A = {1, 2, 3, 4} T ‚" A × A
(a, b) " T "n"N a · n = b.
T
T
T
T
P, T ‚" P P
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
T1 = {(x, y) | (x < y - 1) (" (x = y)}
T2 = {(x, y) | (x > y2) (" (x = y)}
T3 = {(x, y) | (x e" 2y) (" (x = y)}
T4 = {(x, y) | ("n"N x · 3n = y) (" (x = y)}
N
Ä (a, b) " Ä a < b + 2
Ä
N
Ä (a, b) " Ä a = b2

Ä
T1 = {(x, y) " R × R ; x d" y}
T2 = {(x, y) " R × R ; x d" 2y}
T3 = {(x, y) " R × R ; x + y d" 0}
T4 = {(x, y) " R × R ; y = x2}
R+ = (0, +") Ti ‚" R × R
T1 = {(x, y) " R+ × R+; y2 (1 + x2) = 1}
T2 = {(x, y) " R × R+; y2 (1 + x2) = 1}
T3 = {(x, y) " R+ × R; y2 (1 + x2) = 1}
T4 = {(x, y) " R × R; y2 (1 + x2) = 1}
Q = {(x, y) ; x2 + y2 = 1}
Q Q ć% Q Q ć% Q ć% Q
Q-1 Q ć% Q-1
K = {(x, y) ; x2 + y2 d" 1}
K K ć% K K ć% K ć% K
K-1 K ć% K-1
f (x) = 2x2 + x - 2 g (x) = 2 - 3x
f ć% g g ć% f
f ć% g (x) = g ć% f (x)
K
w (x, y) = x y
T = {(x, y) " K × K; w (x, y)}
T
"z"x Źw (x, z) .
"x" Źw (y, x) '" Źw (x, y) '" x = y

y
T
Z
m Ä n a" n3 e" m
m Ä n a" n4 - m4 = n - m
m Ä n a" n3 e" m (" n = m
à ‚" Q × Q
à = {(x, y) " Q × Q | x2 = y2}
Ã
Ä A
Ä
Ä
Ä
Ä
 N Ä
(a, b) " Ä a" a = b2
Ä
A = [0, 1) B = (2, 5]
f : A B
f-1
A = [2, 4) B = [1, ")
f : A B
f-1
f : R R
x2 - 2 dla x < -1
f (x) =
-3 - 2x dla x e" -1
f
f ć% f
f-1
x2 - 6x + 10 dla x < 2
f (x) =
6 - 2x dla x e" 2
f
f-1
f ć% f
f : R R
2x + 1 dla x d" -1
f (x) =
x2 + 2x dla x > -1
f
f ć% f
f-1
f : R R
f(x) = (x - 2)2
f-1 f
f-1
(0; 5) f
(0; 1) f
(1; 4) (1; 3)
f(2) = 2
 A = {1, 2, 3, 4, 5, 6, 7}
A 2, 3 5
A 2 3
1 2 3 4 5 6 7
g =
2 1 4 3 6 7 5
1 2 3 4 5 6 7
h =
6 4 1 3 7 2 5
S7
g h
g h gh
g h gh
g-1 g2
g3
T = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)} |T | = 8
{(1, 1), (2, 2), (3, 3), (4, 4)} " T
(a, b) " T '" (b, a) " T Ò! "m,n"N am = b '" bn =
a Ò! abmn = ab Ò! mn = 1 Ò! m = 1 Ò! a = b
(a, b) " T '" (b, c) " T Ò! "m,n"N am = b '" bn = c Ò!
amn = c Ò! (a, c) " T
(1, 2) " T '" (2, 1) " T
(1, 2) " T '" (1, 1) " T
P = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}
(x = y)
x d" y - 1
(x, y) " T1'"(y, x) " T1'"x = y

y d" x - 1
x + y d" y + x - 2
(x, y) " T1 '" (y, x) " T1 Ò! x = y
(x, y) " T1 '" (y, z) " T1
10 x = y (y, z) = (y, z) " T1
20 y = z (y, z) = (x, y) " T1
x d" y - 1
30 x = y '" y = z x d" y - 1 d"

y d" z - 1
z - 2 < z - 1 x d" z - 1 (y, z) = (x, y) " T1
 Ä n < n + 2
Ä 4 < 3+2'"3 < 4+2 Ò! (4, 3) " Ä '"(3, 4) "
Ä
Ä 4 < 3 + 2 '" 3 < 2 + 2 Ò! (4, 3) " Ä '" (3, 2) " Ä
(4, 2) " Ä
(2, 4) " Ä (4, 2) " Ä
(1, 4) " Ä (4, 1) " Ä
(4, 1) " Ä (1, 2) " Ä (4, 2) " Ä
T1
T2 (-1, -1) " T2 (-2, -1) " T2
(-1, -2) " T2 (2, 1) " T2 (1, 2) " T2
(4, 2) " T2 (2, 1) " T2 (4, 1) " T2
T3 (2, 2) " T3 x + y = y + x
(-2, 1) " T3 (1, -2) " T3
(4, -5) " T3 (-5, 4) " T3 (4, 4) " T2
T4 (-1, -1) " T4 (2, 4) " T2
(4, 2) " T2 y = x2 x = y2
y = y4
y = 0 y = 1
x = y
(2, 4) " T2 (4, 16) " T4 (2, 16) " T4
1
T1 y =
1+x2
1
T2 y =
1+x2
" "
T3 (1, 2) (1, - 2) " T3
T4 (0, 1) (0, -1) " T3
T1 ‚" T2 ‚" T4 T1 ‚" T3 ‚" T4
Q
Qć%Q = {(x, y) ; "z(x, z) " Q, (z, y) " Q} = {(x, y) ; "zx2 + z2 = 1, z2 + y2 = 1} =
{(x, y) ; x2 = y2, x2 d" 1}
Q ć% Q ć% Q = Q
Q-1 = Q
Q ć% Q-1 = Q ć% Q
K
Kć%K = {(x, y) ; "z(x, z) " K, (z, y) " K} = {(x, y) ; "zx2 + z2 d" 1, z2 + y2 d" 1} =
{(x, y) ; x2 d" 1, y2 d" 1}
(1, 1), (-1, 1), (1, -1), (-1, -1)
K ć% K ć% K = K ć% K
K-1 = K
K ć% K-1 = K ć% K
f ć% g(x) = 2(2 - 3x)2 + (2 - 3x) - 2 = 18x2 - 27x + 8
g ć% f(x) = 2 - 3(2x2 + x - 2) = -6x2 - 3x + 8
18x2 - 27x + 8 = -6x2 - 3x + 8 x1 = 0 x2 = 1
"x "y w(x, y) '" w(y, x) Ò! x = y
"x "y Źw(y, x)
"x (x, x) " T
/
Ä
2 Ä 7 '" 7 Ä 9 '" (2, 9) " Ä
/
x2 = x2 Ò! (x, x) " Ã Ã
(x, y) " Ã Ò! x2 = y2 Ò! y2 = x2 Ò! (y, x) " Ã Ã
(x, y) " Ã '" (y, z) " Ã Ò! x2 = y2 '" y2 = z2 Ò! x2 = z2 Ò! (x, z) " Ã Ã
[q] = {q, -q} [0]
 Ä = {(x, y) " A × A | "c x - y = 25c}
Ä = {(x, y) " A × A | "c x - y = 4c}
{1} {2} A \ {1, 2}
Ä = {(x, y) " A × A | x = y (" (x > 2 '" y > 2)}
Ä (4, 2) " Ä '" (2, 4) " Ä
/
Ä (a, b) " Ä Ò! a e" b
Ä (16, 4) " Ä '" (4, 2) " Ä '" (16, 2) " Ä
/
f f(x) = ax + b
f(0) = a0 + b = 5 f(1) = a1 + b = 2 f(x) = -3x + 5
y = -3x + 5
y-5
x =
-3
y-5
5-x
f-1(y) = f-1(x) =
-3 3
log((0, 1]) = (-", 0]
g log h
[2, 4) (0, 1] (-", 0] [1, ") f = h ć% log ć% g
h(x) = -x + 1 g(x) = -1x + 2 f(x) = h ć% log ć% g(x) =
2
h ć% log(-1x + 2) = -log(-1x + 2) + 1
2 2
y = -log(-1x + 2) + 1
2
-y - 1 = log(-1x + 2)
2
10-y-1 = -1x + 2
2
-2 · 10-y-1 + 4 = x
f-1(y) = -2 · 10-y-1 + 4 f-1(x) = -2 · 10-x-1 + 4
x < -1 Ô! y > -1
f ć% f(x) = f(x2 - 2) = -3 - 2(x2 - 2) = -2x2 + 1
x e" -1 Ô! y d" -1
f ć% f(x) = f(-3 - 2x) = (-3 - 2x)2 - 2 = 4x2 + 12x + 7
-2x2 + 1 dla x < -1
f ć% f (x) =
4x2 + 12x + 7 dla x e" -1
x < -1 Ô! y > -1
y = f(x) = x2 - 2
x2 = y" 2
+
"
x = - y + 2 x = y + 2 x < -1
x e" -1 Ô! y d" -1
y = f(x) = -3 - 2x
-3-y
x =
2
-3-y
dla y < -1
2
"
f-1 (y) =
- y + 2 dla y e" -1
-3-x
dla x < -1
2
"
f-1 (x) =
- x + 2 dla x e" -1
 2x - 6 dla x < 2
f (x) = < 0 x < 2 Ò! f(x) > 2
-2 dla x e" 2
x e" 2 Ò! f(x) d" 2
1
3 - x dla x < 2
2
"
f-1 (x) =
3 - x - 1 dla x e" 2
-2x2 + 12x - 14 dla x < 2
f ć% f (x) =
4x2 - 12x + 10 dla x e" 2
4x + 3 dla x d" -1
f ć% f (x) =
x4 + 4x3 + 6x2 + 4x dla x > -1
x-1
dla x d" -1
2
"
f-1 (x) =
x + 1 - 1 dla x > -1
f(1) = f(3)
f(0) = 4, f(5) = 9 i f(2) = 0 f ((0 , 5)) = (0 , 9)
f-1(0) = 2, f-1(1) = {1 , 3} f-1 ((0 , 5)) = (1 , 3)
3
f(x) = 1 + (x - 1)
4
x dla x d" 2
f(x) =
1
2 + (x - 2) dla x > 2
2
{1, 4, 6, 7} 24
{1, 4, 5, 6, 7} 5!
g = (1, 2)(3, 4)(5, 6, 7) h = (1, 6, 2, 4, 3)(5, 7)
gh = (1, 7, 6)(2, 3) rz g = NW W {2, 2, 3} = 6 rz h = NW W {5, 2} =
10 rz gh = NW W {3, 2} = 6
(-1)g = (-1)4 = 1 g
(-1)h = (-1)5 = -1 h
(-1)gh=(-1)9 = -1 h
g-1 = (1, 2)(3, 4)(5, 7, 6) g2 = (5, 7, 6) g3 = (1, 2)(3, 4)
(1, 2, 3, 4, 5, 6, 7, 8, 9)
(1, 2, 3, 4, 5, 6, 7)(8, 9)
(1, 2, 3, 4, 5)(6, 7, 8)