Consider the set ℕ× ℕ. We will informally think of the pair (x, y) ∈ ℕ× ℕ as representing the integer x − y. Of course, there are many different ways of representing the same integer. To fix this, we define a relation ∼ on ℕ × ℕ by (x, y) ∼ (n, m) if and only if x + m = n + y. a) Prove that ∼ is an equivalence relation. b) We shall abbreviate ℕ× ℕ / ∼ by the symbol ℤ*. Prove that the function f : ℤ* → ℤ given by f([x, y]) = x − y is well-defined and bijective.

Consider the set ℕ× ℕ. We will informally think of the pair (x, y) ∈ ℕ× ℕ as representing the integer x − y. Of course, there are many different ways of representing the same integer. To fix this, we define a relation ∼ on ℕ × ℕ by (x, y) ∼ (n, m) if and only if x + m = n + y. a) Prove that ∼ is an equivalence relation. b) We shall abbreviate ℕ× ℕ / ∼ by the symbol ℤ. Prove that the function f : ℤ → ℤ given by f([x, y]) = x − y is well-defined and bijective.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 8E: In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R...

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