Equivalence Relations (special conditions)

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Equivalence Relations (special conditions)

Post by yoyobarn » Fri Feb 17, 2012 1:09 am

It is known that (symmetry and transitivity implies reflexive) is false, even though it seemingly seems that if x~y, then x~y (symmetry) and if x~y and y~x then x~x (transitivity), seems to imply that x~x (reflexive).

Is the reason because that if x~y is false, then the statements of symmetry and transitivity are trivially true, and hence x~x may be false?

Another question: are there additional conditions we can impose to make symmetry and transitivity imply reflexivity?


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Re: Equivalence Relations (special conditions)

Post by quinquenion » Fri Feb 17, 2012 1:29 am

Yep, it has to do with sets with only single elements. If for all x there exists a y such that y≠x and x~y, then symmetry and transitivity imply reflexivity.

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