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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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
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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