Question

Show all necessary work. Print your answers clearly 20 Show that the relation Ron ROR: the set of real numbers) given by Ry ill an equivalence relation. av

Answer 1

show that the relation R on R (R: the set of real mumbers) given by a Ry iff n²=y2 is an equivallence relation. Answer'- First we stake the defination of Equivalence Relation. Equivalence Relation bet, s be a non-emply sel and | be a binany relation on s. The relation sis said to be reflexive if (a, a)EP for all a insie, apa holds for all a ins. opa The relations is said to symmetric it for any two elemento agbins, (a,b) € 8 =) (b, a) EP ie, apb => bpa The relation p is said to be transative if for any three asbe ins (a, b)EP and (6, e) EP (a, e) EP lien apb, bpe =) ape. The relation Pons is said to be an equivallence relation on s, if p is reflexive, symmetrice and transatine. The relation Now, here, R is defined as * yeR. хку і х==y* The set of real numbers R is we know that, non-empty.

(0 het ack Ther, a²=22 ara heldp tack R ip reflexive (11) Let, agbelr and a Rb holds. Then, a2=6² => b² = 92 i bra holds . aRb => brata, bER i r is symmelpie heby a, b, c fR and arb and brc holds. o Now, arb holds ) a=62 -70 s and are holds => 6 = 22 - ) From and 6) a = P2 h otels => are holds arb and bre > are, tabeer . R is transative.

and .. R is an transative reflexive, symmetrie relation on R. so by is an equivallence Relation defination, R on R.