an equivalence relation on a set is a set of ordered pairs of elements of such that
- for all (reflexive property).
- implies (symmetric property).
- and imply (transitive property).
when is an equivalence relation on a set , it is customary to write instead of . also, since an equivalence relation is just a generalization of equality, a suggestive symbol such as , , or is usually used to denote the relation. using this notation, the three conditions for an equivalence relation become ; implies ; and and imply . if is an equivalence relation on a set and , then the set is called the equivalence class of containing .
the equivalence classes of an equivalence relation on a set constitute a partition of . conversely, for any partition of , there is an equivalence relation on whose equivalence classes are the elements of .
here is reflexive, antisymmetric and transitive so it is an equivalence relation
let first we check for reflexivity: let then we know and therefore and therefore is reflexive second we check for symmetry: let so that and therefore by definition of we know , but only because doesnt mean because x might be 0, but we know because therefore therefore by definition of we get therefore is symmetric third we check for transitivity: let such that therefore by definition of the relation we get and we know therefore by definition of we get therefore is transitive therefore is an equivalence relation