Final answer:
The Cut Property states that if we have two nonempty and disjoint sets A and B, where their union is the set of real numbers, and all elements in A are less than all elements in B, then there exists a real number c such that x is less than or equal to c for all x in A, and x is greater than or equal to c for all x in B. (a) We can prove this using the Axiom of Completeness by assuming that there is no real number c that satisfies the property, and showing that this leads to a contradiction. (b) Assuming R possesses the Cut Property, we can prove the existence of sup E for a nonempty set E that is bounded above. (c) The Cut Property distinguishes the real numbers from the rational numbers, as it is not valid when R is replaced by Q.
Step-by-step explanation:
The Cut Property of the real numbers states that if we have two nonempty and disjoint sets A and B, where their union is the set of real numbers, and all elements in A are less than all elements in B, then there exists a real number c such that x is less than or equal to c for all x in A, and x is greater than or equal to c for all x in B.
(a) To prove this using the Axiom of Completeness, we can assume that A and B are nonempty, disjoint sets such that A∪B = R and a < b for all a ∈ A and b ∈ B. Let's suppose that there is no real number c that satisfies the property. Then either there exists an element x in A such that x > c for all c in R, or there exists an element y in B such that y < c for all c in R. In both cases, this would contradict the fact that A and B are nonempty, disjoint sets, as there would be an element in one set that is greater than an element in the other set. Therefore, the Cut Property holds.
(b) To show the other way around, assume that R possesses the Cut Property and let E be a nonempty set that is bounded above. We want to prove that sup E exists. According to the Cut Property, if A is the set of all elements in E that are less than or equal to a certain value x, and B is the set of all elements in E that are greater than x, then there exists a real number c such that x ≤ c whenever x ∈ A and x ≥ c whenever x ∈ B. This means that c is an upper bound for A and a lower bound for B. Since A and B are subsets of E, c is an upper bound for E as well. Now, let D be the set of all upper bounds for E. By the Axiom of Completeness, D has a least upper bound, which we can denote as sup E. Therefore, sup E exists.
(c) The Cut Property can be used as a fundamental axiom that distinguishes the real numbers from the rational numbers. For example, let's consider Q, the set of rational numbers. If we take A to be the set of all rational numbers less than 2 and B to be the set of all rational numbers greater than or equal to 2, it can be shown that there is no rational number c that satisfies the Cut Property. This is because between any two rational numbers, there is always another rational number. Therefore, the Cut Property is not a valid statement when R is replaced by Q.