Show that the axiom A12' implies axiom A12, that is, if A12' is true and {A, B} be a Dedikind's cut then there is a unique real number c such that (a) if a < c then a E A, and (b) if b >c then be B.
Show that the axiom A12' implies axiom A12, that is, if A12' is true and {A, B} be a Dedikind's cut then there is a unique real number c such that (a) if a < c then a E A, and (b) if b >c then be B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3.
Show that the axiom A12' implies axiom A12, that is, if A12' is true and {A, B} be
a Dedikind's cut then there is a unique real number c such that (a) if a < c then a E A, and (b)
if b >c then be B.
![1.6 Completeness Axiom
The completeness axiom distinguishes the real numbers from any other ordered field. There are
a number of versions of this axioms. In this lecture note, we will only discuss two of them.
Dedekind Completeness Axiom
A12: Suppose A and B are two (non-empty) sets of real numbers with the properties:
(1) if a E A and b E B then a < b, and
(2) every real number is in cither A or B (in symbols, AUB= R),
then there is a unique real number c such that
(a) if a <c then a E A, and
(b) if b> e then be B.
Note that every number less than c belongs to A and every number greater than c belongs to
B. Morcover, cither c E A or cE B by (2). Hence, if c E A then A = (-00, c] and B = (c, +o0).
The pair {A, B} is called a Dedekind's cut.
Least Upper Bound Completeness Axiom
A12': Suppose S is a nonempty set of real numbers which is bounded above. Then S
has a least upper bound in R.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10896b01-e502-4220-abe7-000f56152cae%2Fb6a54258-d6e5-47ab-aae4-94ba41822502%2Fomuufil_processed.png&w=3840&q=75)
Transcribed Image Text:1.6 Completeness Axiom
The completeness axiom distinguishes the real numbers from any other ordered field. There are
a number of versions of this axioms. In this lecture note, we will only discuss two of them.
Dedekind Completeness Axiom
A12: Suppose A and B are two (non-empty) sets of real numbers with the properties:
(1) if a E A and b E B then a < b, and
(2) every real number is in cither A or B (in symbols, AUB= R),
then there is a unique real number c such that
(a) if a <c then a E A, and
(b) if b> e then be B.
Note that every number less than c belongs to A and every number greater than c belongs to
B. Morcover, cither c E A or cE B by (2). Hence, if c E A then A = (-00, c] and B = (c, +o0).
The pair {A, B} is called a Dedekind's cut.
Least Upper Bound Completeness Axiom
A12': Suppose S is a nonempty set of real numbers which is bounded above. Then S
has a least upper bound in R.
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