Question

7.3.13 Recall the description of the real projective line (page 122): if Am is the line through the origin with gradient m, then P(R2) = {Am m e RU }}. Define a relation R2 = R2\ {(0,0)} by (a,b)~(c,d) ad = bc N on (a) Prove that is an equivalence relation. (b) Find the equivalence classes of Am? Ho do the equivalence classes differ from the lines

Answer 1

P(R) =Am forcm amx RR1C0.0 wturce Am iz d the defined b Me rrelaion on a,b)n (e,d) ad=be Ca) i ut have ab= a.b (a,b)n(a,b) Teflexive The rrelation 1 (a,k) (C d) ad bc cb da B1mmetric abned) and (C,d)(e,g) ad be and ef = de 13 multiplication and b to the fiTst & 2 n adf bef and bef bde tespectively adt - bde af be nis Trean ziHve. equivale tence the trelation 4relation an - nce

(b) he equi valence classes of defined cl (aib)= as' )ERSOO (2) different Rines at us Conidere two Am and An LOpe and ferm =ma Am is and An wherce n 1 I in u-and arce Varuables direch ons. For farb, if e) elCa,b) then bama and n the equivalence clan fCa,b), hore arce Same pTcesent eshose lo pe caib), ines arce as those s with eut Am Contains all thoge. lin Ces auy real mumber) m