R ‚" X × X X = "




"x " X : xRx

"x, y " X : xRy =Ò! yRx

"x, y, z " X : xRy '" yRz =Ò! xRz


x R ‚" X × X

X = "

[x]R := {y " X : xRy}



"x " X : [x]R = "


[x]R = X
x"X

[x] )" [y] = " =Ò! [x] = [y]


[x] = [y] =Ò! [x] )" [y] = "



R


X/R := {[x]R : x " X}



X










"




<"
l <" k Ð!Ò! l k













"








A(0, 0) B(1, 1) C(1, 2) D(2, 3)
- -

AB = CD

- -

AB = CD = [1, 1]




- -

AB CD











"











N


N × N
(m1, n1) <" (m2, n2) Ð!Ò! m1 + n2 = n1 + m2


N


N/<" a" Z










[(m1, n1)] + [(m2, n2)] := [(m1 + m2, n1 + n2)]
[(m1, n1)] · [(m2, n2)] := [(m1m2 + n1n2, m1n2 + m2n1)]



N



(m1, n1)


[(m1, n1)]







[(m1, n1)] (m1, n1)

(m 1, n 1) [(m2, n2)] (m2, n2)


(m 2, n 2)
m1 + n 1 = n1 + m 1
m2 + n 2 = n2 + m 2




[(m1, n1)] + [(m2, n2)] = [(m 1, n 1)] + [(m 2, n 2)]
[(m1, n1)] · [(m2, n2)] = [(m 1, n 1)] · [(m 2, n 2)]

[(m1 + m2, n1 + n2)] = [(m 1 + m 2, n 1 + n 2)]
[(m1m2 + n1n2, m1n2 + m2n1)] = [(m 1m 2 + n 1n 2, m 1n 2 + m 2n 1)]


(m1 + m2, n1 + n2) <" (m 1 + m 2, n 1 + n 2)
(m1m2 + n1n2, m1n2 + m2n1) <" (m 1m 2 + n 1n 2, m 1n 2 + m 2n 1)



"

Z" := Z \ {0} Z × Z"
(m1, n1) <" (m2, n2) Ð!Ò! m1n2 = m2n1



Z


Z

<"a" Q




[(m1, n1)] + [(m2, n2)] := [(m1n2 + m2n1, n1n2)]
[(m1, n1)] · [(m2, n2)] := [(m1m2, n1n2)]







m
[(m, n)] a"
n
m1 m2 m1n2+m2n1
[(m1, n1)] + [(m2, n2)] a" + =
n1 n2 n1n2
m1 m2 m1m2
[(m1, n1)] · [(m2, n2)] a" · =
n1 n2 n1n2