1. Prove the following propositions:(a) For all real numbers x and y, if x is rational and y is irrational,then x+y is irrational.(b) For all nonzero real numbers x and y, if x is rational and y isirrational, then x/y is irrational.2. If n is a positive integer, then n4 - n is divisible by 4. [Proof ofExhaustion]3. There are no positive integers a and b such that 18a + 6b = 1.(a) Write this statement as: if P then 18a + 6b/ 1.(b) What are the first two sentences of your contradiction proofof this claim?(c) Find a contradiction.(d) Conclude the proof.

According to Bartleby Guidelines we need to answer only one question and its sub questions so i have…

1. Prove the following propositions: (a) For all real numbers x and y, if x is rational and y is irrational, then x+y is irrational. (b) For all nonzero real numbers x and y, if x is rational and y is irrational, then x/y is irrational.

1. Prove the following propositions: (a) For all real numbers x and y, if x is rational and y is irrational, then x+y is irrational. (b) For all nonzero real numbers x and y, if x is rational and y is irrational, then x/y is irrational.

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter18: Deterministic Dynamic Programming

Section: Chapter Questions

Problem 10RP

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Question

1. Prove the following propositions:
(a) For all real numbers x and y, if x is rational and y is irrational,
then x+y is irrational.
(b) For all nonzero real numbers x and y, if x is rational and y is
irrational, then x/y is irrational.
2. If n is a positive integer, then n4 - n is divisible by 4. [Proof of
Exhaustion]
3. There are no positive integers a and b such that 18a + 6b = 1.
(a) Write this statement as: if P then 18a + 6b/ 1.
(b) What are the first two sentences of your contradiction proof
of this claim?
(c) Find a contradiction.
(d) Conclude the proof.

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