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LN6(Equivalence Relation) - Equivalence Relation Reflexive...

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Equivalence Relation Reflexive, Symmetric, and Transitive
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Definition . A binary relation on a set is called an equivalence relation if it is reflexive, symmetric, and transitive. Example . Let and
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The table below gives the properties of the five relations. Relation Reflexive Symmetric Transitive No Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Therefore, and are equivalence relations on while and are not.
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Definition . Let be an equivalence relation on a set . For each , we define the equivalence class of , denoted by the symbol , to be the following subset of : Example . Let and consider the equivalence relation . Then the equivalence classes are:
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Note that there are just two distinct equivalence classes because . The other equivalence class is .
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