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.
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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![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,
E
[a] <q [b] ⇒ there is c E M \ NM such that a + c = b.
Show that <Q 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%2Fb3c324b8-221c-4254-804f-e49057f61437%2F7rj0k86_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,
E
[a] <q [b] ⇒ there is c E M \ NM such that a + c = b.
Show that <Q 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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
