�� 7 7 7 7 7 7 7 7  u � v R S T <� d" D(R) = {x: "y(x, y) " R} D"(R) = {y: "x(x, y) " R} R R �" D(R) � D"(R) "y(x, y) " R "x(x, y) " R x " R y " R (x, y) " R xRy xRy x R y "u x "u y �! x " u '" y " u '" (x " y (" x = y) �"u x �"u y �! x " u '" y " u '" x �" y =u x =u y �! x " u '" x = y 7 7  R R-1 = {(y, x): (x, y) " R} R S RS xRSy �! "z(xRz '" zSy) R u "x"u(xRx) u "x"u�(xRx) u "x"u"y"u(xRy �! yRx) u "x"u"y"u((xRy '" yRx) �! y = x) u "x"u"y"u"z"u((xRy '" yRz) �! xRz) u "x"u"y"u(xRy (" yRx) =u �"u u "u 7 7  R x y x R y =u u � {v} e f g h f y xfy f(x) fx f: u �! v u " x �! f(x) " v f u v f: u �! v f u v f u v u v vu vu vu = {f " P(u � v): f: u �! v} 7 7  f u �" D(f) f|u = {(x, y) " f: x " u} f[u] = D"(f|u) u f f[u] {fx}x"u {f(x): x " u} x �" D(f) x " u f[ u] = {f[x]: x " u} x �" D(f) x " u u = " f[ u] �" {f[x]: x " u} u �" D(f) v �" D(f) f[u] \ f[v] �" f[u \ v] u �" v �" D(f) f[u] �" f[v] f u �" D(f) fx = f[u] fx = f[u] x"u x"u 7 7  f f-1 f: u �! v D"(f) = v f: u �! v x = y �! f(x) = f(y) x, y " u f[u] = v f x �" D(f) x " u u = " f[ u] = {f[x]: x " u} u �" D(f) v �" D(f) f[u] \ f[v] = f[u \ v] u " x �! P(x) " P(P( u)) 7 7  R u u R <�R x <�R y �! x R y '" x = y <�R R x R y �! y R x R R v �" u x R y y R x x, y " v v �" u R v �"u =u u u "u u 7 7  R u v �" u x " u x v x " v �"y"v(y <�R x) v x " v �"y"v(x <�R y) v x " v "y"v(x R y) v x " v "y"v(y R x) v "y"v(x R y) v "y"v(y R x) v v v R R R 7 7  u u u u u u R u u x, y " u x <�R y z " u x <�R z z <�R y (v, w) u R v*"w = u v)"w = " x " v y " w x <�R y R u u (v, w) v w (u, R) R u 7 7  (u, R) (v, S) (u � v, RS) RS (x, y) RS (z, t) �! (x <�R z) (" (x = z '" y S t), RS (u � v, " ) " SR SR (x, y) " (z, t) �! (y <�S t) (" (y = t '" x R z), SR " SR 7 7  (u, R) (v, S) f: u �! v x R y �! f(x) S f(y) x, y " u (u, R) (v, S) f: u �! v (u, R) (v, S) (u, R ) C" (v, S) (u, R) C" (v, S) (v, S) C" (w, T) (u, R) C" (w, T) (u, x, R) = {y " u: y <�R x} x " u v �" u u = v x " u v = (u, x, R) x, y " u x <�R y (u, x, R) = ( (u, y, R), x, R) x R y (u, x, R ) �" (u, y, R) 7 7  (u, R) (1) x " u (u, R) C" ( (u, x, R), R) (2) x, y " u ( (u, x, R), R) C" ( (u, y, R), R) �! x = y (1) x " u (u, R) C" ( (u, x, R), R) f: u �! (u, x, R) v = {f(y): f(y) = y '" y " u} f(x) x " (u, x, R) f(x) = x f(z) / z <�R f(z) f(z) = z z = f-1(z) z " v f(z) = v f(z) R z f(z) <�R z f(f(z)) <�R f(z) f(f(z)) = f(z) f(f(z)) " v f(z) = v f(z) R f(f(z)) (2) ( (u, x, R), R) C" ( (u, y, R), R) x = y x <�R y (u, x, R) = ( (u, y, R), x, R) y <�R x (u, y, R) = ( (u, x, R), y, R) (1) 7 7  (u, R) (v, S) (u, R) C" (v, S) f: u �! v f, g: u �! v f = g x " u f(x) = g(x) z f(x) = g(x) x <�R z f(z) = {y " v: "x<� zf(x) <�S y} = {y " v: "x<� zg(x) <�S y} = g(z) R R (u, u) f: u �! u 7 7  (u, R) (v, S) (1) (u, R) C" (v, S) (2) x " u ( (u, x, R), R) C" (v, S) (3) y " v (u, R) C" ( (v, y, S), S) f f = {(x, y) " u � v: ( (u, x, R), R) C" ( (v, y, S), S)}. f (x, y) " f (x, z) " f y = z f C" ( (v, y, S), S) C" ( (v, z, S), S) f: D(f) �! D"(f) x R y �! (u, x, R) �" (u, y, R) �! (v, f(x), S) = f[ (u, x, R)] �" f[ (u, y, R)] = (v, f(y), S) �! f(x) S f(y) D(f) u x " D(f) y " u y <�R x ( (u, y, R), R) C" ( (v, g(y), S), S) 7 7  g: (u, x, R) �! (v, f(x), S) f y " D(f) D"(f) v D(f) = u D"(f) = v D(f) = u D"(f) = v g: D(f) *" {x} �! D"(f) *" {y} g = f *" {(x, y)} x = (u \ D(f)) y = (v \ D"(f)) (u, x, R) = (D(f) *" {x}, x, R) C" (D"(f) *" {y}, y, S) = (v, y, S) 7 7