I am stuck on 3 proofs for my discrete math class. Any help would be greatly appreciated. 1. Prove that if R is a partial order, then R^{-1} is a partial order. 2.Prove that if R_1 and R_2 are equivalence relations on the set A, then R_1 cap R_2 is an equivalence relation.

Maliyah Robles 2022-07-16 Answered
I am stuck on 3 proofs for my discrete math class. Any help would be greatly appreciated.
Prove that if R is a partial order, then R 1 is a partial order
Prove that if R 1 and R 2 are equivalence relations on the set A, then R 1 R 2 is an equivalence relation.
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Answers (1)

Dalton Lester
Answered 2022-07-17 Author has 12 answers
Step 1
Here is part of (1). The rest of (1) is similar, so is (2). Problem (3) is really a completely different topic, I suggest you delete it and ask a separate question.
Let R be a partial order: therefore R is reflexive, transitive and antisymmetric. We prove that R 1 is transitive.
Step 2
So, suppose that x R 1 y and y R 1 z. By definition of inverse this means that yRx and zRy. Since R is transitive we have zRx, and using the definition of inverse again, x R 1 z. We have proved that if x R 1 y and y R 1 z then x R 1 z; by definition, R 1 is transitive.
Observe that this proof really uses pretty much nothing except various definitions. So I hope this underlines the importance of knowing the definitions properly.

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