Equivalence Relations question

  • Thread starter Ed Quanta
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  • #26
HallsofIvy
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Originally posted by modmans2ndcoming
when I made the post, I had this type of reflexive set in mind:

{ (1,1) , (2,2) }

so I most certainly did not have all cases that are reflexive in mind.

I most certainly know that the class

{ (1,1), (2,2), (2,3) } is not an equivilence....but then if you read carfuly, you would see I used the term "vaccuos" in relation to symetric and transitive, meaning that no ordered pair (like the(2,3) I used above) exists in the set, so all elements are examples of reflexivity, nothing more.

reflexive is a bidirectional proposition, so you MUST have at least one ordered pair in the set, and it must represent for all x : (x,x)

but Symetric is an implication, so it the antecedent is false, it is symetric.

and Transitive is an implication so in the same way as symetric, it will be true.
Then you should have said "identity relation" rather than "reflexive relation". What you orignally said was "a reflexive relation is an equivilence relation because it is reflexive, and vaccuosly transative and vaccuosly symetric."
The "vacuously" part did not clarify, it's just wrong: not all reflexive relations are transitive or symmetric.
 
  • #27
Janitor
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I'm with Halls

"Identity relation" sounds like a better term for what modman was talking about.
 
  • #28
HallsofIvy
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And, of course, the information that every "identity" relation (x is related to y if and only if x= y) is an equivalence relation is not news. Identity is the prototype of all equivalence relations.
 

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