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Partial order and equivalence relations

Posted on 2011-09-28
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Hi guys: Can any one please tell me with example the difference between both of them Thanks.

A relation R on a set S is an equivalence relations if it satisfied all three of the following properties.

1. Reflexivity. For any a belongs S, a R a.
2. Symmetry. For any a, b belongs to S, a R b <--> b R a.
3. Transitivity. For any a,b, c belongs to S, if a R b and b R c, then a R c.

In other words. an equivalence relation is a relation that is reflexive, symmentric, and transitive.
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Question by:mustish1
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TommySzalapski earned 2000 total points
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The traditional = operator is an equivalence relation
Reflexive: Yes a=a
Symmetric: Yes if a=b, then we know b=a
Transitive: Yes if a=b and b=c, then we know a=c

Partial order is a set where some elements are less than others but others are not.

The <= operator is a partial ordering.

Also, a class schedule with prerequisites is partially ordered. You need to take CS53 and Math103 before you take Stat232 but you can take CS53 and Math103 in any order.
So CS53 precedes Stat232 and Math103 precedes Stat232, but we can't say CS53 or Math103 precedes the other.
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by:mustish1
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wow. Thanks Tommy thats is very easy. In exam can i use that example or i have to do some kind of maths equations in order to prove that.
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by:TommySzalapski
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That example would probably be fine if they ask for one. There really aren't any partially ordered sets in basic math that aren't also totally ordered. (Remember, a total order is also a partial order). Graph theory is a good place for partial order. You just need to draw a directional graph that has to cycles and is not fully connected. Like this one http://en.wikipedia.org/wiki/Partially_ordered_set
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