Question

Just number 5 please

4. Show that congruence modulo n is an equivale 5. Consider the unit circle C with equation modulo n is an equivalence relation UI arcle C with equation x2 + y2 = 1 in the plane. Thus C = {(x, y) ERX R: x2 + y2 = 1}. Define a relation on C as follows: For (x, y) EC, antipodal point (-x, -y). In symbols, llows: For (x, y) E C (x, y) is related to itself and to its allo (x, y) ~ (x, y); (x, y) ~ (-x, -y). (a) Show that is an equivalence relation. Think of C as a rubber band and imagine gluing together the equivalent antipodal points of C. Describe the resulting figure. (Much more will be said on this topic in Chapter 7.)

Answer 1

2.5 the consider the Unit Circle x² + y2 ad in Plane. c=1 (L, DEIRXIR : x2 + y2 , ? Define a Relation n on c. asa (36,9)n (,y (6c, J) (-x-9) that is; every element of C, only scelated to itself and it antipodal point (-2,-9) that (as show in is an Equivalence Relation * Reflexive Let (IJEC then By Def" of w(Relation) we know that (x,y) a (x,y) 4 (4) EC son is a Reflexive Relation ® * Symmetry - thet (34,70) and (x2,92) € C such that (2 9 ) (2, J2) • We know that (x,,J,)~ (22,42) If and only if one of following holds

(021,9,) = (22,427 OR (-2,-40) = (12,42) & 4 = 42 sa=X2 OR -21-22 2 -4,-92 (22,42) = (1,9,) OR (-30, ,-4,) (X2,92) - (24,"I) so given Relation is symmetric Transitive :- Let 24,6 (0,7) (22,42), (863,93) + Consider . (x4,5,)m (82,12) And (X2,92) ~ (X3,93) 51=X2 OR 3,5-X2 0 And y, 22 OR Y,Fayz o And 32 = 23 OR 12 = -xz ☺ And y = 43 OR Y = -93 - → By Def" of given ② Relation.

By eq" O and ☺ 24 = 72 OR 3,5-tz And % = %3 OR %=-63 We can conclude that Oye Hyor. 83 5 Similarly By eq o and we can conclude that Yo = Y3 OR Y =-43 By © and ☺ (om, 30) n (23,93) so we proved that (0,409 7(22,92) and (22,922 4(23,9) emplies (0,9,)~ (2393) for (,403,293,4),(2,55) EC so Given Relation is Recanstive © 2) By ® ® Equivalence ® is is Relation an :

(B) Think of c as a rubber Band and Imagine gleing together the equircalent antipodal Point of c. Describle the for resulting figure vce (8,9) (362,422 st 1-82,92) (21,-8, 9 4 clearly the resulting is a point. figure

Thank You !?