Step 1/6
1. First, we need to define an order on R. We can do this by defining a relation ≤ on R as follows: For any two cuts A and B in R, we say that A ≤ B if and only if A is a subset of B. This relation is reflexive, antisymmetric, and transitive, so it is a partial order on R.
Step 2/6
2. Next, we need to define addition and multiplication operations on R. Let A and B be two cuts in R. We can define their sum, A + B, as the set of all elements a + b, where a ∈ A and b ∈ B. This sum is also a cut, so it belongs to R.
Step 3/6
3. Similarly, we can define the product of two cuts A and B, denoted by A × B, as the set of all elements a × b, where a ∈ A and b ∈ B. This product is also a cut, so it belongs to R.
Step 4/6
4. Now, we need to check if R, with the defined order and operations, satisfies all the axioms for a real number system. The axioms for a real number system are:
a. Associativity of addition and multiplication
b. Commutativity of addition and multiplication
c. Existence of identity elements for addition and multiplication
d. Existence of inverse elements for addition and multiplication
e. Distributive law of multiplication over addition
Step 5/6
5. We can prove each of these axioms for R using the definitions of addition and multiplication on cuts that we have provided. For example, to prove the associativity of addition, we can show that for any three cuts A, B, and C in R, (A + B) + C = A + (B + C). This can be done by showing that the elements in both sets are the same.
Answer
6. Once we have proved all the axioms for a real number system, we can conclude that R, with the defined order and operations, is a real number system. This means that the set of Dedekind cuts, R, can be used to represent the real numbers, and it satisfies all the properties that we expect from a real number system.
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