Q2 plz 1. List the axioms of addition and multiplication for integers.2. Using the axioms of the previous problem, write proofs forthe folloiwng statements:(i) For all integers x,y, (x+y)(x-y) = x^2 - y^2(ii) For all integers x, y, (x+y)(x^2 - xy + y^2) = x^3 + y^3.

Answered: 1. List the axioms of addition and…

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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Question

Q2 plz

1. List the axioms of addition and multiplication for integers.
2. Using the axioms of the previous problem, write proofs for
the folloiwng statements:
(i) For all integers x,y, (x+y)(x-y) = x^2 - y^2
(ii) For all integers x, y, (x+y)(x^2 - xy + y^2) = x^3 + y^3.

Expert Solution

Step 1

What is Integer:

In mathematics, an integer is a grouping of positive and negative numbers. Like whole numbers, integers do not include the fractional component. Therefore, numbers that can be positive, negative, or zero but are not fractions are said to be integers. All mathematical operations, such as addition, subtraction, multiplication, and division, can be performed on integers. "Z" stands for an integer, and examples are 1, 2, 5, 8, -9, -12, etc.

To Prove:

We prove that for all integers $x and y$ ,