# Equivalence relation

by andlook
Tags: equivalence relation
 P: 33 Hey I can't see how x~x+1 is an equivalence relation on the real numbers? I don't understand what the relation is. Can anyone help?
 P: 607 Perhaps you want the equivalence relation generated by this... x ~ x+1 ~ x+2 ~ ... Or perhaps not. Without context, we can only guess.
 P: 38 A relation is a rule that relates elements of one set (in this case the real numbers) to elements of another set (in this case, ALSO the real numbers). An equivalence relation is one from a set to itself that has three properties: The relation must be reflexive (x must be related to x) The relation must be symmetric (if x is related to y, then y is related to x), and The relation must be transitive (if x is related to y, and y is related to z, then x is related to z) The relation you have, where x is related to x+1 is not an equivalence relation. 1 is not related to 1, since 1 is only related to 2 (not reflexive). 1 is related to 2, but 2 is not related to 1 (it's related to 3) (not symmetric). 1 is related to 2 and 2 is related to 3, but 1 is not related to 3 (not transitive). It actually fails every single one of the properties.