Remark 0.1. Let (F,+,·,<) be an ordered field. Building on the previous exercise,we can define the set of integers ZF in the field F viaZp = NF U{0}U{-n : n e NF},where 0 denotes the additive identity in the field.Moreover we can define an equivalence relation on the set of pairs ZFx (ZF\{0})via (a, b)define the set of rational numbers QF in the field F as the set of all equivalence~ (c, d) if and only if a · d = b. c. Then, proceeding as in lecture, we canclasses:QF ={ : ; is the equivalence class of a pair (a, b) E Zp X (ZF\ {0})}.Continuing as in lecture, we can define the operations of addition and multiplication,as well as an order relation on QF with respect to which QF is an ordered field.Let F be an ordered field with the least upper bound property. Provethat there is a unique function o : Q→ QF that satisfies the following properties:Ф(р + 9) — Ф(р) + ф(q),$(p · q) = ¢(p) · ø(g),if p < q then o(p) < ¢(q)for any p, q E Q.Hint: When constructing ø, work your way up from N, to Z, and then to Q.

An ordered field F with least upper bound property To prove that exist a unique function s.t. (i)…

Let F be an ordered field with the least upper bound property. Prove that there is a unique function o : Q → QF that satisfies the following properties: o(p q) = 0(p) · ở(q), $(p +q) = ¢(p) + ¢(q), if p < q then ø(p) < ¢(q) %3D for any p, q E Q. Hint: When constructing o, work your way up from N, to Z, and then to Q.

Let F be an ordered field with the least upper bound property. Prove that there is a unique function o : Q → QF that satisfies the following properties: o(p q) = 0(p) · ở(q), $(p +q) = ¢(p) + ¢(q), if p < q then ø(p) < ¢(q) %3D for any p, q E Q. Hint: When constructing o, work your way up from N, to Z, and then to Q.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Remark 0.1. Let (F,+,·,<) be an ordered field. Building on the previous exercise,
we can define the set of integers ZF in the field F via
Zp = NF U{0}U{-n : n e NF},
where 0 denotes the additive identity in the field.
Moreover we can define an equivalence relation on the set of pairs ZFx (ZF\{0})
via (a, b)
define the set of rational numbers QF in the field F as the set of all equivalence
~ (c, d) if and only if a · d = b. c. Then, proceeding as in lecture, we can
classes:
QF =
{ : ; is the equivalence class of a pair (a, b) E Zp X (ZF\ {0})}.
Continuing as in lecture, we can define the operations of addition and multiplication,
as well as an order relation on QF with respect to which QF is an ordered field.
Let F be an ordered field with the least upper bound property. Prove
that there is a unique function o : Q→ QF that satisfies the following properties:
Ф(р + 9) — Ф(р) + ф(q),
$(p · q) = ¢(p) · ø(g),
if p < q then o(p) < ¢(q)
for any p, q E Q.
Hint: When constructing ø, work your way up from N, to Z, and then to Q.

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