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I'm having trouble understanding whether or not this relation would be considered antisymmetric and transitive. The a relation R on the set of real numbers by (x,y) ϵ R if and only if x-y=0. If I am understanding this correctly the this set would be correct:

R= { (1,1) (2,2) (3,3) (4,4) (5,5) }

It is clearly reflexive but I'm having trouble understanding whether or not it is antisymmetric and transitive.

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Edit: Im sorry, i thought you defined the relation with the set you wrote. I am now looking into it further (with your full definition of $R$).

Edit 2: Well, What i wrote still holds for the arithmetic definition of your relation $R$. Try show transitivty with the definition of $R$ and the axioms of The field $\mathbb R$ . (Hint: $(x,y) \in R $ iff $ x=y$)

Using negation is always a useful tool. This relation is transitive because it's not not-transitive.

Formally speaking: $(a,b),(b,c) \in R $ yields $(a,c) \in R$ Which is clearly the case since the negation is Not true.

Try the same in order to understand if it is anti-symmetric

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