Is R an equivalence relation? If so, prove it discrete math and if not, explain why it is not. Let R be a relation on Z defined by (x.y) \in R if and only if 5(x-y)=0. Formally state what it means for R to be a symmetric relation.

Haven 2021-08-06 Answered
Let R be a relation on Z defined by (x.y) R if and only if 5(x-y)=0. Formally state what it means for R to be a symmetric relation. Is R an equivalence relation? If so, prove it discrete math and if not, explain why it is not.
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Khribechy
Answered 2021-08-07 Author has 100 answers
Let R be a relation on Z defined by x, y Z. Then (x, y) R if 5(x-y)=0
Symmetric let (x,y)Ri.e.5(xy)=0
if R symmetric (x-y) Then (y,x)R means 5(y-x)=0
yx=0y=x
as 5(xy)=05(yx)=0
5(yx)=0(y,x)R
so R is symmetric.
Reflexive: xZ,xx=05(xx)=0
so (x,x)R
Hence R is reflexive.
Transitive: Let (x,y)Rand(yz)R so
That 5(xy)=0and5(yz)=0
add both 5(xy)+5(yz)=0
5(xy+yz)=0
5(xz)=0
i.e.(x,z)R
Hence R is transitive.
So, that R is an equivalence selation.
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