Prove the following statements using direct and indirect proof methods. a.) For all integers a, b, c, the product (a−b)(a−c)(b−c) is even. b.) The reciprocal of any irrational number is irrational.
Prove the following statements using direct and indirect proof methods. a.) For all integers a, b, c, the product (a−b)(a−c)(b−c) is even. b.) The reciprocal of any irrational number 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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Prove the following statements using direct and indirect proof methods.
a.) For all integers a, b, c, the product (a−b)(a−c)(b−c) is even.
b.) The reciprocal of any irrational number is irrational.
Expert Solution
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Step 1
a.) For all integers a, b, c, the product (a−b)(a−c)(b−c) is even.
The differences of any two even numbers is even and difference of any two odd number is even
Also product of two or three numbers , in which one number is even , is even .
now, here make cases that
i) a,b,c are even or odd
ii) any two of a,b,c are odd or even
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