A Dedekind cut in is a pair of subsets of satisfying the following:
- If and , then
- contains no largest element.
A real number is a Dedekind cut in and the set of real numbers is denoted .
Note that the cut is ordered and the elements of (as in Lower) are all smaller than the elements of (as in Upper). In the above definition, for a cut we have . Given any rational number an example of a cut is one of the form
which is called a rational cut. This gives an interpretation of rational numbers as cuts and therefore every rational number is also a real number. The real number zero is defined as the rational cut
Note that in a rational cut , the set contains a smallest element, namely . A real number is irrational if the set contains no smallest element. An example of a cut that defines an irrational number is
We now use cuts to define arithmetic operations and order relations.
We first use cuts to define the familiar arithmetic operations for real numbers, including addition, subtraction, multiplication, and division. For cuts and , define addition as follows:
Observe that cut addition is well-defined and satisfies (commutativity), and (associativity). The additive inverse of is with
Then subtraction is defined by and for any cut , we have The absolute value is defined by
and multiplication is defined by
Then addition and multiplication satisfy the properties (commutativity), (associativity), and (distributivity).
For a real number with , the multiplicative inverse is defined by , where
For , .
For rational cuts, all of the above arithmetic operations are consistent with arithmetic operations over the rationals.
Given real numbers and is less than or equal to denoted if The inequality is strict if
This ordering on the real numbers satisfies the following properties:
- and implies
- Exactly one of or holds.
- implies .
All of the above show that is an ordered field. Note that according to our definitions, is not a real number because is not a cut. Furthermore, there are infinitely many elements in the set , which is the set of irrational numbers.
is an upper bound for a set if each satisfies . If is an upper bound for and every upper bound satisfies , then is the least upper bound for . Note that if a least upper bound exists, then it is unique.
Let be a Dedekind cut. Show that given any rational number , there exist rational numbers and such that .
By definition, and are nonempty, so there exist rational numbers and . Consider the sequence of rational numbers defined by
Note that for and . Then by choosing such that the rational numbers satisfy the conditions and .
Show that for any nonempty subset of with an upper bound, there exists a least upper bound for .
A nonempty subset of is a set of cuts . Let
Then is a cut and is an upper bound since implies for all . Since an upper bound for exists, let be any upper bound. Then by definition of an upper bound, contains implying . Therefore, is the least upper bound for .
Suppose and are real numbers. Prove the following statements:
- If for every then
- If for every then
By the ordering principle, we have either or . If , then let be any number in the interval . This gives a contradiction. Therefore, .
If , then let be any number in the interval . This gives a contradiction. Therefore,