SOLUTION: Consider the set Z of all Integers and an integer m > 1. For all integers x and y  Z, if x – y is divisible by m, then show that this defines an equivalence relation on Z

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Consider the set Z of all Integers and an integer m > 1. For all integers x and y  Z, if x – y is divisible by m, then show that this defines an equivalence relation on Z      Log On


   

Question 622759: Consider the set Z of all Integers and an integer m > 1. For all integers x and y  Z, if x – y is divisible by m, then show that this
defines an equivalence relation on Z. An equivalence relation is reflective, symmetric, and transitive.
Answer by solver91311(24713) About Me  (Show Source):
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Let be the set of all integers and let , and accept the notation to mean that is divisible by .

Prove that the set defines an equivalence relation:

1. Since ,



is reflexive.

2. Let









Thus is symmetric.

3. Let and

and

and





Thus is transitive.

is reflexive, symmetric, and transitive is an equivalence relation.


John

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