Math, asked by Shubhendu8898, 1 year ago

What is Dekekind's Property. How is it equivalent to completeness axiom in R

Answers

Answered by HariesRam
3

Answer:

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

Answered by ʙʀᴀɪɴʟʏᴡɪᴛᴄh
6

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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