Math 580 (03/14/12)
Lecturer : Gangyong Lee
Algebra
1. An equivalence relation R on a set X is a subset of X X satisfying the following
three conditions:
(i) Reexivity: xRx for all x X .
(ii) Symmetry: If xRy , then yRx for all x, y X .
(iii) Transitivit
Math 580 (02/01/12)
Lecturer : Gangyong Lee
1-4 Algebra
Denitions:
GY = cfw_f G | f (y ) = y y Y
O(2) = cfw_ , rL | 0 < 2, L is a line going through the origin = Isom(R2 )cfw_(0,0) ,
the group of all isometries R2 xing (0, 0).
SO(2) = cfw_ | 0 < 2
R+ =
Math 580 HW9 (03/07/12)
Lecturer : Gangyong Lee
1. Consider a relation on R2 dened by
(a, b) (c, d) i a2 + b2 = c2 + d2
(i) Show that is an equivalence relation on R2 .
(ii) What are the equivalence classes and interpret them geometrically?
2. (i) Find th