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A relation R on the set of complex numbers is defined by z_1R z_2 if and only if (z_1-z_2)/(z_1+z_3)is real . Show that R is an equivalence realtion. |
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Answer» Solution :Here `z_1 Rz_2 hArr(z_1 -z_2)/(z_1+z_2)` REAL (i) Reflexive `z_1Rz_2 hArr (z_1-z_1)/(z_1+z_2)` is real `rArr-(z_2 -z_1)/(z_1+z_2)` is real `rArr z_2 Rz_1` `thereforez_1 Rz_2 rArrz_2Rz_1` THEREFORE it is SYMMETRIC (iii) Transitive `z_Rz_2` `rArr(z_1-z_2)` is real and `z_2Rz_3` `rArr(z_2-z_3)/(z_2+z_3)` is real Here LET`z_1 =x_1+ iy_1 , z_2 = x_2 + iy_2 and z_3 = x_3 + iy_3` `therefore(z_1-z_2)/(z_1+z_2) is real rArr((x_1-x_2)+i(y_1-y_2))/((x_1+x_2)+i(y_i+y_2)` is real `rArr ({(x_1-x_2)+i(y_1-y_2)}.{(x_1+x_2)-i(y_2+y_2)})/((x_1+x_2)^2+(y_1+y_2)^2)` `rArr (y_1-y_2)(x_1+x_2)-(x_1-x_2)(y_1+y_2)=0` `rArr2x_2y_1-2y_2x_1=0` `rArr(x_1)/(y_1)=(x_2)/(y_2) ....(i)` SIMILARLY`z_2Rz_3` `rArrx_2/y_2=x_3/y_3 rArr z_1 R z_3` HenceR is and equivalence realation |
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