two sets A, B can be regarded as the "same" (but beware of the examples below).

A mapping from A to B is injective or one-to-one if no two elements of A map to the same element of B. If A and B are finite sets this forces B to be at least as big as A. If A and B are finite sets of the same size and f:A->B is injective then f must also be surjective, and so bijective. Indeed, two finite sets have the same size if and only if there exists a bijection from either one to the other.

A mapping f:A->B is surjective or onto if every element of B is f of something in A. If this is not true, all we have to do is replace B by the image f(A), and then f defines a surjective mapping A->f(A). If A and B are finite sets and f:A->B is surjective then A must be at least as big as B.

Examples: Let A = B = {1,2,3,4}. A bijection from A to itself is called a permutation. There are 4!=24 different ones. One of them maps 1 to 2, 2 to 3, 3 to 4 and 4 to 1. It is called a "4-cycle".

Let A = B = R be the set of real numbers. Then f(x)=x³ is a bijection but g(x)=x² is neither injective nor surjective. The mapping exp that sends x to e^x is injective even though its image (0,oo) is in some sense smaller than its domain R=(-oo,oo).

If A = B = Z is the set of integers then the map d (for "doubling") that sends n to 2n is also injective. If E is the set of even integers (it is legitimate to call this set 2Z) then d actually defines a bijective mapping from Z to E even though E is a proper subset of Z.

Paradoxically, it is even possible to define a bijection from Z (or indeed the set N of positive integers) to the set Q of all rational numbers. Consequently these two infinite sets, Z and Q, have the same "size" (the correct word is cardinality). But in this case, there is no way in which the bijection can respect distances as we are requiring in our treatment of Euclidean geometry.

Postulate B1 tells us that once we have fixed a point on a line as the "origin" then there are exactly two points of a given distance along the line from the origin (one to each side, just as in R). In particular, there are points a distance π=3.14159... from O.

Postulate B3 tells us that once we have fixed a half-line r (for "ray") then there are exactly two half-lines making an angle of π/2 from it. Moreover, Postulate B4 tells us that these two rays are actually part of the same line, which is said to be perpendicular to the line of which r is half.