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.Com
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) (Show Source): You can put this solution on YOUR website!
Let 
be the set of all integers and let 
, and accept the notation )
to mean that 
is divisible by 
.
Prove that the set \,|\,x,\,y\,\in\,\mathbb{Z},\,x\ \equiv\ y\,(\text{mod}\,m)\})
defines an equivalence relation:
1. Since 
, )
\ \in\ R)
is reflexive.
2. Let \ \in\ R\ \Rightarrow\ x\ \equiv\ y\,(\text{mod}\,m))
)
)
\ \in\ R)
Thus \ \in\ R\ \Rightarrow\ (y,x)\ \in\ R\ \Rightarrow R\ )
is symmetric.
3. Let \ \in\ R)
and \ \in\ R)
\ \ )
and )
and
\ +\ (y\ -\ z)\ =\ m(q\ +\ q'))
\ =\ m(q\ +\ q'),\ q\ +\ q'\ \in\ \mathbb{Z})
Thus \ \in\ R\ \text{and}\ (y,z)\ in\ R\ \Rightarrow\ (x,z)\ \in\ R\ \Rightarrow R\ )
is transitive.
is reflexive, symmetric, and transitive 
is an equivalence relation.
John
My calculator said it, I believe it, that settles it
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