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section12_1 - Section 12 The Completeness Axiom The...

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Unformatted text preview: Section 12 The Completeness Axiom The Completeness Axiom In the last section, we discussed 15 axioms that make R into an ordered field. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 17 Section 12 The Completeness Axiom The Completeness Axiom In the last section, we discussed 15 axioms that make R into an ordered field. Recall also that Q was an ordered field. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 17 Section 12 The Completeness Axiom The Completeness Axiom In the last section, we discussed 15 axioms that make R into an ordered field. Recall also that Q was an ordered field. In this section, we discuss the one final axiom that makes R complete; and so differentiates it from Q , since Q is not complete ( p p = 2 Q for all primes p ). MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 17 Section 12 The Completeness Axiom The Completeness Axiom In the last section, we discussed 15 axioms that make R into an ordered field. Recall also that Q was an ordered field. In this section, we discuss the one final axiom that makes R complete; and so differentiates it from Q , since Q is not complete ( p p = 2 Q for all primes p ). This axiom is appropriately called the Completeness Axiom MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 17 Section 12 The Completeness Axiom The Completeness Axiom In the last section, we discussed 15 axioms that make R into an ordered field. Recall also that Q was an ordered field. In this section, we discuss the one final axiom that makes R complete; and so differentiates it from Q , since Q is not complete ( p p = 2 Q for all primes p ). This axiom is appropriately called the Completeness Axiom It says Every nonempty subset of R that is bounded above has a least upper bound. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 17 Section 12 The Completeness Axiom The Completeness Axiom In the last section, we discussed 15 axioms that make R into an ordered field. Recall also that Q was an ordered field. In this section, we discuss the one final axiom that makes R complete; and so differentiates it from Q , since Q is not complete ( p p = 2 Q for all primes p ). This axiom is appropriately called the Completeness Axiom It says Every nonempty subset of R that is bounded above has a least upper bound. To understand this better, we now discuss what it means for a subset to be bounded above and what a least upper bound is. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 17 Section 12 The Completeness Axiom Upper and Lower Bounds Consider the set T = f 2 ; 4 ; 6 ; 8 g . MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 17 Section 12 The Completeness Axiom Upper and Lower Bounds Consider the set T = f 2 ; 4 ; 6 ; 8 g . We will use this set to explore the concept of upper bound....
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section12_1 - Section 12 The Completeness Axiom The...

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