▁is ▁the ▁relation ▁S ▁defined ▁by ▁setting ▁x S z ▁to ▁be ▁true ▁for ▁a ▁pair ▁of ▁elements ▁x ▁and ▁z ▁in ▁X ▁whenever ▁there ▁exists ▁y ▁in ▁X ▁with ▁x R y ▁and ▁y R z ▁both ▁true . ▁R ▁is ▁ide mp ot ent ▁if ▁R   =   S . ▁ ▁Equ ival ently , ▁relation ▁R ▁is ▁ide mp ot ent ▁if ▁and ▁only ▁if ▁the ▁following ▁two ▁properties ▁are ▁true : ▁R ▁is ▁a ▁trans itive ▁relation , ▁meaning ▁that ▁R   ∘   R   ⊆   R . ▁Equ ival ently , ▁in ▁terms ▁of ▁individual ▁elements , ▁for ▁every ▁x , ▁y , ▁and ▁z ▁for ▁which ▁x R y ▁and ▁y R z ▁are ▁both ▁true , ▁x R z ▁is ▁also ▁true . ▁R   ∘   R   <0xE2> <0x8A> <0x87>   R . ▁Equ ival ently , ▁in ▁terms ▁of ▁individual ▁elements , ▁for ▁every ▁x ▁and ▁z ▁for ▁which ▁x R z ▁is ▁true , ▁there ▁exists ▁y ▁with ▁x R y ▁and ▁y R z ▁both ▁true . ▁Some ▁authors ▁call ▁such ▁an ▁R ▁a ▁dense ▁relation . ▁Because ▁ide mp ot ence ▁incorpor ates ▁both ▁trans itivity ▁and ▁the ▁second ▁property ▁above , ▁it ▁is ▁a ▁stronger ▁property ▁than ▁trans itivity . ▁ ▁Ex amples ▁For ▁example , ▁the ▁relation   < ▁on ▁the ▁rational ▁numbers ▁is ▁ide mp ot ent . ▁The ▁strict ▁ordering ▁relation ▁is ▁trans itive , ▁and ▁whenever ▁two ▁rational ▁numbers ▁x ▁and ▁z