Consider an infinite set of undefined elements S and the undefined relation R that satisfies the following axioms: Axiom 1. If a, b e S and a Rb, then a #b. Axiom 2. If a, b, c e S, a Rb, and bRc, then a Rc. (i) Show that an interpretation with S as the set of integers and a Rb as "a is less than b" is a model for the system. (ii) Would S as the set of integers and a Rb interpreted as "a is greater than b" also be a model? (iii) Are the models in parts (a) and (b) isomorphic? (iv) Would S as the set of real numbers and a Rb interpreted as "a is less than b" be another model? (v) Is the model in part (d) isomorphic to the model in part (a)?
Consider an infinite set of undefined elements S and the undefined relation R that satisfies the following axioms: Axiom 1. If a, b e S and a Rb, then a #b. Axiom 2. If a, b, c e S, a Rb, and bRc, then a Rc. (i) Show that an interpretation with S as the set of integers and a Rb as "a is less than b" is a model for the system. (ii) Would S as the set of integers and a Rb interpreted as "a is greater than b" also be a model? (iii) Are the models in parts (a) and (b) isomorphic? (iv) Would S as the set of real numbers and a Rb interpreted as "a is less than b" be another model? (v) Is the model in part (d) isomorphic to the model in part (a)?
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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Transcribed Image Text:Consider an infinite set of undefined elements S and the undefined
relation R that satisfies the following axioms:
Axiom 1. If a, b e S and a Rb, then a #b.
Axiom 2. If a, b, c e S, a Rb, and bRc, then a Rc.
(i) Show that an interpretation with S as the set of integers and
a Rb as "a is less than b" is a model for the system.
(ii)
Would S as the set of integers and a Rb interpreted as "a is
greater than b" also be a model?
(iii) Are the models in parts (a) and (b) isomorphic?
(iv) Would S as the set of real numbers and a Rb interpreted as
"a is less than b" be another model?
(v) Is the model in part (d) isomorphic to the model in part (a)?
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