{ x, y " R2 : (x + y > 2) (" (x < 3)}
{ x, y " R2 : (x2 - 1 = 0) '" (y = x + 7)}
{ x, y " R2 : (x2 + y2 = 1) (2x = y)}
{ x, y " R2 : (x > y) ((2x d" y) (x2 + y2 = 3))}
{ x, y " R2 : (x < y) x < (x+y )}
2
{ x, y " R2 : 1 > 2}
{ x, y " R2 : 2 � 2 = 4}
{ x, y " R2 : (|x| < |y|) "! (-y < x < y)}
{ x, y " R2 : (x2 + y2 > 1) [(x2 + y2 d" 2) '" (Ź(x � y = 0) |y| = |x|)]}
{ x, y " R2 : ((x2 + y2 = 4) (y > -1 '" y = 1)) (x2 + y2 = 9)}

{ x, y " R2 : (x � x < 0) (x � x > 0)}
{ x, y " R2 : (x > y) (y + x > 0)}
{ x, y " R2 : (x � x + y � y > 1) (y + x > 0)}
{x " R : "y(x2 + y2 d" 1)}
{x " R : "y(x + y2 e" 3)}
{x " R : "x(x2 e" 0)}
{ x, y " R2 : "x(y + x2 = 3)}
 {x " R : "y"z(z > 0 '" (x - z)2 + (y - log z)2 d" (z + 1)2)}
{x " R : "y"z(z > 0 '" (x - z)2 + (y - log z)2 d" (z + 1)2)}
1
{ x, y " R2 : x2 + y2 > 1 "z(x2 + (y - z)2 d" )}
4
{ x, y " R2 : "z(y2 + (x - z)2 = 1) "z((x - z)2 + (y - z2)2 = 1)}

{ x, y " R2 : "x(x + y < 0) (y + x < 0)}
{z " R : "x"y(x2 + z2 d" (2y + 1)2)}
{z " R : "y"x(x2 + z2 d" (2y + 1)2)}
{z " R : "y"x(x2 + z2 d" (2y + 1)2)}
{z " R : "x"y(x2 + z2 d" (2y + 1)2)}
{z " R : "x"x(x = 1)}
{z " R : "x"x(x = 1)}
{x " R : "x"x(x = 1)}
{z " R : "x"y(y - |x| d" z d" y + |x|}
{z " R : "y"x(y - |x| d" z d" y + |x|}
{z " R : "y"x(y - |x| d" z d" y + |x|}
{z " R : "x"y(y - |x| d" z d" y + |x|}
a, b " P (N)
a )" b = " a *" b = b
"c(a )" c = c (a = c (" Ź"d(d = " '" d ą" c)))

+, �, 0, 1, =
a b c
a b
 a
x y
n n
a b c
f
{a, b} f
+, �, 0, 1, d"
f
a
f a
d"
f
a f
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a
a b
a b
a b, c
f
f
 x y
 A
A
A x " A x x
3
A x " A x x
3
n W (n) U(n)
n W (n)
 n W (n) U(n)
n W (n)
ą ŹŹą
"
"
a
a
 x � y = 0 x + y = y
N, +, �, 0, 1 v(x) = 1, v(y) = 2
w(x) = 0, w(y) = 3
P (N), *", )", ", N v(x) = {2, 3, 5}, v(y) =
{4, 6, 8} w(x) = {2, 3}, w(y) = {2, 3, 5}
P (x) (R(x, y) ŹP (y))
A A
A = N, P , RA n " P n
RA v(x) =
2, v(y) = 2 w(x) = 0, w(y) = 7
B B
B = N, P , RB P = {2, 7} RA = { 2, 3 , 3, 5 }
v(x) = 2, v(y) = 5 w(x) = 7, w(y) = 3
R(f(x)) (Q(g(x)) '" ŹR(f(g(y))))
R
" a " RR a
" a " QR a
" fR(a) = a2 gR(a) = |a|
R(f(x)) (" Q(y) (R(f(y)) x = y)
(R(x) Q(f(x))) '" (R(f(x)) Q(y))
(R(x) R(y)) (Q(f(x)) R(f(y)))
Q, +, �, 0, 1 R, +, �, 0, 1
N, +, 0 N, �, 1
P2, P2, Ą" P2 R2
P2, Ą" P3, Ą" P2 P3 R2
R, +, �, 0, 1 P (N), *", )", ", N
N, d", 0 Z, d", 0
 N, +, 0 {a, b}", �, �
N, d" Z, d"
G �G
" �G G
" �G P (n) P
G
" �G G
�G R
K1, K2, K3, K4
G
R(x, y) '" K1(x) '" K2(y) x y
(p r) '" (q s) '" (Źp (" Źs) (Źp (" Źq)
 ((p q) (" r) '" (Źp r)
(p q) (" (q r) (" (r p)
((p (" q) r) (p r) (" (q r)
(p q) '" (Źp r) (r Źq)
((Źp q) r) Ź(p q)
(((p q) r) Źp) Źq
((p q) p) q
p (" (Źp '" q) (" (Źp '" Źq)
(p q) (" (p Źq)
(p '" q r) (p '" Źr) Źq
q (" r (p (" q p (" r)
(p (" q (" r) '" (q (" (Źp '" s)) '" (Źs (" q (" r) q
{p Źq, q Źr, r Źp}
{p q, q r, r (" s "! Źq}
{Ź(Źq (" p), p (" Źr, q Źr}
{s q, p (" Źq, Ź(s '" p), s}
p q, q |= p
p '" q Źr, p |= r Źq
p q, p (q r) |= p r
p (q r), p q |= q r
(p q) r, Źp |= r
(p q) r, Źr |= p
(p q) r, Źq |= Źr
p q, r Źq |= r Źp
ą v
v(p) v(q) v(r)
v(p) = v(q) = v(r)

 (p '" q '" Źr) (" (p '" Źq '" r)
f : {0, 1}k {0, 1} ą
{p1, . . . , pk}
v v(ą) = f(v(p1), . . . , v(pk))
ą f
k
ą
(Źp '" ą) (" (Źq '" Źą '" r) (" (p '" Źq) (" (p '" q '" Źą)
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((r (Źq '" p)) ą) (ą '" (p q) '" r)
(ą p) (p q)
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p, q, r
ą Źą
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Źq (p q) '" r
ą
ą
p (" ą ą '" p
(ą p) '" (Źą q)
((ą '" q) Źp) ((ą p) Źq)
ą
(ą '" p) "! (p '" q) (ą (" p) "! (p (" r)
(q ą) "! (q (p '" r)) (ą q) "! (Ź(p (" r) q)
(p ą) "! (q (Źp (" r)) ((r q) p) "! (Źp Źą)
p ą1, . . . , ąn, �1 . . . , �m
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(p '" ą1) (" � � � (" (p '" ąn) (" (Źp '" �1) ("
� � � (" (Źp '" �m)
 p ą1, . . . , ąn, �1 . . . , �m
(ą1 '" � � � '" ąn) (" (�1 '" � � � '" �m)
(p("ą1)'"� � �'"(p("ąn)'"(Źp("�1)'"� � �'"(Źp("�m)
q1, . . . , qn-3 n e" 4
ą1, . . . , ąn (ą1 ("� � �("ąn)
(ą1 (" ą2 (" q1) '" (ą3 (" Źq1 (" q2) '" � � � '" (ąn-2 (" Źqn-4 ("
qn-3) '" (ąn-1 (" ąn (" Źqn-3)
� �
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P �
�
(ą (" � (" ł) ą, � ł
�
�
ą
� �ą pą = p (" ą (� ł)ą = �ą łą
|= � |= �ą
v (p) = v(p (" ą) v (�) =
v(� (" ą) �
G �G
" �G G
" �G P (n) P
G
" �G G
� � Ń
" � Ń Ń �
" Ń
� �
�(p) p
q �(p) �(q)
�(p) p q
�(p), �(q) |= p "! q
� p q
�(p) |= p "! �
P (x, y) x
y
W (x, y) x y
 N = N, MN , DN , ZN , JN MN DN
ZN JN
a, b, c " MN a � b = c
a, b, c " DN a + b = c
a " ZN a = 0
a " JN a = 1
N
v(x) = a, v(y) = b, v(z) = c
a2 + 2b2 = c - 3
a b c
a
a b
A
A = N, P , QA
A
a, b " P a + b e" 6
a, b " QA b = a + 2
"xP (x, y) "xQ(x, y)
"xP (x, y) "xQ(x, y)
"xP (x, y) "xQ(x, z)
v(y) = 7 v(z) = 1 u(y) = 3, u(z) = 2
"y("x(r(z, f(x, y)) r(z, y)))
"y("x(r(z, f(x, y))) r(z, y))
"x(Źr(x, y) "z(r(f(x, z), g(y)))
A = Z, fA, rA v(z) = 5 w(z) = 7
fA(m, n) = min(m, n) m, n " Z rA e"
fA(m, n) = m2 + n2 m, n " Z rA d"
fA(m, n) = 5mn m, n " Z rA
"y("z(r(z, x) '" r(z, y)) r(x, y))
 A = N, rA rA
A = N, rA rA
v(x) = 3 w(x) = 6 u(x) = 14
"x(Źr(x, y) "z(r(f(x, z), g(y)))) A = Q, fA, gA, rA
fA rA gA(q) = q + 1 q " Q
v(y) = 0 w(y) = -1 u(y) = 2
P (x, f(x)) "x"yP (f(y), x)
N, +, 0
N2, #, 0, 0 # a, b # c, d = a + c, b + d
Ł r, s " ŁR R, S " ŁR g " Ł2
1 2
� �
� A = A, RA, SA, rA, sA, gA
RA SA
� A = A, RA, SA, rA, sA, gA
sA rA � rA gA
"x"y"z(R(x, y) '" R(y, z) R(x, z)) '" "x"y"z(S(x, y) '" S(y, z) S(x, z))
'" "x"y"z((R(x, y) (" S(x, y)) '" (R(y, z) (" S(y, z)) '" ŹR(x, z) '" ŹS(x, z)))
"x(s(x) "! "y"z(x = g(y, z) '" r(y) '" r(z)))
Ł R, S
f � �
� A = A, RA, SA, fA
RA fA SA
� A = A, RA, SA, fA
SA RA f
Ł R, S
f � �
 � A = A, RA, SA, fA
RA f SA
� A = A, RA, SA, fA
SA f RA
Ł f " Ł1 �
A = A, fA
A = A, fA
A = A, fA f
A = A, fA f
Ł f " Ł1 �
N, s s
N, q q(n) = n2 + 1 n " N
N, s s
N, g g(n) = n2 mod 7 n " N
L f g
r �
A = A, fA, gA, rA
fA gA
fA
rA
L f g
r �
A = A, fA, gA, rA
fA gA
fA
rA
Ł r s
f
A = A, rA, sA, fA
 rA sA
rA sA
B, fB h : A, fA B, fB
rA
idA )" rA fA
"x"y(r(x, y) '" s(x, y) "z(r(x, z) '" s(z, y)))
ŹĄ"
["xr(x, x)] '" ["x"y(r(x, y) r(y, x))] '" ["x"y"z(r(x, y) '" r(y, z) r(x, z))]'"
'"["x"x "y"y (r(x, x ) '" r(y, y ) r(f(x, y), f(x y ))] rA
A, fA
"x"y(r(x, x) '" r(y, y) f(x, x) = f(y, y))
Ł R " ŁR r " ŁR f " Ł2 �
2 1
�
� A = A, RA, rA, fA
RA
� A = A, RA, rA, fA
RA rA fA
Ł S " ŁR P " ŁR f " Ł1
2 1
� �
A
� A = A, P , SA, fA

A A
SA ą" P � fA(P )
A
� A = A, P , SA, fA
SA fA
"x"y(S(x, y) P (x) '" "z(P (z) '" f(z) = y));
"x"y(f(x) = f(y) x = y) '" "x"y((f(x) = y S(y, x)) '" (S(y, x) f(x) = y)).
Ł S, P " ŁR f " Ł1 � �
1
A
� A = A, P , SA, fA

A A
SA )" P = " fA(SA) ą" P
A
� A = A, P , SA, fA
fA|SA fA SA
 Ł R S
f
g � �
� A = A, fA, gA, RA, SA
RA � SA f
RA )" SA
� A = A, fA, gA, RA, SA
f g ć% f ć%
"x"y(R(x) '" S(y) R(f(x, y)) '" S(f(x, y))
"x("y"z(x = f(y, z)) "! "y"z(x = g(f(y, z))))
Ł R S
f
g � �
� A = A, fA, gA, RA, SA
RA)"SA f
RA � SA
� A = A, fA, gA, RA, SA
f g
Ł R S
�1 �2 �3
�1 A = A, RA, SA
RA
�2 A = A, RA, SA
RA SA SA
RA
�3 A = A, RA, SA
SA RA
["xR(x, x)]'"["x"y(R(x, y) R(y, x))]'"["x"y"z(R(x, y)'"R(y, z) R(x, z))]'"
["x"y"z(ŹR(x, y)'"ŹR(x, z)'"ŹR(y, z))]'"["x"y"z"u(R(x, y)("R(x, z)("R(x, u)("
R(y, z) (" R(y, u) (" R(z, u)]
"x"y("z(R(x, z) '" S(z, y)) "! "z(S(x, z) '" R(z, y)))
"x"y(S(x, y) "! R(x, y) (" R(y, x))
 A, rA
rA = B � C B, C
A
A
SA P
SA
A
P SA
Ł r s
f
A = A, rA, sA, fA
rA sA rA )" sA
f rA *" sA
rA A A
rA fA A
A � A fA
"x"z"y(r(x, y) '" s(y, z) r(x, z) '" s(x, z))
"x("y"z(x = f(y, z)) "! "y(r(x, y) (" s(x, y)))
"x(Ź"yr(x, y) (" "y"z(r(x, y) '" r(x, z) '" Ź(y = z)))
"x"y r(x, y) '" "x"y"x "y (r(x, y) '" r(x , y )
"x "y (r(x , y ) '" f(x , y ) = f(f(x, y), f(x , y ))))
Ą"
1
N, d" {m - | m, n " N - {0}}, d"
n
N, + Z, +
N, d" Z, d"
Ł R
� �
� A = N, d" B = N - {0}, |
|
m|n n = k � m k " N - {0}
 � C = {a, b}", d"
w d" v v = w � u u " {a, b}"
D = {a, b}",
w v |w| d" |v|
�
"x"y(R(x, y) (" R(y, x))

�
"x"y(R(x, y) '" R(y, x) x = y)
Ł +
0 � �
� A = Z, +, 0 B = N, +, 0
� B = Z, +, 0 C = Q, +, 0
� {a, b}", �, � |= � N, +, 0 |= �
� {a, b}", �, � |= � {a, b, c}", �, � |= �
N N
{a, b}", �, � {a, b, c}", �, �
"y(r(x) r(y))
"y(r(x) '" r(y) r(f(x, y))
"x(q(x, y) q(f(x), y)
"x"y(r(x) '" r(y) r(f(x, y))
"x(r(x) (" p(x)) ("xr(x) (" "xp(x))
"x"y(q(x, y) (" "yŹq(x, y))
 "x"y"z(q(x, z) '" Źq(x, y))
"x"y(Ź(x = y) "z(P (x, z) '" P (y, z))
"x"y"z(Ź(y = z) '" P (x, y) '" P (x, z))
Ł P, Q " ŁR f " Ł1
1
"x"y(P (x)("Q(y)) "y(P (f(y))("Q(y))
"y(P (f(y)) ("Q(y)) "x"y(P (x)("Q(y))
C
C = N, P , QC, fC
N fC 7
C C
P QC P = QC = N
B B
B = N, P , QB, fB QB P = {0, 1, 2} fB
B, 0 |= P (x)
B, 0 |= "y(P (x) (" Q(y)) B, 4 |= P (f(y)) (" Q(y) 4 " QB
B
fB(4) = 5 " P
A
A = A, P , QA, fA A |= "y(P (f(y)) ("
Q(y)) "x"y(P (x) (" Q(y))
A
fA(a) " P
a " A A, fA(a) |= P (x) A, fA(a) |= "y(P (x)("Q(y))
A |= "x"y(P (x) (" Q(y))
A
fA(a) " P A, a |= P (f(y)) a " A
A |= "y(P (f(y)) (" Q(y))
A, a |= P (f(y))("Q(y)
a " A A, a |= P (f(y)) a " A
A, a |= Q(y) b " B A, a, b |= P (x) (" Q(y)
A, b |= "y(P (x) (" Q(y)) A |= "x"y(P (x) (" Q(y))
" (Q R) Q, "x(P (x) Q) R |= R
" "x(P (x) Q) Q |= Q
"
"
"
R(x) x M(x, y) x
y
"x(R(x) "yM(x, y))
"xR(x). "y"xM(x, y)
"x(R(x) "yM(x, y)), "xR(x) |= "y"xM(x, y).
A = A, RA, MA RA
a " RA A, a |= "yM(x, y)
b A, a, b |=
M(x, y) a, b " MA A, b |= "xM(x, y) b
A |= "y"xM(x, y)
K(x) x D(x) x W (x)
x
 K(x) x P (x) x D(x) x
�1
"x(K(x) D(x)) '" Ź"x(P (x) D(x)) "x(P (x) ŹK(x)).
A
A = N, KA, DA, P n
" x " KA x
" x " DA x
A
" x " P x
A |= "x(K(x) D(x))
A |= Ź"x(P (x) D(x))
A, 6 |= P (x) A, 6 |= ŹK(x) A |= "x(P (x) ŹK(x)) A |= �1
�2
"x(K(x) D(x)) '" Ź"x(P (x) D(x)) "x(P (x) '" ŹK(x)).
A
A = A, KA, DA, P
A |= "x(K(x) D(x)) A |= Ź"x(P (x) D(x)) A |= �2
A |= "x(K(x) D(x)) A |= Ź"x(P (x) D(x))
KA ą" DA a A, a |= P (x) D(x)
A
a " P a " DA KA ą" DA a " KA
A, a |= P (x) '" ŹK(x) A |= "x(P (x) '" ŹK(x))
A |= �2
K(x) x M(x) x D(x)
x
 ("xP (x) (" "x"yQ(x, y)) "x"y(P (x) (" Q(x, y))
"x"yP (x, y) "x"yP (x, y)
"x"yP (x, y) "y"xP (x, y)
("xR(x) "yS(y)) "x"y(R(x) S(y))
("xR(x) "yS(y)) "x(R(x) S(x))
("xP (x) Q(y)) "x(P (x) Q(y))
(P (x) "yQ(y)) "y(P (x) Q(y))
("xP (x) Q(y)) "! "x(P (x) Q(y))
("xP (x) Q(x)) "! "x(P (x) Q(x))
"y"z(P (y) Q(z)) ("yP (y) "zQ(z))
("yP (y) "zQ(z)) "y"z(P (y) Q(z))
A A A
A = N, P , QA P = QA = {13} A |= "yP (y) P = "

A |= "zQ(z) QA = N A |= "yP (y) "zQ(z)

A |= "y"z(P (y) Q(z)) A, v |= "z(P (y) Q(z)) v(y) = 7
"yP (y) "zQ(z) Ź"yP (y)(""zQ(z)
"yŹP (y) (" "zQ(z)
"y(ŹP (y) (" "zQ(z))
"y"z(ŹP (y) (" Q(z))
"y"z(P (y) Q(z))
"y"z(P (y) Q(z)) "y"z(P (y) Q(z))
R S
"x (R(x) (" S(x)) ("x R(x) (" "x S(x))
("x R(x) (" "x S(x)) "x (R(x) (" S(x))
"x(P (x) "yQ(y)) "x"y(P (x) Q(y))
"x("yQ(y) P (x)) "x"y(Q(y) P (x))
"x("yQ(y) P (x)) "x(Q(x) P (x))
" |= "x(P (x) "yQ(y)) "! "x(ŹP (x) (" "yQ(y))
" |= "x(ŹP (x) (" "yQ(y)) "! "x"y(ŹP (x) (" Q(y)) y ŹP (x)
" |= "x"y(ŹP (x) (" Q(y)) "! "x"y(P (x) Q(y))
A A
A = N, P , QA P = " QA =
{2, 3, 5, 7} A |= "yQ(y) QA = N A, {x 4} |= "yQ(y) P (x)

A |= "x("yQ(y) P (x))
a " N A, {x a, y 5} |= Q(y) P (x)
A, {x a} |= "y(Q(y) P (x)) A |= "x"y(Q(y) P (x))
A
A = A, P , QA A |= "x("yQ(y)
P (x)) A |= "x("yQ(y) P (x)) "x(Q(x) P (x))
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