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Show that the order (Q,

Show that the order (Q,

Database System Concepts
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.
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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