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Let n be the structure (R; +,.). (The language is assumed to have equality and the parameters V, +, -_. N is the structure whose universe is the set R of real numbers and is such that + and.R are the usual addition and product operations.) a) Give a formula that defines in the interval [0, ∞). the set {2}. b) Give a formula that defines in c) Show that any finite union of intervals whose endpoints are algebraic is definable on n.

Let n be the structure (R; +,.). (The language is assumed to have equality and the parameters V, +, -_. N is the structure whose universe is the set R of real numbers and is such that + and.R are the usual addition and product operations.) a) Give a formula that defines in the interval [0, ∞). the set {2}. b) Give a formula that defines in c) Show that any finite union of intervals whose endpoints are algebraic is definable on n.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 23E
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Mathematical Logic First-order or predicate logic. Please be as clear as possible. Explain your answer. Thank you. R are Let I be the structure (R; +,.). (The language is assumed to have equality and the parameters V, +, . . N is the structure whose universe is the set R of real numbers and is such that +® and the usual addition and product operations.) a) Give a formula that defines in the interval [0, ∞). the set {2}. b) Give a formula that defines in c) Show that any finite union of intervals whose endpoints are algebraic is definable on n. (The converse is also true; these are the only definable sets in the structure. But we will not prove this fact.) Please be as clear as possible. Explain your answer. Thank you very much.
Transcribed Image Text:Mathematical Logic First-order or predicate logic. Please be as clear as possible. Explain your answer. Thank you. R are Let I be the structure (R; +,.). (The language is assumed to have equality and the parameters V, +, . . N is the structure whose universe is the set R of real numbers and is such that +® and the usual addition and product operations.) a) Give a formula that defines in the interval [0, ∞). the set {2}. b) Give a formula that defines in c) Show that any finite union of intervals whose endpoints are algebraic is definable on n. (The converse is also true; these are the only definable sets in the structure. But we will not prove this fact.) Please be as clear as possible. Explain your answer. Thank you very much.
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