The integers are more than a set; the operation of addition is also a crucial part of the concept of the integers. Define an operation ⊕ : ℤ × ℤ → ℤ by [(x, y)] ⊕ [(n, m)] = [(x + n, y + m)]. a) Prove that ⊕ is well-defined. b) Prove that ⊕ is associative and commutative. c) Prove that ⊕ is “the same operation as” +. In other words, for any [(x, y)], [(n, m)] ∈ ℤ, prove that f([(x, y)] ⊕ [(n, m)]) = f([(x, y)]) + f([(n, m)]).
The integers are more than a set; the operation of addition is also a crucial part of the concept of the integers. Define an operation ⊕ : ℤ × ℤ → ℤ by [(x, y)] ⊕ [(n, m)] = [(x + n, y + m)]. a) Prove that ⊕ is well-defined. b) Prove that ⊕ is associative and commutative. c) Prove that ⊕ is “the same operation as” +. In other words, for any [(x, y)], [(n, m)] ∈ ℤ, prove that f([(x, y)] ⊕ [(n, m)]) = f([(x, y)]) + f([(n, m)]).
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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The integers are more than a set; the operation of addition is also a crucial part of the concept of the integers. Define an operation ⊕ : ℤ × ℤ → ℤ by [(x, y)] ⊕ [(n, m)] = [(x + n, y + m)].
a) Prove that ⊕ is well-defined.
b) Prove that ⊕ is associative and commutative.
c) Prove that ⊕ is “the same operation as” +. In other words, for any [(x, y)], [(n, m)] ∈ ℤ, prove that f([(x, y)] ⊕ [(n, m)]) = f([(x, y)]) + f([(n, m)]).
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