{ } (comma) A B 7
A { , , , , , , } B { 2, 1, 0, 1, 2} B {0, 1, 1, 2, 2}
( ) B
()
C {2,5, 2, 3, 3, 2} C 3 2, 3 5
C {2, 3,5}
{ s , e
( “” )
{ 2, 1, 0, 1, 2} {0, 1, 1, 2, 2} C {2, 3,5} D {2,5, 2, 3} C D {a,
b, c, d, e} {a, e, i, o, u}
3 “...”
E 3 33 E {4,5,6, 7, 8, ..., 32}
E 6 E 29 9, 10, 11, 12, 31
(Finite Set)
“ X” n(X) n(A) 7 , n(B) 5 , n(C) n(D) 3 n(E) 29
(Null Set Empty Set) { }
n( ) 0
K
{ , 0, 1, {2, 3, 4}, {5, {6}},(7,8)} 6 , 0, 1, {2,3,4}, {5,{6}},
(7,8) .. 1 1 1
{(1,2),(2, 1), {1,2}, {2, 1}} 3 (1,2), (2,1) {1,2} .. 1,2 2,1 1,2
2,1
{2, 4, ...} 6, 8, 10, … 8, 16, 32, …
( ) (Infinite Set) F 2, G 0 1
F {1, 0, 1, 2, 3, ...} G
F G
G G { x | 0 x 1} < < “ x () 0 x 1< < ( x)” x
1.1
x x A { , , , , , , }
() 0 {0} 1
“” .. “” { } n( ) 0 n({ }) 1
x x 7 B { 2, 1, 0, 1, 2}
. 2Z { x | x 4}
2x x 4 x –4, –3, –2, –1, 0, 1, 2, 3, 4 2x Z {0, 1, 4,9, 16}
“”
“”
:
A , 2 B , 3 C , 0.5 G ,
2.5 B , 4 C , 1.5 G “” “” “2 B”, “3 C”, “2.5 B”,
() (Relative Universe) U “ () U U ”
H { x | x 2} > { 2, 1, 0, 0.5, 1, 2, 4.5, 7} U H {2, 4.5, 7} U H
{2, 3, 4, 5, 6, ...}
“” ( R )
“” ()
“X Y” X Y X Y X {1, 2} {0, 1, 2} 1 2 {0, 1, 2}
: () ()
A {m, p, r, w} A
{m} {p} {r} {w} {m, p} {m, r} {m, w} {p, r} {p, w} {r, w} {m, p, r}
{m, p, w} {m, r, w} {p, r, w} {m, p, r, w}
A 16 ( ) B B A 16
1. A 2. A A 3. n n2
.. A 16 42 16
“ A B A B ”
A B (A B B A)
(Proper Subset) n2 1 ( ) A A A A ( < )
(Power Set) A P(A)
A n P(A) n2 A {m, p, r, w} P(A) {m} {p} {r} {w} {m, p} {m, r} ...
{m, p, r, w} { , , , , , , , , } 4n(P(A)) 2 16
“B A” (B A ) “B P(A)” (B P(A) ) “A 16 ” “n(P(A)) 16 ”
1.2
. A {a}
12 2 {a} P(A) { , {a}}
. B {a, b}
22 4 , {a} , {b} {a,b} P(B) { , {a}, {b}, {a, b}}
{a,b, c} A “ a A b A c A ” 3
K
K
kanuay.com
19
. C {2, 3,5}
32 8 , {2} , {3} , {5} , {2, 3} , {2,5} , {3,5} {2, 3,5} P(C) { ,
{2}, {3}, {5}, {2, 3}, {2,5}, {3,5}, {2, 3,5}}
. D
1.3 E {6, 7} P(E) P(P(E))
P(E) {6} {7} {6, 7} { , , , }
P(P(E)) { } {{6}} {{7}} {{6, 7}} { , {6}} { , {7}} { , {6, 7}}
{{6}, {7}} { , , , , , , , , , {{6}, {6, 7}} {{7}, {6, 7}} { , {6},
{7}} { , {6}, {6, 7}} { , {7}, {6, 7}} , , , , , {{6}, {7}, {6, 7}}
{ , {6}, {7}, {6, 7}}, }
1.4 F { , 1, {2, 3}}
. 1 F .. 1 F .. ( 1 )
F .. F ..! () { } F .. { } F .. ( F) 2 F .. {2, 3} F .. ( 2 3 F)
{2, 3} F .. {{2, 3}} F .. ( {2, 3} F)
. 1 P(F) .. (1 P(F) ) {1} P(F) .. {1} F
P(F) .. F { } P(F) .. { } F ( F ) {2, 3} P(F) .. {2, 3} F {{2, 3}}
P(F) .. {{2, 3}} F
1.5 A, B A {1, 3,5, 7} B {1,2, 3, 4,5,6, 7}
. X X P(A)
1 X P(A) X A X 42 16
2 4 4 4 4 4 1 4 6 4 10 1 2 3 4
16
11
2 X , {1} , {3} , {5} , {7} , {1, 3} , {1,5} , {1, 7} , {3,5} , {3,
7} , {5, 7} 11
. Y A Y Y B
A Y 1, 3, 5, 7 Y () Y B 2, 4, 6 Y ( 1, 3, 5, 7 Y B )
2, 4, 6 Y {2, 4,6}
32 8
8 Y {1, 3,5, 7} , {1, 3,5, 7,2} , {1, 3,5, 7, 4} , {1, 3,5, 7,6} ,
{1, 3,5, 7,2, 4} , {1, 3,5, 7,2,6} , {1, 3,5, 7, 4,6} {1, 3,5, 7,2,
4,6}
. (1) (1.1) {a, {b, c}} (1.9) {a, {b, c}} (1.2) a {a, {b, c}}
(1.10) a {a, {b, c}} (1.3) b {a, {b, c}} (1.11) b {a, {b, c}} (1.4)
{a} {a, {b, c}} (1.12) {a} {a, {b, c}} (1.5) {b} {a, {b, c}} (1.13)
{b} {a, {b, c}} (1.6) {b, c} {a, {b, c}} (1.14) {b, c} {a, {b, c}}
(1.7) {{b, c}} {a, {b, c}} (1.15) {{b, c}} {a, {b, c}} (1.8) {a,
{b, c}} {a, {b, c}} (1.16) {a, {b, c}} {a, {b, c}} (2) (2.1) { , 0,
1, {1}, {0, 1}} (2.7) { , 0, 1, {1}, {0, 1}} (2.2) {0} { , 0, 1,
{1}, {0, 1}} (2.8) {0} { , 0, 1, {1}, {0, 1}} (2.3) {1} { , 0, 1,
{1}, {0, 1}} (2.9) {1} { , 0, 1, {1}, {0, 1}} (2.4) {0, 1} { , 0,
1, {1}, {0, 1}} (2.10) {0, 1} { , 0, 1, {1}, {0, 1}} (2.5) {0, {1}}
{ , 0, 1, {1}, {0, 1}} (2.11) {0, {1}} { , 0, 1, {1}, {0, 1}} (2.6)
{{0, 1}} { , 0, 1, {1}, {0, 1}} (2.12) {{0, 1}} { , 0, 1, {1}, {0,
1}}
kanuay.com
21
(3) A {{ }, a, b, {a}, {a, b}} (3.1) { } A (3.3) {{a}, b} A (3.2) {
} A (3.4) {a, b} A {a, b} A (4) A { , a, {b}, {a, b}} (4.1) P(A)
(4.6) a P(A) (4.2) { } P(A) (4.7) {a} P(A) (4.3) P(A) (4.8) {b}
P(A) (4.4) { } P(A) (4.9) {{b}} P(A) (4.5) { , a, {b}} P(A) (4.10)
{ , a, {b}} P(A) (5) A { , 1, 2, 3, {1}, {1, 2}, {1, 2, 3}} (5.1) {
, {1}, {1, 2}} P(A) (5.3) {{1}, {2}, {3}} P(A) (5.2) { , {1}, {1,
2}} P(A) (5.4) {{1}, {2}, {3}} P(A) (6) B { , {0}, { }} P(B) ( )
(6.1) _____ B (6.5) {0} _____ B
(6.2) _____ P(B) (6.6) {0} _____ P(B) (6.3) { } _____ B (6.4) { }
_____ P(B)
(7) (7.1) (7.5) P ( ) (7.2) (7.6) P ( ) (7.3) { } (7.7) { } P (
)
(7.4) { } (7.8) { } P ( ) (8) (8.1) n(A) 5 A 32 (8.2) n(A) 5 A 32
(8.3) n(A) 5 A 32
(8.4) n(A) 5 A 32 (9) A 511 A 511 5 *
* .4 .6 “”
Math E-Book Release 2.7pre
n n!(n r)! r!r
x! x! 1 2 3 ... x 7 3
5 6
1 2 3
35
(10) S {1, 2, 3, 4, 5, 6, 7} n(X) n(Y) X { A P(S) | 1 A 7 A } Y { A
X | A 6 }
. -
(Venn-Euler Diagram) U ( ) A, B, C
A B A B A B ( disjoint sets)
K
kanuay.com
23
U
0 2 4
1.6 {0, 1,2, 3, 4, ..., 11}U A 5, B 9 C A, B A, B, C
A {0, 1,2, 3, 4} , B {1, 3,5, 7, 9} C {2, 3,5, 7, 11} A, B A, B,
C
, , , , , , 4
1. (Union: ) A B A B ( “A B”)
A B B
2. (Intersection: ) A B A B ( “A B”)
A B A B A
U
U
B
A
A B { x | x A x B } A B { x | x A x B } A B A cup B A B A cap B AB
A B ()
3. (Complement: ' ) A' U A cA A A ' { x | x A }
4. (Difference Relative Complement: ) B A B A B A “ A B” B
B A B A' B A { x | x B x A } n(B A) n(B) n(A) n(B A) n(B) n(A
B)
n(B) 9 n(A) 4 n(B A) 9 4 5 A B n(A B) n(A B) 3 n(B A) 9 3 6 n(A B)
4 3 1
kanuay.com
25
U
1.7 A {2, 3,5, 7} B { , 1,2, {3, 4},5} { , 1,2, 3, 4,5,6, 7, {1,2},
{3, 4},(5,6)} U A B , A B , A ' , B' , B A A B
A B { , 1,2, 3,5, 7, {3, 4}} A B {2,5} A ' { , 1, 4,6, {1,2}, {3,
4},(5,6)} B' {3, 4,6, 7, {1,2},(5,6)} B A { , 1, {3, 4}} A B {3,
7}
1.8
. A B
. (A C) B
A C +++++ B ++ ++
. C (A B)'
C +++ A B +++++ (A B)' + ( C (A B) )
. (A C') B
A C ' A C B ++++
1.
(A B) (B A) (A B)
.. (A B) (B A) (A B) A B
. [ A (A B)] ' [(A C) C ] '
A (A B) A (A B) A (A C) C (A C) C C .. [A] ' [C] ' A' (C ') ' A' C
C A
C A
1.10 (A B' C ') (A' B C ') (A B' C) (A' B C)
[(A B' C ') (A B' C)] [(A' B C ') (A' B C)] () [ A B' (C ' C)] [ A'
B (C ' C)]
[ A B' ] [ A' B ] U U [ A B'] [ A' B ] ..
(A B') (A' B) .. (A B) (B A) “” A B' C ' “ A B C” A' B C ' A B' C
A' B C +++
. n(P(C))
. n(P(C) C)
*** n(P(C) C) 64 6 58 C P(C) n(P(C) C) C P(C)
“” ( )
{ } “” .. C ( C) “” C
0 “” .. ()
{{ }, 0} “” C () “” { } C 0 C
{ , {0}} “” .. {0} C
{{ , {0}}} “” .. { , {0}} C C ()
C P(C) 4 n(P(C) C) 64 4 60
. n(C P(C))
n(C P(C)) 6 64 P(C) C . 4 n(C P(C)) 4 n(C P(C)) 6 4 2
( .) n(C) 6 n(P(C)) 64 . n(C P(C)) 4 (.) n(C P(C)) 2 (.) n(P(C) C)
60
. n [(P(C) C) (C P(C))]
. . () 60 2 62 ( )
1.12 A, B A {1, 3,5, 7} B {1,2, 3, 4,5,6, 7} ( . . . )
. Y A Y Y B
Math E-Book Release 2.7pre
28
1.5 (A Y B ) A Y 1, 3, 5, 7 Y ( A Y ) {1, 3,5, 7} 42 1 15
Y B 2, 4, 6 Y ( 1, 3, 5, 7 Y B ) 32 8
15 8 120 . Z {1,2, 3} Z Z A
. ... {1,2, 3} Z 1, 3 Z ( {1, 2, 3} Z ) 2 Z ( Z A ) 22 1 3
Z A 5, 7 Z ( 1, 3 Z A ) 22 4
3 4 12 . Z {1,2, 3} Z Z A
{1,2, 3} Z 1, 2, 3 Z Z A 5, 7 Z ( Z Z A )
22 4
. (11) A, B A B A B x A y B (11.1) {x} B (11.3) {A} {B} (11.2) {y}
A (11.4) {A} {B} (12) (12.1) A B B C A C
kanuay.com
29
(12.2) A B B C A C (12.3) A B B C A C
(13) A (13.1) { x | x A } A (13.3) { x | {x} A } {A} (13.2) { x | x
A } {A} (13.4) { x | {x} } (14) A B {0, 1, 2, 3, 4, 5} A B {1, 3,
5} B C {2, 3, 5} A C {0, 1, 2, 3, 5} A C {0, 3, 5} . A B ' {0} . B
C ' {1} . A C ' {1} . B A ' {2, 4} (15) C' B'
{ x | 1 x 10 } < <U I I B { x | x 3 } C { x | x 5} < (16)
A {0, 1} B {0, {1}, {0, 1}} (16.1) A P (B) (16.2) {1} P (A) P (B)
(16.3) n(P (A B)) n(P (A B)) (17) (17.1) ' U (17.7) A A ' (17.2) '
U (17.8) A A ' U (17.3) A (A B) (17.9) A U A A ' U (17.4) B (A B)
(17.10) A A A
(17.5) (A B) A (17.11) A A (17.6) (A B) B (17.12) A B A B '
(18) (18.1) A B P (A) P (B) (18.2) A B A B (18.3) A B A B (18.4) A
B B C B A ' C ' U (18.5) A B B C A C (19) A, B (19.1) A B A B
(19.5) x A x A B (19.2) A B B A (19.6) x A x A ' B '
(19.3) A B A B ' (19.7) x A x A ' B ' (19.4) (A B) ' B ' A (19.8) x
A x (A ' B ') '
Math E-Book Release 2.7pre
30
(20) (20.1) A (A B) (20.6) (A B) B (20.2) (A B) B (20.7) (A B) B
(20.3) (A B) B (20.8) A (A B) (20.4) A (A B) (20.9) (A B) (B A')
(20.5) A (A B) (21) (21.1) A C B C A B (21.2) A C B C A B (21.3) A
C B C A B (21.4) A' B' A B
(22) A B A 3 (23) (23.1) (A B) (B A) (A B) (23.2) [A (A' B)] [B (B'
A')] (23.3) [(A B) (B A)] A' A' [(A B) (B A)]
(23.4) [(A B)' (B C ')] [(D E) (C ' E')] (A E') ' (24) (24.1) (A B
C) (A ' B C) (B' C ') U (24.2) (A B C D') (A ' C) (B' C) (C D) C
(24.3) P (A B) P (A B) (24.4) P (A B) P (B A) { } (24.5) A B P (A
B) P (A) P (B) (25) A {0, 1, 2, 3} , B {{0}, 1, 2, {3}} C {0, {1},
{2}, 3} (25.1) P(A) P(B) P(C ') { , {1}, {2}, {1, 2}} (25.2) P(A)
P(B') P(C) { , {0}, {3}, {0, 3}} (25.3) P(A') P(B) P(C) { , {0}}
(25.4) P(A) P(B') P(C ') { } (26) n( ) 35U , n(A) 22 , n(B) 18 n(A'
B')
A B A B ( A B A B A B )
K
kanuay.com
31
(27) n(A) a , n(B) b , n(C) c , n(D) d n(A B) b , n(B C) c n(C D) d
n(A B C D) n(A B C D) (28) A,B, C P (C) { , {a}, {c}, C} n(P (A)) 8
, n(P (B)) 16 C A , C B , {b, d, e} A B , b A B '
. d (A B ') ' . e (C B ') ' . b (A ' B ') ' . {b, e} (A ' B)
'
(29) A { , 1, {1}} A B ' (29.1) n [ P (A) P (B) ] 8 (29.3) P (A B)
{ } (29.2) {1} P (A B) (29.4) P (B A) { } (30) A { , { }, 0, {0},
{1}, {0, 1}} [ P (A) A ] [ A P (A) ] (31) A (31.1) A B {1, 2, 3, 4,
5} B {1, 3, 5} (31.2) A B {1, 2, 3, ..., 15} B {2, 4, 6, 8, 10}
(32) A {1, 2, 3, 4, 5, 6, 7} B {1, 2, 3} X (32.1) B X A (32.2) X A
B X (33) B A n(A) 10 , n(B) 4 n(C) (33.1) C { S | B S A } (33.2) C
{ S A | S B } (34) A {0, 2, 4, 6, 8} B {0, 1, 2} C {1, 2, 3} D {0,
2, 3} X X A (34.1) B C ' X (34.3) B D X (34.2) B C ' X (34.4) B D X
(35) {1, 2, 3, 4, ..., 8}U A {1} U B {2, 4, 6} C {1, 7} D (B' C) D
A
Math E-Book Release 2.7pre
= + –
= + + – – – +
= –
(36) { x | 2 x 6} < <U I I 2A { k | k } U B { k | k } U
C { X | A B X X A B } (37) A {a, b, c, d, f} B {a, c, d, e} X X A B
A B X (38) A {1, 3, 5, 7, 9} kS { B A | n(B) k } n(S) 1 4 52 3S S S
S S S (39) A, B U n( ) 100U , n(A') 40 , n(B) 55 n(A B') 32 n(A'
B')
.
-
n(B A) n(B) n(A B) ” -
n(B A) n(B) n(A B)
A B
2 n(A B) n(A) n(B) n(A B) 3 n(A B C) n(A) n(B) n(C) n(A B) n(A C)
n(B C) n(A B C)
2–3
x 80 (30 18 20) 12
kanuay.com
35
A
x y B
1.17 A, B, C n(A B) 92 , n(A C) 79 , n(B C) 75 , n(A B C) 32 , n((A
B) C) 18 , n((A C) B) 6 n((B C) A) 2 n(A B C)
n(A B C) m z m 92 , y m 79 , x m 75
m (m 75) (m 79) (m 92) 18 32 6 2 m
m n(A B C) 94
. (40) 300 100 100 150 (41) 50 32 6 1 (42) 80 35 27 32 (43) 20 ( )
17 15 (44) 20 7 (45) 34 5 1 3
Math E-Book Release 2.7pre
kanuay.com
37
Math E-Book Release 2.7pre
38
() (1) (1.2), (1.6), (1.9), (1.12), (1.15), (1.16) (2) (2.2), (2.5)
(2.6) (3) (3.1), (3.3) (4) (4.6), (4.8) (4.10) (5) (5.3) (6) P(B) ,
{ }, {{0}}, { {{ }}, { , {0}}, { , { }}, {{0}, { }}, { , {0}, { }}
} (6.1) (6.4) (6.5) (6.6) (7) (7.1), (7.7) (8) (8.1), (8.4) (9) 9 ,
126 (10) 32, 6 (11) (11.1), (11.4) (12) (12.1) (13) (13.3) (14) .
(15) {1,2, 4,5,6, 7,8,9, 10} (16) , , 16–2=14 (17) (18) (18.3),
(18.5) (19) (19.3), (19.4) (19.6)
(20.1) A B (20.2) A B (20.3) (20.4) A B (20.5) A (20.6) A B (20.7)
(20.8) A B (20.9) (21) (21.4) (22) A B A B (23.1) A B (23.2) B
(23.3) B ' (23.4) (A E)' (24) (25) (25.3) (26) 13 (27) d, a (28) .
(29) (29.4) (30) 61+3=64 (31.1) 8 (31.2) 32 (32.1) 16 (32.2) 112
(33.1) 64 (33.2) 960
(34.1) 16 (34.2) 16 (34.3) 8 (34.4) 24 (35) 16 (36) 4 (37) 56 (38)
31 (39) 13 (40) 50 (41) 11 (42) 66 (43) 6 (44) 11 (45) 13 (46) 9
(47) 130 (48) 6 (49) 385 (50) 26, 12, 24 (51) 15 (52) 50, 95, 40,
230, 115, 255 (53) 7, 19 (54) 20 (55) 13 (56) . (57) 22–13=9 (58)
26
kanuay.com
39
() (1) 2 a {b, c} “” () 8 (1.2) (1.6)
4 , {a} , {{b, c}} {a, {b, c}} “” ( ) 8 (1.9) (1.12) (1.15)
(1.16)
(1.1) (1.3) b ( {b, c} ) (1.4) (1.5) (1.7) (1.8) 1.3 (1.10) (1.11)
(1.13) 1.3 (1.14) b c
(2.1) ( 5 ) (2.2) {0} (2.3) {1} () (2.4) {0, 1} () (2.5) (2.6) ..
2.2
(2.7) (2.8) “0 ” (2.9) “1 ” (2.10) “ 0 1 ” (2.11) “ 0 {1} ” (2.12)
“ {0, 1} ”
(3.1) { } A ( A 5 { } ) (3.2) { } A A ( A ) (3.3) {{a}, b} A {a} A
b A (3.4) .. {a, b} A ( A ) {a, b} A a A b A {a, b} A
(4.1) .. P(A) A (4.2) .. { } P(A) { } A A A (4.3) ! .. (4.4) .. { }
P(A) P(A) 4.1 (4.5) .. { , a, {b}} P(A) { , a, {b}} A A a A {b}
A
(4.6) .. P(A) (4.7) .. {a} P(A) {a} A a A (4.8) .. {b} P(A) {b} A b
A (4.9) .. 4.8 {b} A (4.10) .. { , a, {b}} P(A) P(A) a P(A) {b}
P(A) ( 4.1, 4.6, 4.8)
Math E-Book Release 2.7pre
40
(5.1) { , {1}, {1, 2}} P(A) { , {1}, {1, 2}} A A {1} A {1, 2} A ..
(5.2) { , {1}, {1, 2}} P(A) P(A) ( A ) {1} P(A) ( {1} A 1 A ) {1,2}
P(A) ( {1,2} A 1 A, 2 A ) .. (5.3) {{1}, {2}, {3}} P(A) {{1}, {2},
{3}} A {1} A {2} A {3} A .. (5.4) {{1}, {2}, {3}} P(A) {1} P(A),
{2} P(A), {3} P(A) {1} A, {2} A, {3} A 1 A, 2 A, 3 A ..
(6) P(B) , { }, {{0}}, {{ }}, { , {0}}, { { , { }}, {{0}, { }}, { ,
{0}, { }} }
(6.1) “ B ” .. B “ B ” .. ( B ) (6.2) “ P(B) ” .. B B “ P(B) ”
..
(6.3) “ { } B ” .. B { } () “ { } B ” .. B B (6.4) “ { } P(B) ” ..
{ } B B “ { } P(B) ” .. P(B) B
(6.5) “ {0} B ” .. B {0} () “ {0} B ” .. 0 B (6.6) “ {0} P(B) ” ..
{0} B “ {0} P(B) ” .. 0 P(B) 0 B
(7.1) .. (7.2) .. 02 1 () .. “ () ”
(7.3) .. ( “”) (7.4) .. (7.2)
(7.5) .. P( ) ( 7.2 ) (7.6) .. (7.7) .. { } P( ) { } ( 7.1 ) (7.8)
.. { } P( ) P( ) ( 7.2 )
(8.1) .. A n(A) 52 2 32 (8.2) .. () 31 (8.3) .. P(A) 1 P(A) 32
(8.4) .. 8.3
(9) n(A)2 512 n(A) 9
A 5 9
9! 1265! 4! () ( “”)
kanuay.com
41
B
C
A
B
(10) X “ A ” (1) A P(S) .. A S (2) 1 A 7 A .. 1 A 7 A ( A ) 5 2, 3,
4, 5, 6 {2,3,4,5,6} 5n(X) 2 32
Y A ( 32 X) 6 .. {1}, {1,2}, {1,3}, {1,4}, {1,5} {1,2,3} 6 n(Y)
6
(11.1) x A x B (11.2) y B , y A () (11.3) {A} {B} A {B} {B} B
(11.4) A B {A} {B}
(12.1) ( ) (12.2) .. B {A}, C {B} C {{A}} A C (12.3) .. A, B, C A
B, B C A C
(13.1) (13.2) () (13.3) {x} A x A (13.1) A (13.4) {x} x x
(14) A B, A C, B C A B C {3, 5} 0, 1, 2
A C {0, 1, 2, 3, 5} A C 4 B
. B C ' {1, 4}
(15) {1,2, 3, ..., 10}U B {3,6,9} C {1, 2, 3, 4, 5} .. C ' B' (C B)
' C B {3} {1,2, 4,5,6, 7,8,9, 10}
(16.1) A P(B) A B A B 1 B
(16.2) P(A) P(B) P(A B) P({0}) { , {0}} {1} { , {0}}
(16.3) A B {0, 1, {1}, {0, 1}} 4n(P(A B)) 2 16 A B {0} 1n(P(A B)) 2
2 .. 16 2 14
A
C
B
(17.1) (17.2) .. U (complement) (17.3) (17.6) (17.7) (17.11) A A
(17.8) (17.10) (17.12) (!)
17.3 17.12
(18.1) () (18.2) .. A B (18.3) .. A B A B .. A B (18.4) A B A B B C
B B C (B C ) A ' C ' (A C)' ' U .. ( A C ) (18.5) A B A B B C B C A
C A C .. “ A B B C A C ” .. A C
(19.1) .. A B A B A B ( A, B ) (19.2) .. A B A B B A (19.3) .. A B'
A (B') ' A B (19.4) .. B' A B' A' (B A)'
(19.5) .. A B x B (19.6) “ x A x (A B) ' ” x A x A B x A x (A B) '
.. (19.7) “ x A x (A B) ' ” .. x B (19.8) “ x A x A B ” .. x
B
(20) ( )
A B A B B A
(20.1) A (A B) A B (20.2) (A B) B A B (20.3) (A B) B (20.4) A (A B)
A B (20.5) A (A B) A (20.6) (A B) B A B (20.7) (A B) B (20.8) A (A
B) A B (20.9) B A' B A (A B) (B A ')
(21.1) .. C U A B (21.2) .. C (21.3) .. C U (21.4)
(22) A B A, B (A B )
kanuay.com
43
A E ( (A E))' (A E)'
(24.1) B C 2 [(A A') B C] (B C)'
U
(B C) (B C)' U .. (24.2) C C [(A B D') A' B' D] A, B, D C [(A B D')
(A B D ')' ] C
U
..
(24.3) .. (A B) (A B) P( ) P( ) (24.4) .. A B B A P(A B) P(B A)
(24.5) A B P(A) P(B) P(A) P(B) P(B) ......(1) A B A B B P(A B) P(B)
.....(2) .. (1) (2)
(25) P( ) P( ) P( ) P( ) (25.1) .. A B C ' {1, 2}
(25.2) .. A B ' C {0, 3} (25.3) .. A ' B C P(A ' B C) { } (25.4) ..
A B ' C '
(26) n(A' B ') n(A B)' n(A B) .. B A n(A B) n(A) 22 n(A B)' 35 22
13
(27) n(A) a, n(B) b n(A B) b B A (B A ) .. D C B A n(A B C D) n(D)
d n(A B C D) n(A) a
(28) P(C) C {a, c} n(P(A)) 8 n(A) 3 n(P(B)) 16 n(B) 4 C A C B A {a,
c, } B {a, c, , } {b, d, e} A B b A B ' A {a, c, b} B {a, c, d, e}
. d A' B ( B A) .. . e C ' B ( B C) .. . b A B .. . .. (A' B)' A B'
A B {b}
(29) A B ' A B A B ( A B A ) (29.1) 3n[P(A B)] n[P(A)] 2 8 ..
(29.2) {1} P(A B) {1} A B 1 A B ( 1 A ) .. (29.3) P(A B) P( ) { }
.. (29.4) B A
Math E-Book Release 2.7pre
32 55
(30) A P(A) 3 , { } {0} .. 6n(P(A) A) 2 3 61 n(A P(A)) 6 3 3 61 3
64 P(A) A A P(A) ()
(31.1) A {2, 4, {1,3,5} } 32 8 (31.2) A {1, 3, 5, 7, 9, 11, 12, 13,
14, 15, {2,4,6,8,10} } 52 32
(32.1) X {1, 2, 3, {4,5,6,7} } 42 16 (32.2) X { {1,2,3} , {4,5,6,7}
} 3 4(2 1) (2 ) 112
(33.1) n(C) 6S 2 64 (33.2) n(C) S 4 6(2 1) (2 ) 960 ( 32)
(34.1) {0} X {0, 2, 4, 6, 8} X 42 16 (34.2) {0} X X {0, 2, 4, 6, 8}
.. X {0, 2, 4, 6, 8} {0} X ( 34.1) 5 42 2 16 (34.3) {0, 2} X {0, 2,
4, 6, 8} X 32 8 (34.4) {0, 2} X X {0, 2, 4, 6, 8} .. X {0, 2, 4, 6,
8} {0,2} X ( 34.3) 5 32 2 24
32.2 33.2 34.2 34.4
(35) {3, 5, 8} D {2, 3, 4, ..., 8} D 42 16
(36) { 2, 1, 0, 1, 2, ..., 6} U A {0, 1, 4} ( 2k U ) B {0, 1, 2}
{0, 1} X {0, 1,2, 4} .. n(C) X 22 4
(37) X {a, b, c, d, e, f} {a, c, d} X X { {a,c,d} , {b,e,f} } 3 3(2
1) (2 ) 56
(38) n(A) 5 1S {B | B A, n(B) 1} , 2S {B | B A, n(B) 2} , ... 3S {B
| B A, n(B) 3 } 1 2 3 4 5S S S S S S = { A } .. 5n(S) 2 1 31
(39) n(A B') n(A B) 32 n(B) 55
n(A' B') n(A B)' n( ) 100 U .. 100 32 55 13
n(A ') 40
(40) A B n(A) 100 n(B) 150 .. “ 100 ” n(A B) 300 100 200
n(A B) n(A) n(B) n(A B) 200 100 150 n(A B)
n(A B) 50 50
(41) .. x 50 32 6 1 11 11
(42) “ 35 ”
35 “ 27 ”
27 32 32 3 2 94
80 ( 80) 14 n(A' B') n(A B)' 80 14 66
.. n(A B) 80 32 48 48 35 27 n(A B) n(A B) 14 n(A B)' 80 14 66
(43) “ ” ()
20 , 17 15 2( ) 20 17 15 52 26
() 26 ( ) 26 20 6
(44) 20 , 2( ) 7
2 20 .....(1) 2 7 .....(2) 2 (1 ) (2) ; 3 3 33 11
(45) “ ” ( “” “”, “” “” )
34 .....(1) 5 .....(2) 1 .....(3) 3 .....(4)
(2), (3) (1) (4) 5 ( 5 1) 34 5 3 7 , 9 1 7 5 1 13
Math E-Book Release 2.7pre
100 60
24 11
600 500
(46) ( 43 ) “ ” ()
7 , 6 5 2( ) 18 .. 9
(47) 30 .. 50 30 20 35 .. 60 35 25 55 .. 80 55 25 3 2 120 2
70
100 20 60 10 .. 160 130
1. 2. “” “”
(48) 16 .....(1) 7 21 7 14 .....(2) 2 30 24 .. 30 24 6
(49) 200 30 2 230 15 215 .. 600 215 385
48 49 , 47)
(50) 4 ! ( , 0 )
6 8 3 ( 5 ) 2 4 18 4 14
( ) ( ) 4 2 6 14 26 ( ) 4 2 6 12 3 4 26 2 24
kanuay.com
47
14 4 B
(51) A, B, C , , .. n(A) 140 , n(B) 195 , n(C) 155 , n(A B) 50 ,
n(B C) 45 , n(A C) 60 n(A B C) 20
3 n(A B C) 140 195 155 50 45 60 20 355 355
370 15
n[(A B C)' ] n( ) n(A B C) U 370 355 15
(52) ( 1.14 1) 50 70 25 95 40 120 25 70 15 230 n(A') n( ) n(A) 370
140 230 U 30 40 25 20 115 50 120 70 15 255
(53) A, B, C ,
42 1 n(A B C) 42 3 .. 42 29 22 21 10 12 15 n(A B C)
n(A B C) 7 7
14 4 1 19
(54) 3 180 95 92 125 52 43 57 x x 20 y n(A C) 20 43 20 23 z n(A B)
20 52 20 32 .. 95 20 23 32 20
(55) 3 , , n(A B C)' 100 n(A B C)
.. 3 39 n(A B C)
32 35 20 87
Math E-Book Release 2.7pre
4
(56) x, y x y 20 23 22 11 9 100 (x 20 23 22) (y 20 23 11) 6 x 5, y
10
70 , 64 65 . . 70 64 65 199 . ( 5 ) . 5 10 9 24
(57) 3 (, , ) 3 n(A B C) 60 8 20 15 22 6 10 11 x .. x 22 () 40 4 15
13 18 8 6 9 y .. y 13 x y 9
(58) {0, 1, 2, ..., 100}U A { x | x 2 } B { x | x 3 } C { x | x 5
}
n(A' B' C ') n(A B C)' n( ) n(A B C) U n(A B C) 3
n(A) 2 .. 51 n(B) 3 .. 34 n(C) 5 .. 21
n(A B) 2 3 6 .. 17 n(A C) 2 5 10 .. 11 n(B C) 3 5 15 .. 7 n(A B C)
2 3 5 30 .. 4
51 34 21 17 11 7 4n(A B C) 75 n( ) 101U n(A' B' C ') 101 75
26
(58) “A 6 B 8 ” A B ?
.. “ 6 8 ” “ 48 ” ! 6 8 !
. . . “ 24 ”
K
“” ( )
{..., 3, 2, 1, 0, 1, 2, 3, ...} I
( 0) (Rational Number) Q
ab{ | a,b Q I b 0 }
( N I ) ( I Q )
R ea
2 1.41421.. 3 1.73205.. 3.14159.. 3 2 1.25992..
(Real Number) ( ) R
1. (, ) 3 “”
2. 0 ( 0 ) ( “”)
Q I
2.1 ( A, B, C, D)
A B C Q D I
. N A
B C D
. A
2 2 0 B 2 2 2 C D
. { x | x 0 }< A
B C D () 2
Math E-Book Release 2.7pre
. 22{1.414, }7 A B
C 1.414 2 227 () D
. { 1, 0, 1} A 1 1 2
B C D
. { 10 x | x } I {0, 10, 20, 30, ...}
A B C D
.
(Identity) a a e
a e e a a a 0 0 a a .. 0 a 1 1 a a .. 1
(Inverse) a a i
a i i a e a ( a) ( a) a 0 .. a –a 1 1a aa ( ) ( ) a 1 .. a 1a
1a ( “a ” “a ”)
0 0 1
.
a e a a e 2 a .. e 2 e a a e a 2 a .. e 2 2
. a
2 a i 2 a i 2 2 .. i 4 a ( i a 2 i 4 a ) a 4 a
2.3 x y x y 3
.
a e a a e 3 a .. e 3 e a a e a 3 a .. e 2a 3 “”
. a
a a [2] (Symmetric Property)
a b b a [3] (Transitive Property)
a b b c a c [4]
a b a c b c a b ac bc
(, )
K
[2] a –a a ( 0) 1a
[3] (Closure Property) a b a+b a b ab
[4] (Commutative Property) a b b a ab ba
[5] (Associative Property) a (b c) (a b) c ( a b c ) a(bc) (ab)c (
abc)
[6] (Distributive Property) a(b c) ab ac (a b)c ac bc
[7] “ a 0 a R a R ”
[1]
() 0 0
[1] (b ) 1a b a b ( )
“ a b c c a bc ”
[2] (a ) 1a 0
[3] (a, b ) 1a bab
[4] (b, c ) a acb bc
[5] (b, c ) a d ac bdcb bc a d adcb bc
[6] (b, c, d ) a/b ac bc a acb/c b a/b adc/d bc
1. ()
a b a c b c a c b c a b
2. () () 0
a b ac bc a b a bc c c 0
3. 0 ()
ac bc a b c 0
4. 2
a b 2 2a b 2 2a b “ a b a b ”
.
56
(2) a, b, c R (2.1) a b a b 1 (2.2) a b 0 a 0 b 0 (2.3) b 0 a cb b
a c (2.4) b, c 0 a acb b c
(3) . . 2 . . 4 (4) (4.1) (4.2) (4.3) 0 (4.4) 0 (5) A { x | x } N Q
B A N (5.1) A B (5.2) A B (6) A “ x A y A xy 1 xy A ” . 0 . . . 0
(7)
1 6 5
6 5
(8) . (a b) a c . (b c) b a . (a b) (c b) b . (c a) (b a) b
* a b c a a b c
b b c a
c c a b
K
kanuay.com
57
(9) . x y 3 xy (x y) . x y 2(x y) 3 x y . 3 1xy x yx y . 1x yx y 2
xy (10) a b 3ab (a b) x (y z) (z y) x (11) A A a b2a b ab2a b
(11.1) A (11.2) A
.
(Polynomial) x p(x)
.
kanuay.com
59
(Remainder Theorem) “ p(x) x c p(c)”
3 2p(x) 4x 2x x 3 x 3 3 2p(3) 4(3) 2(3) (3) 3 96 x 2
3 2p( 2) 4( 2) 2( 2) ( 2) 3 39
1 0 ( x )
2.5
. 3 22x x 6x 1 x 2 3 22x x 6x 1 x 2 3 22(2) (2) 6(2) 1 1
. 3 22x x 6x 1 x 1 3 22x x 6x 1 x 1 3 22( 1) ( 1) 6( 1) 1 4
p(x) 2Ax Bx C 3 3 3
“ x 3” p(0) 3 2A(0) B(0) C 3 “ x 1 12” p(1) 12 2A(1) B(1) C 12 “ x
2 25” p(2) 25 2A(2) B(2) C 25
A 2 , B 7 , C 3 ... 2p(x) 2x 7x 3 p(x) x 3 22(3) 7(3) 3 42
. p(x) x c c
1 p(x) x c ... x c 2p(x) 2x 7x 3 (2x 1)(x 3) p(x) x c c –1/2 c
–3
2 p(x) x c ... p(c) 0 () 22c 7c 3 (2c 1)(c 3) 0 c –1/2 c –3
. p(x) x c 7 c
1 “p(x) x c 7” p(c) 7 22c 7c 3 7 22c 7c 4 (2c 1)(c 4) 0 c 1/2 c
–4
2 “p(x) x c 7” “p(x) 7 x c ” ( 38 5 3 38 3 5 ) 2p(x) 7 2x 7x 4 (2x
1)(x 4) c 1/2 c –4
0 x c p(x)
“ p(x) x c p(c) = 0” (Factor Theorem)
p(x) k p(k) = 0 x k x k p(x)
2.7 3 22x x 25x 12
3 2p(3) 2(3) (3) 25(3) 12 0 x 3
x 3
23 22x x 25x 12 2x 7x 4x 3 3 2 22x x 25x 12 (x 3)(2x 7x 4) 3 22x x
25x 12 (x 3)(2x 1)(x 4) .. 3 22x x 25x 12 (x 3)(2x 1)(x 4)
kanuay.com
61
x k p(x) k
1. a 0a ()
b na ( x ) k ab
2. k ( ab )
x k ( abx )
2.8 3 22x x 25x 12 ()
12 1, 2, 3, 4, 6, 12 2 1, 2 k x k ..
31 11, –1, 2, –2, 3, –3, 4, –4, 6, –6, 12, –12, , ,2 2 2 32
k p(k) 0 .. 3 2p( 4) 2( 4) ( 4) 25( 4) 12 0 x 4
x 4
23 22x x 25x 12 2x 7x 3x 4 3 2 22x x 25x 12 (x 4)(2x 7x 3) 3 22x x
25x 12 (x 4)(2x 1)(x 3) .. 3 22x x 25x 12 (x 4)(2x 1)(x 3) 3
“”
() “ x c ()” (Synthetic Division)
4 3 2x 3x 4x x 6 x 2
1. ( 1, 3, 4, 1, 6 ) c ( 2)
2 1 3 4 1 6
2 1 3 4 1 6 2
1
2 1 3 4 1 6 2 2 4 10
1 1 2 5 4
2.9 32x 7x 6 x 1
.. 22x 2x 5 11
2.10 4 33x 7x 4x 4 33x 7x 4x 0
x
4 3 3 23x 7x 4x (x)(3x 7x 4)
4 () 1, 2, 4 3 () 1, 3
1 2 41, 2, 4, , ,3 3 3 ...
0 ...
3 23x 7x 4 (x 1)(x 2)(3x 2) 4 33x 7x 4x x (x 1)(x 2)(3x 2) .. 4 33x
7x 4x 0 2{0, 1, 2, }3
1. ( 2 ) 2.
2 2 5 11
2 3 4 4 0 6 4
3 2 0
.
(12) 3 24x 21x 26x 17 x 4 a 3 23x 13x 11x 5 x 3 b b a (13) x 1 2x
2a x 2 x a a (14) 4 3 2x x 3x x 1 3 22x x 75x a x 5 a (15) x 2 3 2
a4x ax x 2b 21a x x b a b (16) 2x 2x 3 4 3 2x ax bx 3x 4 2x x 2 3
2x 10x cx d a b c d (17) (17.1) 3x 7x 6 0 (17.2) 3 2x 4x x 6 0
(17.3) 3 26x 11x 4x 4 0 (17.4) 4 3 2x x 11x 5x 30 0 (17.5) 5 4 3
23x 13x 7x 17x 6x 0 (17.6) 6 5 4 3 2x 2x 14x 40x 11x 38x 24 0
(17.7) 6 5 4 3 2x x 7x 9x 6x 28x 24 0 (18) 6 5 4 3 23x 2x 64x 96x
27x 98x 40
(19) ... 3x 7x 6 , 3 23x 7x 4 4 3x 3x 6x 4 (20) ... 3 2x 2x 5x 6 3
2x x 10x 8 (21) 2 2 2 2x a b 2abx b 0 (21.1) a (21.2) b (21.3) a
(21.4) b
2/5 ( 0) “x – 2/5” “5x – 2” 5
K
kanuay.com
65
[1]
a b b a R a b a b R
[2] (Transitive Property) a b b c a c
[3] a b a c b c
[4] (Trichotomy Property) “ a, b R a b a b a b ” a b a b< ( “a
b”) a b a b> ( “a b”)
[5] a b c a b b c a b c < a b b c< a b c< a b< b c a b
c< < a b< b c<
(Interval) “ 2 5”
66
a b (a, b) { x | a x b } [a, b] { x | a x b }< < (a, b] { x |
a x b } < [a, b) { x | a x b }<
(a, ) { x | x a } [a, ) { x | x a }> ( , a) { x | x a } ( , a] {
x | x a }< ( , ) R
(Number Line) ()
A, B, C , , ,
2.11 A [1, 4] B ( 2, 3) A B A B (A B)'
A B [1, 3) A B ( 2, 4] (A B)' ( , 2] (4, )
() ( R )
–2 1 3 4
–2 1 3 4
–2 1 3 4
K
kanuay.com
67
2.12 A [ 2, ) B ( 2, 3] A B B A
A B { 2} (3, ) B A
2x a x b
a 0> b 0> 2 2(a , b ) a 0 b 0 2 2(b , a ) a 0 b 0> 0
( 0 ) 2a 2b x ( 4, 3) x
1, 0, 1 0 9 16
2x [0, 16)
x 2x ()
2.13 2x . x (2,5) 2x (4,25) . x ( 5, 2) 2x (4,25) . x ( 2,5) x 0 0
4 25 .. 2x [0,25) . x ( 5,2) 2x [0,25)
2.14 x . x (2,5) x (2,5) . x ( 5, 2) x (2,5) . x ( 2,5) x [0,5) . x
( 5,2) x [0,5)
–2 1 3
68
() a x b c y d
x y (a c, b d) ( )
x y (a d, b c) y d y c x
xy ac, ad, bc, bd xy x ( 1, 3) y ( 5, 4) 5, 4, 15, 12 xy ( 15,
12)
x/y ( 0 ) x ( 1, 3) y (2, 4) 1/2, 1/4, 3/2, 3/4 x/y ( 1/2,
3/2)
2.15 2 x 3 < 1 y 5 < 2x y
2x [0,9] 2x y 0, 0, 9, 45 .. 2x y [ 9, 45) .. 2x y [0, 45)
1. ()
a b a c b c a c b c a b
2. () () “ ”
a b ( c 0 ) ac bc ( c 0 ) ac bc a b ( c 0 ) a/c b/c ( c 0 ) a/c b/c
ac bc ( c 0 ) a b ( c 0 ) a b
3. ()
a b ( a, b 0 ) 2 2a b ( a, b 0 ) 2 2a b
4. ()
0 a b c d ... 0 () a, b, c, d, …
2x x 6 0 (x 3)(x 2) 0 ( ) x
x 3 –2 0 x 3
x 3 x –2 3
x x –2
x –2 ( , 2) (3, )
1. 2x x 6 0 ( 2, 3) 2. 2x x 6 0 > ( , 2] [3, ) 2x x 6 0 < [
2, 3] “”
–3 0 1 2
)
1 2 3
(x c )(x c )(x c )... (x d )(x d )...
2
x (x 2)
1. 2 ( 2) 2
“ 0” (x–2)(x–3) = 0 x = 2 3 .. (x–2)(x–3) < 0 x < 2 3 !
!
K
kanuay.com
71
< <
<
(Least Upper Bound Axiom)
S {0, 1, 2, 3, 4, ...} 0 0.5 1.8 S 0
(a, b) (a, b] [a, b] b ( , b) ( , b] b (a, ) [a, ) ( , )
“ R
R ” Q
(22.4) a 0 , b 0 a b 2 2 b a 1 1
a ba b
(23.1) a ba b2
(23.2) a b ca c3
(23.4) ab bc
(24) 7 x 5 3 y 6 (24.1) 2x y (24.2) 2xy (25) 6 x 2 2 y 3 (25.1) xy
(25.3) x/y (25.2) x y (26) 20 . 5 . (27) A 4 3x 2 13 < B 11 x 4x
1 2x 7 < A B' (28) (28.1) 2x x 2< (28.2) x(2x 1) 1 >
(28.3) 3 26x 11x 2x 0 (29) m n 2x 6x 7 0 < m n (30) m 221 5x 6x
0 > n 2 23x 1 1 x 3x m n (31) . 220 3x 2x 0 > 13 . 23x 7x 30
0 7 (32) a b 22x 4x 5 0 (32.1) {ab} {a, b} (32.2) {a b} {a,
b}
Math E-Book Release 2.7pre
74
(36.2) (A ' B ') ' A 4(x 2)(x 3)(x 1) 0 B 3(x 4)(x 3)(x 2) 0
>
(36.3) 3
x (x 5)
>
(38) S x 1 2x 2
a S 2a 1 (39) A 2 32x x 0
B 2x 2x 2 1x 2
<
(40.1) 1 2 x 1 3x 1
(40.3) 1 2x 1 2x 2
>
(41) (41.1) 2{ x | x 7 } (41.3) ( 2,6] (3, 8] (41.2) { 1, 5, 7, 9 }
[6, ) (41.4) { x 2n | n } I
kanuay.com
75
(42) a nA { x | x , n }n 1
I
a b (43) 22x 5x 2 5
.
“ (Absolute Value Modulus) a” a “ a a 0” “ a b a b”
5 5 5 0 5 3 3 3 0 3
7 1 6 7 1 6 1 7 6 1 7
>
. 3
. x 4
x x ()
>
. 2x 1
>
. . x “ ” 0 2x 1 0 > x 1/2>
[1] a 0> [2] a a a b b a [3] ab a b nna a
aa b b ( b 0 )
[4] 2 22a a a [5] a b a b < a b a b >
[6] n n
2a a a .. 2a a 2(x 3) 1 x 3 1 .. x 3 1
K
kanuay.com
77
[1] p(x) k “p(x) k p(x) k ”
[2] p(x) k k p(x) k p(x) k< k p(x) k < < p(x) k “p(x) k
p(x) k ” p(x) k> “p(x) k< p(x) k> ”
2.19
. 3x 2 4
3x 2 4 3x 2 4 x 2 x 2/3 .. {2, 2/3}
. 3x 2 4 >
3x 2 4 > 3x 2 4 < x 2> x 2/3< .. ( , 2/3] [2, )
2.20
. 3 x 1
3 x 1 3 x 1 x 2 x 4 .. {2, 2, 4, 4}
. 3 x 1 <
1 3 x 1 < < 3 4 x 2 < < .. 2 x 4< <
x 2 4 .. x [ 4, 2] [2, 4]
()
K
[2] p(x) q(x) p(x) q(x) q(x) ( q(x) 0> )
[3] p(x) q(x) q(x) 0> () q(x) 0 1
2.21 2x 1 3x 2
2 2(2x 1) (3x 2) 2 2(2x 1) (3x 2) 0 (2x 1 3x 2)(2x 1 3x 2) 0 ( x
3)(5x 1) 0 x 3 x 1/5
.. x 3 x 1/5 { 1/5} 2x 1 3x 2 { 3, 1/5} ()
kanuay.com
79
2.22 3x 2 4x 1 <
2 2(3x 2) (4x 1) < 2 2(3x 2) (4x 1) 0 < (3x 2 4x 1)(3x 2 4x
1) 0 < ( x 1)(7x 3) 0 < –1 (x 1)(7x 3) 0 >
.. ( , 3/7] [1, ) 4x 1 0 > x 1/4> [1, ) 3x 2 4x 1 < ( ,
3/7] [1, )
0
<
–1 2
3 ( 1 2 )
2 1 .. x 2 x “ x 2 x x 2 x ” ..
.. x 2 x< 1 “ x x 2 x< < ” .. 2 3
K
–1/2 2
2x 1 x 2 x 3 2 2 –1/2 2 x 1/2 , 1/2 x 2 < x 2>
2.
3
3 0> 1 0 3 0< 3.
x 3 x 2>
2.23 x 3 4 x 1 2
>
x 0 x 1 2 .. x 1>
> >
> > <
(3,5] (3,5] x 1
> >
> > <
[ 7/5, 1) [ 7/5, 1) .. () [ 7/5, 1) (3,5]
. (44) (44.1) n I n 1 nn a a (44.2) a, b 0 a b a b (45) m (45.1) 4x
0.5 m 3 2x 1 0.5
(45.2) x 2 5 mx
x (2,6)
Math E-Book Release 2.7pre
82
(46) x 1 5 y 2 4 x y (47) (47.1) 2x 6 x 8 0 (47.2) x 1 x 1 2 (47.3)
x 4 x 3 1 (48) A 2 3x 2 3 x B 2 3x 2 3x B A ' (49) 28(x 2) 14(x 2)
3 0 (50) 2A { x | x 3x 3 2x 3 } I
5 3xB { x | 2 }x 2
<
(52.3) 1x 0x 2x x 2 0
(53) A x 2 x 42
<
(54) 4x 5A { x | x 5 }2
<R
(54.1) a, b A (a b)/2 A (54.2) a, b A a b A
(55) 2A { x | x 2 14 } R 1B { x | 1 0 }x R
A B '
kanuay.com
83
(56) a, b, c (56.1) 4 x 1 ax b c (56.2) x 10 x 8 ax b c (57) (57.1)
3x 2 4x 1 (57.4) x 1 x 3 x 5
(57.2) x 2 2x 1
>
(57.3) x 7 5 5x 25
(58) x 3 x 2 (59) x (59.1) (1 x )(1 x) (59.2) (1 x )(1 x)
.
, , ... ...
“m n ” n m m (Dividend) n (Divisor) n m m n
“6 2 ” 2 6 “2 6 ” ( 6/2 ) “n m” “m n” m n ( m/n)
m, n ( n 0 )
“n m q m nq ” ( q m/n )
[1] a b b c a c [2] a b a b< [3] a b a c a (bx cy) x y [4] (
n)
a b a bc ( a b na b ) ac b a b c b ( na b a b )
“ (Linear Combination) b c” bx cy ( x y )
2.25 4a (3x 2y) a (4x y) a 22 x
1 .. 4a (3x 2y) a (3x 2y) .. a (3x 2y) a (4x y) a ((3x 2y) 2(4x y))
.. a 11x a 22 x
1. 4. 1. a b a c a (b c) 3. a b na b 2. a b a (b c) 4. na b a
b
5. 8. 5. a (b c) a b a c (.. 2 (3 5)) 6. a (b c) a b (.. 6 (2 3) )
7. na b a b (.. 24 6 ) 8. a b na b (.. 2 6 )
K
kanuay.com
85
2 4a (3x 2y) 43x 2y a m .....(1) ( m ) a (4x y) 4x y an .....(2) (
n ) (1) 2 (2) ; 411x a m 2an 311x a (a m 2n) 2 322 x a (2a m 4n) (
) .. a 22 x
m, n ( n 0 )
“ q, r m nq r 0 r n< ” q (Quotient) r (Remainder) m n
5 17 17 5(3) 2 3 2 5 –17 5 17 5( 4) 3 ( –4 3) –5 17 5 17 5( 3) 2 (
–3 2)
(Even Numbers) (Odd Numbers) “ 2n n ” “ 2n 1 n ” 2 2 1
(Prime Numbers)
“ p p 1 p 1 ”
1 100 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
59, 61, 67, 71, 73, 79, 83, 89, 97
0 2 ( 0) “ 0 ” K
Math E-Book Release 2.7pre
( . )
[1]
p p mn p m p n ( p )
[2] () “ 1 ” 416 2 2 2 2 2 210 2 3 5 7
2 25445 3 3 5 11 11 3 5 11 73 73
“97” 2, 3, 5, 7 “97” () “97”
97 () ≈10 97 10 10
2.26
. 643 .. 2, 3, 5, 7, 11, 13, 17, 19 23
. 1127 .. 2, 3, 5, 7, 11, 13, …, 29, 31
7 ( 23)
kanuay.com
87
. 2431 .. 2, 3, 5, 7, 11, 13, …, 43, 47
11 ( 13 17)
. 4201 .. 2, 3, 5, 7, 11, 13, …, 59, 61
(...: the Greatest Common Divisor: GCD) ... a b
a b “d ... a b d a d b n a n b n d ”
... a b (a, b)
(m, n) 1 m n (Relative Primes) ( m n ) (8, 15) 1 8 15 ()
(...: the Least Common Multiple: LCM) ... a b
a b “c ... a b a c b c a n b n c n ”
... a b [a, b]
... ... ... ...
(a, b, c) ((a, b), c) (a,(b, c)) [a, b, c] [[a, b], c] [a, [b,
c]]
... ...
[1] ... ... (a, b) [a, b] a b ( a b ) ... 252 312 12 ... 6552 12
6552 252 312
[2] ... (a, b) d (a/d, b/d) 1
... ...
...
... a b ( a b )
1. a b a 2. b a ( b) b 3. 4. ... ( 0 )
... 138 182 (182) (138)1 (44) (138) (44)3 (6) (44) (6)7 ( ) (6) (
)3 2 2 ... 2 ( 2 ) ... ... ... (a, b) [a, b] a b
...
...
2.27 ... ... 192 276
(276) (192)1 (84) .....(1) (192) (84)2 (24) .....(2) (84) (24)3 ( )
12 .....(3) (24) ( )2 12
... 192 276 12
... 192 276 192 276 441612
kanuay.com
89
d ... a b “ d a b d ax by x y
1 ” x y ...
2.28 ... 192 276 12 12 192 276
“ = .........” (1) (84) (276) (192)( 1) .....(4) (2) (24) (192)
(84)( 2) .....(5) (3) (12) (84) (24)( 3) .....(6) (6) 24 (5) 12
(84) ((192) (84)( 2))( 3) (84)(7) (192)( 3) 84 (4) 12 ((276) (192)(
1))(7) (192)( 3) (276)(7) (192)( 10)
12 192 276 12 192( 10) 276(7)
. (60) 3 2(19) (288) 5 (61) ... (d) 252 34 d 252 x 34 y x, y (62)
... –504 –38 (63) a ( a 12 ) 7, 9, 12 4 (64) b 7 6 9 8 12 11 (65)
... ... x 128 16 384 x
Math E-Book Release 2.7pre
90
(66) ... 3 ... 30 9 (67) a, b a b , 5 a 3 b a, b ... a, b 165 a b
(68) x, y 80 x 200 x pq p, q x, y ... 15015 y (69) x, y x y (x, y)
9 , [x, y] 28215 x 3 x, y (70) n ... n 42 6 0 042 nq r , 00 r n 0
1n 2r r , 1 00 r r 0 1r 2r 0q , 0r , 1r ... n 42 (71) a b a 1998 b
r 0 r 1998 11998 47 r r 10 r r 1(r, r ) 6 (71.1) (a, b) 6 (71.2)
(a, 1998) 6 (71.3) (b, r) 6 (71.4) (1998, r) 6
kanuay.com
91
() (1) (2) (2.3) (3) . (4) (4.1) (4.3) (5) (6) . (7) 6 5 1 (8) .
(9) . (10) (11.1) (11.2) (12) 1 (13) –3 (14) –81 (15) 7 –39/7 (16)
27 (17.1) { 1, 2, 3} (17.2) { 1, 2, 3} (17.3) {2, 1/2, 2/3}
(17.4) {3, 2, 5, 5} (17.5) { 1, 0, 2, 3, 1/3} (17.6) { 4, 1, 1, 2,
3} (17.7) { 3, 1,2} (18) 2(x 2)(x 4)(x 5)(3x 1)(x 1) (19) (x 1)(x
2) (20) (x 1)(x 2)(x 3)(x 2)(x 4) (21.1) {b, b} (21.2) {0} (21.3)
{0, 2b} (21.4) { a 1, a 1} (22) (22.1) (22.2) (23) (23.4) (24.1) (
6, 46) (24.2) ( 252, 180) (25.1) ( 18, 4) (25.2) ( 9, 4)
(25.3) ( 3, 2/3) (26) [7.5, 10) . (27) 2, 4 (28.1) [ 1,2] (28.2) (
, 1] [1/2, ) (28.3) ( , 2) (0, 1/6) (29) 2 (30) 2 (31) (32) (33) a
( , 2) (3, ) (34) [ 2, 1] [2, ) (35) –5 (36.1) ( , 1) (0, 1) (36.2)
[ 4, ) {1} (36.3) 11 (37) 0 (38) 5 (39) 0 (40.1) ( , 1) (1/3, 1)
(40.2) ( 8, 2] (1, 4] (40.3) (2,5/2) (40.4) (2,8] (41.1) 7 (41.2)
(41.3) 8 (41.4) (42) 0 (43) 5/2 (44) (45.1) 3.5 (45.2) 17/3 (45.3)
96 (46) [0, 12) (47.1) 2, 2, 4, 4 (47.2) [ 1, 1] (47.3) [3,
4]
(48) [ 2/3, 0) (49) –8 (50) 90 (51) 1, 6 (52.1) ( 1/5, ) (52.2) (
4, 1) (5,8) (52.3) ( 1,2) {0}
(52.4) 3 21( 1, 3) [ , )2
(52.5) ( , 2] ( 1, 1) [2, ) (53) (2, ) (54) (55) 7 (56.1) 2, 3, 5
(56.2) 1, 1, 9 (57.1) 3( , ) (1, )7
(57.2) ( , 4) (0, ) (57.3) (2, 4) (6, 12) (57.4) ( 1, 3) (57.5)
1
3(( , 1] [ , 3]) { 2, 1}
(58) ( , 1/2) (5/2, ) (59.1) ( , 1) ( 1, 1) (59.2) (1, ) (60) 1
(61) 2 (252)(5) (34)( 37) (62) 2 ( 504)( 4) ( 38)(53) (63) 256 (64)
251 (65) 48 (66) 21 (67) 3 (68) 105, 165 (69) 495, 513 (70) 210
(71) (71.2) (71.4)
Math E-Book Release 2.7pre
92
() (1.1) (1.2) (1.3) a 2 2a a (1.4) a 3 2a a
(2.1) .. a 0 b (2.2) .. a 0 b 0 ( 0 ) (2.3) ( b a c ) (2.4) .. a 0
b c
(3) . ()
. ( 3 5 8 8 ) ( 3 5 15 )
. ( 3 3( ) 04 4
3 4 14 3 )
. 4 4 .. .
(4.1) .. (4.2) .. (a b) c a (b c) (4.3) .. 0 ( 0 0 ) (4.4) .. a b[
] c a [ ]b c
(5) A { x | x x } {1, 4, 9, 16, 25, 36, ...} A .. B A N A
(5.1) A .. B .. 2 2 4 4 B
(5.2) A .. 1 1 2 2 A B .. 2 2 4 4 B
(6) . .. x 2 y xy 1 . .. x 0 y xy 1 . .. xy 1 xy A (1 ) . .. x y xy
1 y ( x, y 0)
(7) a 1/a ..
1 6 5
6 5
1
(8) . (a b) a b a b .. . (b c) b a b b .. . (a b) (c b) b a b .. .
(c a) (b a) c b a ..
(9) . x y y x .. x y
kanuay.com
93
(10) x (y z) x (3yz y z) 3x(3yz y z) x 3yz y z (z y) x (3zy z y) x
3(3zy z y)x 3zy z y x
a b x (y z) (z y) x ( y, z x )
(11.1) A
5 7 62
..
ab ba 2 2 ..
(12) 3 2a 4(4) 21(4) 26(4) 17 7 3 2b 3( 3) 13( 3) 11( 3) 5 8 b a 8
7 1
(13) x 1 2x 2a 2(1) 2a x 2 x a ( 2) a .. 1 2a 2 a a 3
(14)
4 3 2(5) (5) 3(5) (5) 1 569 3 22(5) (5) 75(5) a 650 a .. 569 650 a
a 81
(15) “” 0 3 2(2) a(2) (a/4)(2) 2b 0 .... (1) 2(1/a)(2) (2) b 0 ....
(2)
a 4, b 3 a 4/7, b 5 .. a b 7 39/7
(16) 2x 2x 3 (x 3)(x 1) 4 3 2x ax bx 3x 4 x 3 x 1 ( 0)
4 3 2 4 3 2
a 19/9, b 37/9
2x x 2 (x 2)(x 1) 3 2x 10x cx d
3 2
c 7, d 18 .. a b c d 27
(17.1) –1, –2, 3 () (x 1)(x 2)(x 3) 0 { 1, 2, 3}
(17.2) –1, 2, 3 () (x 1)(x 2)(x 3) 0 { 1, 2, 3}
(17.3) 2, 1/2, –2/3 () ( 2
26x x 2 ) .. (x 2)(2x 1)(3x 2) 0 {2, 1/2, 2/3}
(17.4) 3, –2 () 1 0 –5 2x 5 2(x 3)(x 2)(x 5) 0
(x 3)(x 2)(x 5)(x 5) 0 {3, 2, 5, 5}
Math E-Book Release 2.7pre
(17.6) 1, 1, –1, 2, 3, –4 () 2(x 1) (x 1)(x 2)(x 3)(x 4) 0 { 4, 1,
1, 2, 3}
(17.7) 1, 2, 2, –3 () 1 1 2 2x x 2
2 2(x 1)(x 2) (x 3)(x x 2) 0
{ 3, 1, 2}
(18) 2, 4, –5, –1/3 () 3 0 3 23x 3
.. 2(x 2)(x 4)(x 5)(x 1/3)(3x 3) 2(x 2)(x 4)(x 5)(3x 1)(x 1)
(19) ( )
3x 7x 6 (x 1)(x 2)(x 3) 3 23x 7x 4 (x 1)(x 2)(3x 2)
4 3x 3x 6x 4 (x 1)(x 2)(x 2)(x 2) ... ... 2(x 1)(x 2) x 3x 2
(20)
3 2x 2x 5x 6 (x 1)(x 3)(x 2) 3 2x x 10x 8 (x 1)(x 2)(x 4) ... ...
(x 1)(x 2)(x 3)(x 2)(x 4) 5 3 2x 17x 12x 52x 48
(21.1) a 0 2 2x b 0 (x b)(x b) 0 .. { b, b} (21.2) b 0 2x 0 .. {0}
(21.3) a 1 2 2 2x b 2bx b 0 2x 2bx 0 x(x 2b) 0 .. {0, 2b} (21.4) b
1 2 2x a 2ax 1 0 2(x a) 1 0 (x a 1)(x a 1) 0 .. { a 1, a 1}
(22.1) .. c b a c d ( )( )( ) 0 (22.2) .. 2 1 2( 2) 21
(22.3) (a b)/2 ab
2 2a b 2 ab a 2ab b 4ab ()
2 2 2a 2ab b 0 (a b) 0 a b
(22.4)
aba b
() (b a) 2 2(b ab a ) ab(b a)
( 0) 2 2 2b 2ab a 0 (b a) 0
a b
(23.1) (23.2) .. min maxx X x (23.3) .. 3x ( ) (23.4) .. b 0 ab bc
b ab bc
kanuay.com
95
x
2 2h x h
(24.1) 7 x 5 20 x 49< 3 y 6 6 y 3 .. 26 x y 46 ( 6, 46)
(24.2) 3 y 6 29 y 36 2xy ( 7) 36 2xy 5 36 .. 2252 xy 180 ( 252,
180)
(25.1) xy 12, 18, 4, 6 .. ( 18, 4)
(25.2) x y 8, 9, 4, 5 .. ( 9, 4)
(25.3) x/y 3, 2, 1, 2/3 .. ( 3, 2/3)
(26) ( h . 2x .) x h
2 2 2 220 2x 2 h x 10 x h x 2 2 2100 20x x h x 2 2100 h h
20 20x 5
215 h 4 205 5 < .. 15
4 x 5<
2x [7.5, 10) .
(27) A; 6 3x 15 2 x 5 < < A [2,5) B; “” 11 x 4x 1 10 5x x 2
“” 4x 1 2x 7 2x 6 x 3 < < < () B (2, 3] .. A B' A B {2}
(3,5) A B ' 2 4
(28.1) 2x x 2 0 < .. (x 2)(x 1) 0 < [ 1,2] (28.2) 22x x 1
> 22x x 1 0 > .. (2x 1)(x 1) 0 > ( , 1] [1/2, ) (28.3) ..
x(x 2)(6x 1) 0 ( , 2) (0, 1/6)
(29) 2x 6x 7
2b b 4ac
2a
.. 2 2(x 6x 9) 2 0 (x 3) 2 0 < <
(x 3 2)(x 3 2) 0 < 3 2 x 3 2 < < .. m 3 1 2 n 3 1 4 m n
2
+ – + 3 2 3 2
–5 –3 –1 2
(30) 2x 1 26x 5x 21 0 < .. (3x 7)(2x 3) 0 <
m 1 0 1 2 2
26x x 2 0 (3x 2)(2x 1) 0
0 n 0 .. m n 2
(31) . 2x 1 22x 3x 20 0 < .. (2x 5)(x 4) 0 <
4, 3, 2, ...,2 13 .. . .
2 2 49 4097 7 3 3 36 36x x 10 0 (x x ) 0
2 4097 6 36(x ) 0
409 4097 7 6 6 6 6(x )(x ) 0
7 409 7 409 6 6x ()
27/6 x 13/6 .. 4, 3, 2, ...,2 7 .. .
(32) 2 22x 4x 5 0 x 2x 2.5 0 2 2(x 2x 1) 3.5 0 (x 1) 3.5 0
(x 1 3.5)(x 1 3.5) 0
3.5 1.8 a 0, b 2 {0} {0, 2} { 2} {0, 2}
(33) 3 2(1) a (1) a 2 5
2a a 6 0 (a 3)(a 2) 0
.. a ( , 2) (3, )
(34) (x 1)(x 2)(x 2) 0 >
[ 2, 1] [2, )
(35) 3 2x 2x 5x 6 0 < ..(x 2)(x 1)(x 3) 0 <
A B ( 5, ) A B
4 3 1 0 1 2 5
(36.1) ( , 1) (0, 1) (36.2) A
B
(A ' B ') ' A B [ 4, 1) (1, ) ( [ 4, ) {1} ) (36.3)
3, 2, 0, 1, 5
| 3| | 2| |0| |1| |5| 11
+ – + –4 5/2
–2 1 1 3
+ – + –1/2 2/3
+ – + –3/2 7/3
>
(44.2) a 2, b 3 |a b| 1 |a| |b| 1 +
(45.1) 3 2x 1 0.5 2 2x 1.5 1 x 0.75 ..
4 4x 3 3.5 4x 0.5 3.5 |4x 0.5| 3.5 m |4x 0.5| m 3.5 (45.2) 2 2 2x 5
1 5 6x x x
2 x 6 1 2 2 1
3 31 1x x 2 17
35 6x
m 2x 5 mx 17 3
(45.3) 6 x 5 6 11 x 1 ..
2 20 x 121 25 x 25 96 < < m 2|x 25| m 96
+ – + –1/2 3
+ – + 1/2 2
kanuay.com
99
. .
(46) 5 x 1 5 4 x 6 4 y 2 4 2 y 6 6 x y 12 .. |x y| [0, 12)
(47.1) 2 2x |x|
2|x| 6|x| 8 0 (|x| 4)(|x| 2) 0 |x| 2 4 .. x 2, 2, 4, 4 (47.2)
. x 1 .. ( x 1) ( x 1) 2 2x 2 x 1 . 1 x 1 < .. ( x 1) (x 1) 2 2
2 [ 1, 1) . x 1> .. (x 1) (x 1) 2 2x 2 x 1 {1} .. [ 1, 1] (47.3)
. x 3 .. ( x 4) ( x 3) 1 2x 6 x 3 . 3 x 4< .. ( x 4) (x 3) 1 1 1
[3, 4) . x 4> .. (x 4) (x 3) 1 2x 8 x 4 {4} .. [3, 4]
(48) A; . x 2/3 .. 2 3x 2 3x 2 2
. 2/3 x 0 < .. 2 3x 2 3x 6x 0 x 0 . x 0> .. 2 3x 2 3x 0 0 [0,
) .. A [0, )
B; . x 2/3 .. 2 3x 2 3x 2 2
. x 2/3> .. 2 3x 2 3x 0 0 [ 2/3, ) .. B [ 2/3, ) 2
3B A' B A [ , 0)
(49) 28 x 2 14 x 2 3 0 .. (2 x 2 3)(4 x 2 1) 0 3
2x 2 1 4
3 3 1 1 4 42 2x { 2 , 2 , 2 , 2 }
8
(50) A; .. 2 2 2(x 3x 3) (2x 3) 0
2 2(x 3x 3 2x 3)(x 3x 3 2x 3) 0 2 2(x x)(x 5x 6) 0
x(x 1)(x 2)(x 3) 0 A {0, 1, 2, 3}
B; ( 0 x 2 ) .. |5 3x| |2x 4|
2 2(5 3x) (2x 4) 0 (5 3x 2x 4)(5 3x 2x 4) 0 (1 5x)(9 x) 0
x B {9}
.. A B {0, 1, 2, 3,9} 2 2a 9, b 3 a b 90
–1 1
212 2 xx 3 3( x) x x x
. x 1 1 1 . 0 0 00 .. x 0 00 x 0 . 2 21
2 x 3 x 6 x 6
.. {1, 6}
(52.1) 1 ( 47, 48 ..) x 1/2 .. 2x 1 3x 2 1 5x x 1/5
( 1/5, 1/2) x 1/2> .. 2x 1 3x 2 3 x x 3 [1/2, ) ( 1/5, ) 2 .. (
)
.. 2 2(2x 1) (3x 2)
2 2(2x 1) (3x 2) 0 (2x 1 3x 2)(2x 1 3x 2) 0 ( x 3)(5x 1) 0 (x 3)(5x
1) 0
( , 3) ( 1/5, )
.. 3x 2 0 x 2/3 > > ( 1/5, )
(52.2) .. 3 x 2 6 6 x 2 3 3 x 2 6 4 x 1 5 x 8 ( 4, 1) (5, 8)
3 x 2 x 2 6
(52.3) 1x 0|x|
x 0 ..
21 x 1x 0 0x x
(x 2)(x 1) 0 ( 1, 2)
.. x ( 1, 0) (0, ) x ( 1, 2) “” “” ( 1, 0) (0,2)
kanuay.com
101
3 3x x 0( x 1) 2 x 1
< <
< >
x 1> ..
< <
< >
>
3 21 2[1, 3) [ , )
.. 3 21 2( 1, 3) [ , )
(52.5) 2 x 0> x 0 A x ..
A A2 2 0A 1 A 1
< <
< >
A ( , 1) [2, ) A 0 A [0, 1) [2, )
.. ( x) ( , 2] ( 1, 1) [2, )
< <
x 10 0 x 10 ( , 2) < <
< <
x 52.1 ( 2) B; x 7 ..
7 7 2 2x x 7 2x 7 x ( , )
x 7> .. x x 7 0 7 .. 7
2B ( , )
A B ( ,2] (A B)' (2, )
(54) () 2x |4x 5| |4x 5| 10 < 2x |4x 5| x 5/4 .. 2x 4x 5 6x 5 x
5/6 ( , 5/4)
x 5/4> .. 2x 4x 5 5 2x x 5/2 [ 5/4, ) .. x R
|4x 5| 10 < 10 4x 5 10 15 4x 5 < < < <
15/4 x 5/4 < <
4 4A [ , ]
4 4 4( ) A
Math E-Book Release 2.7pre
102
(55) A; 214 x 2 14 212 x 16 20 x 16< () 4 x 4 .. A ( 4, 4)
B (0, 1)
A B' A B ( 4, 0] [1, 4) 7
(56.1) 34 1
2 2
3 3 3
2 2 24 x 1 5 3 5 2 2 2x 5 2x 3 5
|2x 3| 5 .. a 2, b 3, c 5 (56.2) 10 8
2 1
x 1 9 x 1 9 |x 1| 9 .. a 1, b 1, c 9
(57.1) ( 52.1 2)
2 2 2 2(3x 2) (4x 1) (3x 2) (4x 1) 0 (3x 2 4x 1)(3x 2 4x 1) 0 ( x
1)(7x 3) 0 (x 1)(7x 3) 0
.. 3 7( , ) (1, )
(57.2) () ( x –1)
2 2 2(x 2) (2x 2) (x 2) (2x 2) 0 (x 2 2x 2)(x 2 2x 2) 0 ( x 4)(3x)
0 3(x 4)(x) 0
.. ( , 4) (0, )
(57.3) () |x 7| 5 |5x 25| 5
|x 7| 5 5 x 7 5 2 x 12 |5x 25| 5 5x 25 5 5x 25 5 x 6 x 4
.. (2, 4) (6, 12)
(57.4) 4 x 1 x 1 x 3 x 5 1 x ( 1, 1)
1 x 3< x 1 x 3 x 5 x 3 [1, 3) 3 x 5< x 1 x 3 x 5 x 3 x 5>
x 1 x 3 x 5 x 1 ( 1, 3)
(57.5) 1 57.1, 57.2..
2 2 2 2(x 5x 4) (x x 2) > 2 2 2 2(x 5x 4 x x 2)(x 5x 4 x x 2) 0
>
2( 6x 2)(2x 4x 6) 0 > 2(3x 1) 2(x 3)(x 1) 0 <
1 3( , 1] [ , 3]
.. 0 2x x 2 0 x –2 1 1
3(( , 1] [ , 3]) { 2, 1}
kanuay.com
103
> 2 2
<
>
x x 2
> <
2 2
<
x x 2
<
<<
(58) 2
x 0 | x 3| |x 2|
2 2 2 2( x 3) (x 2) ( x 3) (x 2) 0
( x 3 x 2)( x 3 x 2) 0 ( 2x 1)( 5) 0 (2x 1)(5) 0
x 1/2
x 0> |x 3| |x 2|
2 2 2 2(x 3) (x 2) (x 3) (x 2) 0
(x 3 x 2)(x 3 x 2) 0 ( 1)(2x 5) 0 (1)(2x 5) 0
x 5/2 .. 51
x 0> (1 x)(1 x) 0 (x 1)(x 1) 0 ( 1, 1) [0, 1) .. ( , 1) ( 1,
1)
(59.2) .. (1 |x|)(1 x) x 1 1 x 1, 1 (1 |x|)(1 x) .. x (1, )
(60) 1 3 2(19) (288) 9 9 9 8 8 3 29 8 9 4 6 5 1
2 3 2(4x 1) (58x 2) x 3 2( 1) ( 2) 4 .. x 5
3 2(19) (288) 5 4 4 5 1
Math E-Book Release 2.7pre
104
(61) ... 252 34(7) 14 .....() 34 14(2) 6 .....() 14 6(2) 2 .....()
6 (3) 2 ( ... 2)
, , ...... () 14 252 34( 7) () 6 34 14( 2) () 2 14 6( 2)
.. () () 2 14 (34 14( 2))( 2) 14(5) 34( 2) () 2 (252 34( 7))(5) 34(
2) 252(5) 34( 37) .. 2 252(5) 34( 37)
(62) ... () –504 = –38(14) + 28 28 = –504 + (–38)(–14) () –38 =
28(–2) + 18 18 = –38 + 28(2) () 28 = 18(1) + 10 10 = 28 + 18(–1) ()
18 = 10(1) + 8 8 = 18 + 10(–1) () 10 = 8(1) + 2 2 = 10 + 8(–1) 8 =
2(4) ( ... 2)
.. () (); 2 = 10 + (18 + 10(–1))( –1) = 10(2) + 18(–1) (); 2 = (28
+ 18(–1))(2) + 18(–1) =28(2) + 18(–3) (); 2 = 28(2) + (–38 +
28(2))(–3) = 28(–4) + (–38)(–3) (); 2 = (–504 + (–38)(–14))(4) +
(–38)(–3) = (–504)(–4) + (–38)(53)
(63) a 7, 9, 12 4 .. a 4 7, 9, 12 a a 4 ... 7, 9, 12 ... 7, 9, 12
252 a 4 252 a 256
(64) b 7 6 .. b 1 7 b 9 8 .. b 1 9 b 12 11 .. b 1 12
b b 1 ... 7, 9, 12 ... 7, 9, 12 252 b 1 252 b 251
(65) “ ... ... ” x 128 16 384 x 48
(66) “ ... ... ” 3 30 90 (1,90), (2,45), (3,30), (5,18), (6,15),
(9,10)
9 6 15 .. 21
(67) 165 5 3 11 a 5 ______ , b 3 _______ a, b a b (... 1) a b a 5 b
3 11 33 “a b” 33
5 3
kanuay.com
105
(68) ... x, y 1 ... x, y 15015 x y 15015 3 5 7 11 13
.. x 2 80 x 200 x 13 7 13 11 ()
y 3 5 11 165 y 3 5 7 105
(69) ... 9 3 3 ... 28215 3 3 3 5 11 19 x y 9 x 3 3 ?? y 3 3 x 5 11
3 y
.. x 3 3 5 11 495 y 3 3 3 19 513
(70) ... 42 n ... 6 1r 6 0 1r 2r 12 0 1n 2r r 30 “... ... ”
42 30... 2106
(71) ... ...
a 1998b r 0 r 1998 a 1998 ( b, r)
11998 47 r r 10 r r 1998 r ( 47, r1) .. r r1 ... r r1 6 71.2
71.4
71.1 71.3 b ...
5 14 . 00 00
4 1
4 4 1 14
4 42 1 14
4 42 1 14
84 44 30 00
4 42 1 14
84 446 30 00
26 76 3 24
4 42 1 14
84 446 30 00
3 16 89 .... ....
.. (1) 514 514 2 14 5 ( ) 2 ( 00 00 ) (2) ( 5) ( 5) 2 2 ... 2 (3) 2
2 4 ... 4 5 1 (4) () 2 4 ... ( 14) 114 (5) x 4x x 114 ( 114) ... 41
1 41, 42 2 84, 43 3 129 () 42 2 ... 2 ( 4) 2 42 2 84 114 ( 30) (6)
(4) (5) (22) 44 ( 00) 30 3000 (7) x 44x x 3000 ( 3000) ... 446 6 6
( 44) 6 446 6 2676 3000 ( 324) (8) (226) 452 ( 00) 324 32400 x 452x
x 32400 ( 32400) ... 4527 7 ... 7 ( 452) 7 4527 7 31689 32400 ...
514 22.67... 514 2 5,14 2 () 903601 3 90,36,01 3 ... 225, 3000,
214.7 :]
(Proposition Statement) “ ”, “1 2”, “”
, , , ,
p, q, r, s 2 (True; T) (False; F)
K ~l
“” , “”
p q (p q )
p q (p q )
p q (p q )
p (~ p )
T T T T T T F T F F T F F F F T F T T F T F F F F T T T
1.
p “”, q “” p q “”
2.
p “”, q “” p q “”
3. -
p q “”
)
p “”, q “” p q “”
(, , , )
- , p q q p (p q) r p (q r) ()
(Truth Table) 2n n 1 2 , 2 4 (), 3 8
Math E-Book Release 2.7pre
. [(p r) q] r
[(F T) F] T [T F] T F T F “”
. [(r q) ~p] [(~ q p) r]
[(T F) ~F] [(~F F) T] [F T] [T T] T T T “”
3.2 p q
. [(q s) r] [(p s) t]
r, s, t q (q s) [T r] [(p s) t]
“” [T] [(p s) t] “” .. “”
. [q (s r)] [p (q ~ s)]
r, s p, q [F (.....)] [T (F ....)] [T] [T (F)] [T] [F] .. “”
( ) “” “, ” “-” “”
p ~ r q r [p ((~r) q)] r
kanuay.com
111
2 ()
1. p (q r) (p q) (p r) p (q r) (p q) (p r)
2. p q ~ p q ~ q ~ p p q (p q) (q p) ( )
3.3 p ~ q ~ (p q)
1 p ~ q ~ (p q) “” () p ~ q ~(p q)
p q p ~ q ~ (p q) T T F F T F T T F T T T F F T T
3. ~(p q) ~ p ~ q ~(p q) ~ p ~ q ~(p q) p ~ q ~(p q) ~ p q p ~
q
Math E-Book Release 2.7pre
2 () p ~ q ~p ~ q ~ (p q) ~p ~ q “”
3.4 (p q) r (p ~ q) (p r)
(p q) r ~ (p q) r ~p ~ q r
(p ~ q) (p r) (~ p ~ q) (~ p r) ~p ~ p (~ q r)
“”
. (1) p
T p T p T p T p F p F p F p F p p p p p p T p p p ~ p p ~ p p F p ~
p p p p ~ p
(2) p, r q (2.1) [(p s) (p r)] (p s) (2.2) [(q s) r] [(q s) r]
(2.3) [(r q) (p q)] (p ~ q) (2.4) [(p q) (q r)] ~ s (2.5) [(q p) r]
~(~ r) (2.6) [(p q) ~ r] [(~ p q) r] (2.7) [(p ~ q) ~ r] [(p q) (~
q r)] (2.8) [(p q) ~ r] [(r ~ s) (~ p ~ q)] (2.9) [p (q r)] [(q p)
r] (2.10) [q (p r)] [p (q ~ r)] (2.11) [(~ p ~ q) (~ r ~ s)] [(~ p
r) (s ~ q)]
kanuay.com
113
(3) (3.1) (p ~ q) (p q) q (3.2) (p ~ q) (p q) p (3.3) (~ r p) (~(r
s) (r ~ q)) p, q r, s (3.4) (p q) (s p) (s q) p, r, r q (3.5) (~ q
(p r)) (~ r) p q , q r (3.6) n [(m q) ~ s] q n (3.7) (p r) q p q ,
q r (3.8) (q p) (r s) (p q) (r s) , q s (3.9) r s (p r) (q s) , p q
(3.10) (p r) ~ q (p ~ r) (p q) (3.11) p, q, r (p q) (p r) (3.12) r
p (p ~ r) (q r) (3.13) ((p ~ q) ~ p) (p q) (3.14) p ~(r s) (~ r ~
s)~ p (4) [(p q) (p r)] (s r) . [(p q) (q r)] (r s) . [(~ p q) (~ q
r)] (~ r s) (5) [(p q) (r ~ s)] [(~ p r) (q ~ s)] . ~(p s) ~ r . r
(p ~ q) . (s r) (p q) (6) p q r s . [(p ~ q) (r ~ s)] [(~ p q) (~ r
~ s)] . [(p r) (q s)] [(p ~ q) (r ~ s)] (7) . (p ~ q) (~ r s) . (p
~ q) ~ r (8) “” . . . . (9) . p q ~(~ p ~ q) . ~(p ~ q) ~ q ~ p . ~
p (q p) ~ q p . ~ p q (~ p q) (q ~ p)
Math E-Book Release 2.7pre
114
(10) (10.1) p q . (p q) (q ~ p) . (~ q ~ p) (~ q p) . (p ~ q) (q p)
. (p ~ q) (~ p ~ q)
(10.2) ~ q r((q ~ t) p) ((q ~ t) ~ p) . q ~ t p . (t q) p . t q r .
(t q) r
(10.3) [(q r) (p s) (q ~ r)] [(q ~ r) (p ~ s) (q r)] . p q . p q .
p q . p q (11) “ a 0 b 0 ab 0 ” . a 0> b 0> ab 0< . a
0> b 0> ab 0> . ab 0< a 0> b 0> . ab 0> a
0< b 0< (12) . a a a . a a a . a a a (13) . ~(p ~ r) ~ q q (r
~ p) . p (q r) q (p r) . (p q) r (p ~ q) (p r) (14)
(14.1) (p p) (q q) . p q . p q . p q . p q
(14.2) p q . ~(~ p q) . ~ p q . ~(q ~ p) . q ~ p (15) p q ~(p q) p
(q r)
p q p * q T T F F
T F T F
F F F T
p r (4 )
p r r p p r (r p) (p r) T T T T T T F T F F F T T T T F F F T
T
..
p r (4 )
p r r ~ p p r (r ~ p) (p r)
T T T T T T F F F T F T T T T F F T T T
..
( )
3.6
“-”
“”, “” “” “” “-” ( “” )
K
kanuay.com
117
“-”
[(p q) (q r)] (p r) F T T F T F T T F F
p r q .. “” “”
“” p r ~p r ..
. (16) (16.1) (p q) [(p q) r] (16.2) (p q) [(p q) r] (16.3) [(p q)
(q r)] (p r) (16.4) [(p r) (q r)] [(p q) r] (16.5) [(p r) (q r)]
[(p q) r] (16.6) [(p r) (q s) (p q)] (r s) (16.7) (p q) r (p r)(p
q) (17) (17.1) ~(p ~ q) (p q) (17.2) [(~ p q) p] (p q) (17.3) [(p
q) ~ p] (~ p q) (17.4) (p q) [(q ~ p) (p ~ q)] (17.5) [(p q) (p q)]
[(~ p ~ q) (~ p ~ q)]
Math E-Book Release 2.7pre
118
(17.6) [p (q r)] [(p q) (p r)] (17.7) [p (q r)] [(p q) r] (17.8) [p
(q r)] [(p q) r] (18) (18.1) [(p r) (q r)] (p q) (18.2) [(~ p q) ~
p] (p q) (19) (19.1) (p ~ p) (q ~ q) (19.2) [p (q ~ q)] [~ p (q ~
q)] (19.3) ~(p q) (~ p ~ q) (20) p, q, r . (p q) (~ p ~ q) . (p q)
(~ p q) . ~((p q) r) (~(p q) ~ r) . ((p r) (q r)) ((p q) r) . ((p
q) (p r)) (p (q r)) (21) [(p ~ q) (p ~ r)] [p ~(q r)]
.
1 2 3 np , p , p , ..., p ( “”) “” q
1 2 3 n(p p p ... p ) q
1. 2.
(valid)
(invalid)
1 2 3 n(p p p ... p ) q
3.8 1. 2. 3.
1. p q 2. ~p r 3. ~ q r
(p q) (~ p r) (~ q) r F
T T T F T T T F F
q .. “”
.. p q ~ q ~p
() “” p q ~ p (~ q) q (p)
() p q q r
p r
“”
“”
Math E-Book Release 2.7pre
1. ~ q ~p 3. ~p [ ()] ~ p 2. r [ () ] .. r “”
. (22)
(22.1) 1. p q 2. q s 3. ~ s ~ p
(23) (23.1) 1. x 2 | x 2. x 2 | x x 3. “x x ” 4. x x
(23.2) 1. a a 2. 2a 2 2a 1 3. 2a 2 a 4. 2a 1 a
(22.2) p (r s) ~ p (r s)
()
2. ~ s p 3. q
(25) 1.
2.
2. 3. 4. .
.
“ ” (Open Sentence) x P(x), Q(x), R(x) ( x x xP , Q , R ) , ,
-,
P(x) Q(x) ~ P(x) Q(x)
p
“x 2” “x 2” “ x () 2” ( “” “” )
x x (For All x) x (For Some x) x x (Quantifier)
P(x) “x 2”
x [P(x)] “ x .. x 2” x P(x) x [P(x)] “ x .. x 2”
x P(x)
3.10 P(x) “x 2”
{3, 4,5}U x [P(x)] 3, 4 5 2 x [P(x)] 3 2 ()
{1,2, 3}U x [P(x)] 1 2 ( “x ”) x [P(x)] 3 2 ()
{0, 1,2}U x [P(x)] x [P(x)] x 2
x x (There Exists x) x [P(x)] “ x 2”
All () Exist ()
3.11 { 2, 0, 1, 1.5, 2} U
. x [ x x ] x ()
kanuay.com
123
. x [ x x ] x 2 () . x [ x x ] x 1.5 ()
. x [ x x 3 ] x 1 ( T F ) . x [ x x 3 ] x 2
. x [ x 3 x ] x 0 ( T F )
. x [ x 2x 0.5 ] x 1.5 () x 0 ( 0.5) . x [ x 2x 0.5 ] x ( “”
)
x [...] x [...] x [...] x [...] ( ) 1. R 2.
x [P(x) Q(x)] x [P(x)] x [Q(x)] x [P(x) Q(x)] x [P(x)] x
[Q(x)]
x “”, x “”
x y P(x, y) x y x y P(x, y) , x y x y P(x, y)
x y y x x,y x y y x x,y
x y , x y , y x , y x
x y [...] “ x y ...” x y [ ] x y
y x [...] “ y x ...” y x y x
3.12 P(x) “x ” Q(x, y) “x+y ”
x y [P(x) Q(x, y)] “ x y ... x x+y ” y x [P(x) Q(x, y)] “ y x ... x
x+y ”
() {1,2, 3}U x y [P(x) Q(x, y)] .. (1,2), (2,1), (3,2) y x [P(x)
Q(x, y)] .. (3,1), (2,2), (3,3) [ ]
1. 2.
x [P(x)] x [~ P(x)] x y [P(x) Q(x, y)] x y [P(x) ~Q(x, y)] x [ x x
4 ] x [ x x 4 ]<
K
kanuay.com
125
. (27) { 2, 1, 0, 1, 2} U . x [x 2x 0] . 3 2x [x x 2x x ] . x [ x x
] . x [x x ] (28) P(x) “x ” Q(x) “x ” . x [P(x) Q( 2)] . x [Q(x)
P(0.5)] . x [P(x) ~ Q( )] . x [Q(x) ~ P(22/7)] (29) P(x) “ 2x x ”
Q(x) “x 3 x 1” . x [P(x)] U (0, 1) . x [Q(x)] {2, 3, 8, 10}U (30) 3
2x (x 5x 1 4) x ( 0 x 2)x 1 > (31) N
2[ x (x 1 2x) x (x 1 0)] x 0 >
(32) { 1, 0, 1} U (32.1) 2 2x (x 1) x (x 1) (32.2) 2x (x 1 0) x(x
1) (32.3) 2x (x 1 0 x 1) (32.4) 2x (x 0) x(x 0) (32.5) 2x (x 0 x
0)
32.2 32.3 some 32.2 “” 32.3 x
32.4 32.5 all .. all , some
K
126
(33) { 1, 0, 1} U (33.1) 2x y (x y 2) > (33.2) 2x y (x y 2) >
(33.3) 2x y (x y 2) > (33.4) 2x y (x y 2) > (34) { 1, 0, 1}
(34.1) 2 2x y (x y y x) (34.2) 2 2x y (x y y x) (34.3) 2 2x y (x y
y x) (34.4) 2 2x y (x y y x) (34.5) 2 2x y (x y y x) (35)
(35.1) x y (x y y x) { 2, 0, 2} U (35.2) x y (x y 0) { 2, 2}
U
(36) x y(xy 1) x y (xy y) . . . . (37) R . x y [x y xy] > . x y
[x y 0] < . x y [x y] < . x y [y x] (38) (38.1) x [P(x) ~
Q(x)] (38.2) x [P(x) (Q(x) R(x))] (38.3) ~ x [P(x)] x [Q(x)] (38.4)
x y [(x y 5) (x y 1)] (38.5) x y [x 0 y 0 xy 0] (38.6) x y (xy 0 x
0 y 0) (38.7) x y [(P(y) ~R(x)) (~ Q(x) ~ P(y))] (38.8) x y z (x y
z xy z)< (39) . 22yx [x 5 0] y [ ] 22yx [x 5 0] y [ ] >
. x [x 6] x [x 8] > x [x 6] x [x 8] >
kanuay.com
127
. A {2, 4,6,8, 10, ...} 2 12, 14, 16, … ()
. 1 1 , 1 3 4 , 1 3 5 9 , 1 3 5 7 16 , 1 3 5 7 9 25 n 2n
. 1, 3, 7, 15, 31, … 2, 4, 8, 16 63 ( 32)
. 11 11 121 , 111 111 12321 , 1111 1111 1234321 , … 11111 11111
123454321
. 3 12, 51, 96, 117, 258, 543, 2930, 5022, 7839 … 3 3 3
3.14 ()
. (n 1)2(n 1) 2 n n 1,2, 3, 4 4 1, 9 2, 16 4, 25 8 .. n 7,8,9,
...
. 2n n 5 n n 1,2, 3, 4 2n n 5 5, 7, 11, 17 n 5 2n n 5 25
2. 100 22 70
2 0 2 0
. (1) (2)
. (1) (2)
( “” )
( )
Math E-Book Release 2.7pre
( )
3. (1) (2)
( )
3.16
()
()
()
()
(1) (2) (3)
, , ,
. (40) a (40.1) 1, 3, 5, 7, a (40.5) 3, 1, 1, 3, a (40.2) 2, 7, 12,
17, a (40.6) 31 2 4
4 52 3, , , , a (40.3) 1, 2, 3, 4, a (40.7) 1, 4, 9, 16, a (40.4)
3,6, 12, 24, a (40.8) 3, 3 3, 3 3 3, 3 3 3 3 , a
(40.9) [] 125, 726, a, 40328, 362889
Math E-Book Release 2.7pre
Math E-Book Release 2.7pre
134
() (1) p | F | p | F T | p | p | T p | T | T |~p | T |~p p |~p | T
| F (2) (2.6) (2.10) (3) (3.5), (3.7), (3.9), (3.12) (3.11) T, T, F
(4) . . (5) (6) . . (7) .(p q) (~ r ~ s) . p ~ q r (8) . (9) .
(10.1) . (10.2) . (10.3) . (11) . (12) (13) (14.1) (14.2) . (15)
3:5 (16) (16.1), (16.2) (17) (17.2), (17.7) (18) (19) (19.1) (20) .
. (21) (22) (23) (24.1) s r
(24.2) r (25) (26) . (27) . (28) . (29) . . (30) (31) (32) , , , ,
(33) , , , (34) , , , , (35.1) (35.2) (36) (37) . (38.1) x [P(x)
Q(x)] (38.2) x [P (x) Q(x) ~ R(x))] (38.3) x [P(x)] x [Q(x)] (38.4)
x y [(x y 5) (x y 1)] (38.5) x y [x 0 y 0 < xy 0] > (38.6) x
y(xy 0 x 0 > y 0) > (38.7) x y [P (y) ~ R(x) Q(x)] (38.8) x y
z(x y z < xy z) (39) . . (40.1) –9 (40.2) 22 (40.3) 5 (40.4)
48
(40.5) –5 3 (40.6) 5/6 (40.7) 25
(40.8) 3 3 3 3 3 (40.9) 5047 (41.1) 37 12 444 , 37 15 555 (41.2) 9
9999 89991 , 9 99999 899991 (41.3) 1234 9 11111 5 , 12345 9 111111
6 (41.4) 9 9876 4 88888 , 9 98765 3 888888 (41.5) 11 14 154 , 11 15
165 (41.6) 1089 4 4356 , 1089 5 5445 (41.7) 2 (3) 2 (9) 2 (27) 2
(81) 3 (81 1) , 2 (3) 2 (9) 2 (27) 2 (81) 2 (243) 3 (243 1) (41.8)
6 7 2 (1 2 3 4 5 6) , 7 8 2 (1 2 3 4 5 6 7) (42) (45) (43.1),
(43.3), (44.1), (44.4), (45.1)
kanuay.com
135
() (1)
. “” T T T ( F F )
T p p .. “” ( T T T, T F F ) F p F .. “” p p p .. “” p ~p F ..
“”
. “” F F F ( T T )
T p T .. “” F p p .. “” ( F T T, F F F ) p p p .. “” p ~p T ..
“”
. “-” F T F ( F T T )
T p p .. F p T .. p T T .. p F ~p .. ( T F F, F F T ) p p T .. “-”
p ~p ~p .. “-” ( T F F, F T T )
. “” T, F
T p p .. F p ~p .. p p T .. p ~p F ..
(2.1) [(p s) (p r)] (p s) T T T (2.2) [(q s) r] [.....] T T (2.3)
[(r q) (p q)] [.....] T F F (2.4) [(p q) (q r)] ~ s T T (2.5) [(q
p) r] r T T T T (2.6) [(p q) ~ r] [(~ p q) r] F F F T T F (2.7) [(p
~ q) ~ r] [(p q) .....] F T F . T F (2.8) (p q) ~ r [... ...] F F
(2.9) [p (q r)] [.....] F F (2.10) [q (....)] [p (q ~ r)] F F T F .
T F (2.11) [(~ p ....) (~ r ....)] [.....] T T T
(3.1) (....) (p q) T T T (3.2) (....) (p q) T F . T (3.3) (~ r p)
(....) T T (3.4) r q T r q (p q) (s p) (s q) T T T T T (3.5) p T, q
F, r T (~ q (p r)) (~ r) F T T F (3.6) n F n [....] T (3.7) q F
(....) q F (3.8) q F, s F, r T, p F (q p) (....) T F
Math E-Book Release 2.7pre
136
(3.9) p r T, q s F q F, s F p q T p F r T .. T F F (3.10) p q F p
T, q F (....) ~ q T T (3.11) p q T p T, q T p r F r F (3.12) p T, p
~ r T r F
(3.13) (3.14) p, q, r, s p, q, r, s “”
.
(3.14) “-” p ~(r s) (~ r ~ s)~ p F T F T T r s .. r s p ~ p
..
(4) s r F .. s T, r F p r T .. p T p q T .. q T
. [(....) (q r)] (r s) F F F . [....] (~ r s) T T
(5) p q T .. p q r ~ s F .. r F, s T [(~ p r) ....] T F . ~ (....)
~ r T T . r (p ~ q) T F F . (....) (p q) T T
(6) p q , r ~ s .. . [.... (r ~ s)] [.... (~ r ~ s)] T T T . [....]
[(p ~ q) (r ~ s)] T T T . .
(7) . ~ [(~p ~ q) (r s)] (p q) (~ r ~ s) . ~ [~ (p ~ q) ~ r] (p ~
q) r
(8) p “”, q “” r “” .. (p q) r p q T r F
. p ~ q F . ~ p q F . ~ r ~ q F . p ~ r T .. .
kanuay.com
137
(9) . ~ (~ p ~ q) p q .. . (~ p q) (q ~ p) .. . p (~ q p) ~ q p q p
( .) .
(10.1) . . (p q) (q p) (10.2) {[(q ~ t) (p ~ p)] ~ q} r T
[(q ~ t) ~ q] r [(q ~ q) (~ t ~ q)] r
T (~ t ~ q) r (t q) r
(10.3) [(q r) (q ~ r)] [(p s) (p ~ s) ]
[q (r ~ r)] [(p (s ~ s)] q p F T
(11) . . ab 0, a 0, b 0> < <
.. . . (p q) r . (~p ~ q) ~ r .. . ~ r (~p ~ q) ..
(12) . p (q r) . (~ q ~ r) ~ p . ~ p q r .. . . .
(13) . ~(p ~ r) ~ q ~p r ~ q q (r ~p) .. . p (q r) ~p (~ q r) q (p
r) ~ q (~p r) .. . (p q) r ~p ~ q r (p ~ q) (p r) ~p ~ q ~p r
..
(14.1)
p q p*p q*q (p*p)*(q*q) T T F F
T F T F
F F T T
F T F T
T F F F
(14.2) F*F “” ( “” T, T, T, F ) .. p q ~(p q) . ~(~p q) ~(p
q)
(15)
p q r q*r p*(q*r) T T T T F F F F
T T F F T T F F
T F T F T F T F
F F F T F F F T
F F F F T T T F
: 3 : 5
(16) “-” “”
(16.1) (p q) [(p q) r] F T F T T T T F
(16.2) (p q) [(p q) r] F T F T T T T F
(16.3) [(p q) (q r)] (p r) F T T F T T F F T F q ..
Math E-Book Release 2.7pre
138
(16.4) [(p r) (q r)] [(p q) r] F T T F T T T T T T F r ..
(16.5) [(p r) (q r)] [(p q) r] F T T F T F F F F F F F p q p q
..
(16.6) [(p r) (q s) (p q)] (r s) F T T T F F F F F T T F F p , q
..
(16.7) (p q) r (p r)(p q) F T T F T F F T T T F q ..
(17) “” “”
(17.1) ~(p ~ q) ~(~p ~ q) p q ..
(17.2) (~p q) p (~ p p) (q p) q p T ..
(17.3) (p q) ~p (p ~ p) (q ~p) F q ~p ..
(17.4) (p q) (p q) (q p) (~ p q) (~ q p) ..
(17.5) (p q) (p q) ~(p q) ~(p q) (~p ~ q) (~p ~ q) ..
(17.6) ~p (q r) (~p q) (~p r) (p q) (p r) ..
(17.7) ~p ~ q r ~(~p q) r (p ~ q) r ..
(17.8) “” ()
p q r T T T T F F F F
T T F F T T F F
T F T F T F T F
T F F T F T T F
T F F T F T T F
(18) “” “”
(18.1) [(p r) (q r)] (p q) F F F F F p q r r
(18.2) [(~ p q) ~ p] (p q) F F F T F p q
kanuay.com
139
(19.1) (p ~p) (q ~ q) F F T () ..
(19.2) [p T] [~ p F] p ~p F
(19.3) p q ~p ~ q ~(p q) (~p ~q) ~ F
(20) p, q, r “”
. (p q) (~ p ~ q) F T F F T F T
. (p q) (~p q)
. ~((p q) r) (~(p q) ~ r) F T F F F F F F T r ..
. (~ p r) (~ q r) (~ p ~ q) r (p q) r
. (~ p q) (~p r) ~p (q r) p (q r)
.. . .
(21) ..
(22.1) 1. p q
2. (p q) r 3. ~ (s r) 4. p
s (23.2)
1. p q
4. p q 2.(p q) r r 3. ~ s ~ r
r ~ s ~ s
3. q p r
1. ~p q 2. q ~ r ~p ~ r r p
3.
.. . ~ s p .. ( “”) . s p .. ~p . ~ r ~ s .. “” r . p ~ q ..
~p
(27) . x 2x 0 x 0 . x 2 8 4 2 4 . x 1 . x
(28) . “ x ... x 2 ” ... x 3 T F F
. “ x ... x 0.5 ” ... x 2 F F T
. “ x ... x ” ... T T
. “ x ... x 22/7 ” ... 22/7 x 1 T
