SOLUTION: Given a set S = ℝ, prove that a relation R on S where (a , b) ∈ R and a - b = 0 is an equivalence relation.
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Question 1191720: Given a set S = ℝ, prove that a relation R on S where (a , b) ∈ R and a - b = 0 is an equivalence relation.
Answer by Solver92311(821) (Show Source): You can put this solution on YOUR website!
An equivalence relation is a relation that is Reflexive, Symmetric, and Transitive:
Reflexive: \ \in\ R\ on\ S, a\ -\ a\ =\ 0)
therefore 
is reflexive.
Symmetric: \ \in\ R\ on\ S, a\ -\ b\ =\ 0\ \right\ b\ -\ a\ =\ 0)
therefore is symmetric.
Transitive: \ &\ (b,c)\ \in\ R\ on\ S, \(a\ -\ b\ =\ 0\ \right\ a\ =\ b\ &\ b\ -\ c\ \right\ b\ =\ c)\ \right\ a\ =\ c\ \right\ a\ -\ c\ =\ 0)
therefore is transitive.
R is reflexive, symmetric, and transitive, therefore R is an equivalence relation.
John
My calculator said it, I believe it, that settles it
From
I > Ø
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