Show that the order (Q,
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Please complete the part c of this question.
![Let M be a nonstandard model of PA.
(a) For all a, b = M, define
a~ b ⇒ |a − b| € NM,
where z = |a − b] is the unique element in M such that x + z = y or y + z = x. Show
that is an equivalence relation on M and that it is NOT definable in M.
(b) Let Q = M/~ denote the quotient by this equivalence relation, i.e. Q := {[a] : a € M},
where [a] denotes the equivalence class of a. Define the relation <q on Q as follows:
for all [a],[b] = Q,
[a]<q [b] ⇒ there is c E M \ NM such that a+c = b.
Show that < is well-defined (does not depend on the representatives a, b) and is a
strict linear order on Q.
(c) Show that the order (Q, <q) has a least element but no greatest element, and it is a
dense (in itself), i.e. u <q v ⇒ ]w(u <q w <q v) for all u, v ¤ Q.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b945f66-390e-417d-9fc3-363b543734f1%2F59309a58-e2a3-4f42-9bc4-06a504dc263a%2F886x4q_processed.png&w=3840&q=75)
Transcribed Image Text:Let M be a nonstandard model of PA.
(a) For all a, b = M, define
a~ b ⇒ |a − b| € NM,
where z = |a − b] is the unique element in M such that x + z = y or y + z = x. Show
that is an equivalence relation on M and that it is NOT definable in M.
(b) Let Q = M/~ denote the quotient by this equivalence relation, i.e. Q := {[a] : a € M},
where [a] denotes the equivalence class of a. Define the relation <q on Q as follows:
for all [a],[b] = Q,
[a]<q [b] ⇒ there is c E M \ NM such that a+c = b.
Show that < is well-defined (does not depend on the representatives a, b) and is a
strict linear order on Q.
(c) Show that the order (Q, <q) has a least element but no greatest element, and it is a
dense (in itself), i.e. u <q v ⇒ ]w(u <q w <q v) for all u, v ¤ Q.
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