What is Dekekind's Property. How is it equivalent to completeness axiom in R
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Theorem: If A and B are two non-empty subsets of R such that (i) A∪B=R, (ii) x∈A∧y∈B⇒x<y, then either A has the greatest member or B has the least member.
Proof: By (ii), the non-empty set A is bounded above. If A has the greatest member, it establishes the theorem. If A has no greatest member, then by the completeness axiom, in R, B being the set of upper bounds of A, it has a least member. Hence the theorem is proved
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Dekekind's Property.
If A and B are two non-empty subsets of R such that (i) A∪B=R, (ii) x∈A∧y∈B⇒x<y, then either A has the greatest member or B has the least member.
Dedekind's Property is equivalent to the completeness axiom in R.
How is it equivalent to completeness axiom in R
Intuitively, completeness implies that there are not any “gaps” or “missing points” in the real number line. This contrasts with the rational numbers, whose corresponding number line has a “gap” at each irrational value.
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