A Path through the Lattice Points in a Quadrant
Let be the set of positive integers. The set of lattice points in the first quadrant is the set , where both coordinates are positive integers. Even though is two-dimensional, it is possible to set up a one-to-one correspondence between and , as shown in the picture.[more]
By associating with the lattice point , the path through the lattice points gives an enumeration of the positive unreduced rational numbers. Skipping past the fractions that have a common factor gives a listing of the positive rational numbers. The matching shows that there are as many positive fractions as positive integers.
In spite of that, there are differences between the integers and the rationals; for example, between any two rationals there is another rational.[less]
A set is enumerable (or denumerable or countable or listable) if it can be written as . In other words, there is a one-to-one correspondence between the set and the positive integers.
When there is a one-to-one correspondence between two sets and , the sets are said to be equipotent (or equinumerous or equipollent or of the same cardinality), which is denoted by . The Demonstration shows that .
Let be the set of integers. The mapping that matches with the term of the sequence is a one-to-one correspondence, so .
Use square paper to draw pictures showing that and . In other words, for draw a path that passes through each lattice point above the axis exactly once. The path does not have to be connected. Similarly, for , find a path that passes through every lattice point in the plane exactly once.
It is not true that every infinite set is countable. For example, neither the real numbers nor the complex numbers are countable (but ).