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# 2.2: Equivalence Relations, and Partial order

• • Contributed by Pamini Thangarajah
• Professor (Mathematics & Computing) at Mount Royal University

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Note

For every equivalence relations over a nonempty set  $$S$$, $$S$$ has a partition.

For the following examples, determine whether or not each of the following binary relations on the given set is reflexive, symmetric, antisymmetric, or transitive.  If a relation has a certain property, prove this is so; otherwise, provide a counterexample to show that it does not.  If is an equivalence relation, describe the equivalence classes of Hasse Diagram

Definition

Let S be a non empty set and let $$R$$ be a partial order relation on $$S$$.  Then two elements $$a$$ to $$b$$ of $$S$$ are connected if  $$a R b$$. This diagram is called Hasse diagram.