Please complete the part c of this question. Let M be a nonstandard model of PA.(a) For all a, b = M, definea~ b ⇒ |a − b| € NM,where z = |a − b] is the unique element in M such that x + z = y or y + z = x. Showthat 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

The question introduces a set equipped with a relation . The goal is to analyze the order structure…

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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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Concept explainers

Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.

Question

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.

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