ABSTRACT
The equivalence relation [20] in set X is a binary relation eC , which is
UHÀH[LYHV\PPHWULFDQGWUDQVLWLYH7KHUHODWLRQ eC is:
x 5HÀH[LYHLI xCx e , ( Xx )
x 6\PPHWULFLI xCyyCx ee Ÿ , ( Xyx , )
x 7UDQVLWLYHLI zCxzCyyCx eee Ÿš )()( , ( Xzyx ,, )
Diagonal x' LHVHW
}|),({ XxxxC xe '
LVDWULYLDOH[DPSOHRIHTXLYDOHQFHUHODWLRQ 7KHFRPSOHWHUHODWLRQLH
},|),({2 XyxyxXCe
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Equivalence classes
The equivalence class, x RIHOHPHQWxLQUHJDUGWRHTXLYDOHQFHUHODWLRQ eC LVWKHVHWGH¿QHGDVIROORZV
)(,},|{ XxXyyCxyx e
Two equivalent classes x DQG z ( Xzx , ) are either equivalent or GLVMRLQWHGWKHLULQWHUVHFWLRQLVWKHHPSW\VHW
7KHUHIRUHHTXLYDOHQFHFODVVHV x ( Xx GH¿QHDSDUWLWLRQRIVHWX
(YHU\SDUWLWLRQ },,,{)( 21 nXXXX ! S RIVHW
nn XXXXX *!** 121
GH¿QHVDQHTXLYDOHQFHUHODWLRQLQVHWX(TXLYDOHQFHFODVVHVDUHVXEVHWV iX 7KH SDUWLWLRQV DQG HTXLYDOHQFH UHODWLRQ LQ VHWX XQLTXHO\ GH¿QH HDFK
RWKHUDQGUHSUHVHQWGLIIHUHQWH[SUHVVLRQVRIDVDPHIDFW>@
Quotient set
7KHVHWRIHTXLYDOHQWFODVVHVRIDOOHOHPHQWVRIVHWXLVFDOOHG the quotient
of X by eC DQGLWLVGHQRWHGE\ eCX /
$QHTXLYDOHQFHFODVVRIUHODWLRQ eC in set X, where XLVWKHVHWRIQRGHV
RIJUDSK ),( eCXG UHSUHVHQWVDFOLTXHRIJUDSK*LQWKH%HUJH¶VVHQVH
