For each of the statement below, decide whether it is true or false, and prove your claim. (a) For all a, b ∈ R, if a ∉ Q \ {0} and b ∈ Q then ∉ Q. (b) For all x, y ∈ R, if x is irrational and y is positive and irrational, then x + √y is irrational.

For each of the statement below, decide whether it is true or false, and prove your claim. (a) For all a, b ∈ R, if a ∉ Q \ {0} and b ∈ Q then ∉ Q. (b) For all x, y ∈ R, if x is irrational and y is positive and irrational, then x + √y is irrational.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter7: Real And Complex Numbers

Section7.1: The Field Of Real Numbers

Problem 20E: Give counterexamples for the following statements. If and are irrational, then is irrational. If ...

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Question

For each of the statement below, decide whether it is true or false, and prove your claim.

(a) For all a, b ∈ R, if a ∉ Q \ {0} and b ∈ Q then ∉ Q.

(b) For all x, y ∈ R, if x is irrational and y is positive and irrational, then x + √y is irrational.

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