Consider the structure R = (IR, +, x, <). We say that a set C CR is definable in R if there is a formula p(x) such that c E C if and only if y[c] is true in R. Show that the set C = {0, 1} is definable in R. (Hint: Consider the polynomial whose solution is exactly 0 or 1.)
Consider the structure R = (IR, +, x, <). We say that a set C CR is definable in R if there is a formula p(x) such that c E C if and only if y[c] is true in R. Show that the set C = {0, 1} is definable in R. (Hint: Consider the polynomial whose solution is exactly 0 or 1.)
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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![3. Consider the structure \( \mathbb{R} = (\mathbb{R}, +, \times, <) \). We say that a set \( C \subseteq \mathbb{R} \) is definable in \( \mathbb{R} \) if there is a formula \(\varphi(x)\) such that \( c \in C \) if and only if \(\varphi[c]\) is true in \( \mathbb{R} \). Show that the set \( C = \{0, 1\} \) is definable in \( \mathbb{R} \).
(Hint: Consider the polynomial whose solution is exactly 0 or 1.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1b54d61-1255-406a-9b79-99af3ebb60e9%2F56bab4db-d9d6-4439-91c6-3f6bdde60031%2Fygefjj7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Consider the structure \( \mathbb{R} = (\mathbb{R}, +, \times, <) \). We say that a set \( C \subseteq \mathbb{R} \) is definable in \( \mathbb{R} \) if there is a formula \(\varphi(x)\) such that \( c \in C \) if and only if \(\varphi[c]\) is true in \( \mathbb{R} \). Show that the set \( C = \{0, 1\} \) is definable in \( \mathbb{R} \).
(Hint: Consider the polynomial whose solution is exactly 0 or 1.)
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