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11. ( R2.9pre .4-5-6 ) (Set) , 7, 360 , (Element Member) A, B, C {
} (comma) A B 7 A { , , , , , , }B {2, 1, 0, 1, 2} B {0, 1, 1, 2,
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Math E-BookRelease 2.5 ( ) {2, 1, 0, 1, 2} {0, 1, 1, 2, 2} C {2, 3,
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29 9,10, 11, 12, 31 S {2, 4, ...} 6, 8, 10, 8, 16, 32, (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{, 0, 1, {2, 3, 4}, {5, {6}},(7, 8)}S 6 , 0, 1, {2,3,4},
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(2,1), {1,2} .. 1,2 2,1 1,2 2,1 13. [email protected] () 0 S {0}
1 .. {} n() 0 n({}) 1 ( ) (Infinite Set) F 2, G 0 1 F {1, 0, 1, 2,
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x x A { , , , , , , } 14. 14Math E-BookRelease 2.5 . B {x | x 7} x
x 7 B {2, 1, 0, 1, 2} . Z { x2 | x 4} x2 x 4 x -4, -3, -2, -1, 0,
1, 2, 3, 4 x2 Z {0, 1, 4, 9, 16} () : A , 2 B , 3 C , 0.5 G , 2.5 B
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| x > 2 } 2 () 15. 15 [email protected] . (Subset) X Y X Y X Y
X {1, 2} {0, 1, 2} 1 2 {0, 1, 2}: () () X Y X Y X Y {1, 3} {0, 1,
2} 3 {0, 1, 2} S 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,
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A B (A B B A) 16. 16 Math E-BookRelease 2.5 ( U ) U ( U ) (Proper
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{a, b} 2 P(B) {, {a}, {b}, {a, b}} 17. [email protected] .C {2,
3, 5} 23 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 20 1
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{6, 7}}, {, 6, 7, {6, 7}} } 1.4 F {, 1, {2, 3}} . 1F .. 1 F.. ( 1 )
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5, 6, 7} . X X P(A)1 X P(A) X A X 24 16 2 4 4 4 4 4 0 1 2 3 4 1 4 6
4 1 16 18. 18 Math E-BookRelease 2.5. X X P(A)n(X) < 21 X P(A) X
A 16 ( .) n(X) < 2 4 4 4 0 1 2 1 4 6 11 2 X , {1} , {3} , {5} ,
{7} , {1, 3} , {1, 5} , {1, 7} , {3, 5} , {3, 7} , {5, 7} 11 . Y A
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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,
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1}}(2.4) {0, 1} {, 0, 1, {1}, {0, 1}}(2.10) {0, 1} {, 0, 1, {1},
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{1}, {0, 1}}(2.6) {{0, 1}} {, 0, 1, {1}, {0, 1}}(2.12) {{0, 1}} {,
0, 1, {1}, {0, 1}} 19. 19 [email protected](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},
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( .) n r nn!r (nr)! r ! x! x ! 1 2 3 ... x 7 3 7 7! 12 3 4 5 6 7 3
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E-Book Release 2.5 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:)U A U A A c A A A { x | x A
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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 23. 23 [email protected] 1.7 A {2, 3, 5, 7} B {, 1, 2, {3,
4}, 5} A B , A B , A, B , B A, A BA 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 BA B A +++ B + + (
C ) C . (A C) B U A C +++++ B ++ ++ . C (A B) C +++ A B +++++ (A B)
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(A C)3. P(A) P(B) P(A B) P(A) P(B) P(A B) 24. 24 Math E-Book
Release 2.5 1.9 . (A B) (B A) (A B) U (A B) (B A) (A B) .. (A B) (B
A) (A B) AB A B .[ A (A B)] [(A C) C ] A (A B) A (A B) A (A C) C (A
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A) A B C A B C A B C A B C A B C +++ 25. [email protected] 1.11
C { , {}, 0, {{}, 0}, {, {0}}, {{, {0}}} } . n(P(C))n(C) 6 n(P(C))
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n(P(C)) 64 24 60 . n(C P(C)) 4 (.) n(C P(C)) 2 (.) n(P(C) C) 60 C
P(C) . n [(P(C) C) (C P(C))] . . () 60 2 62 ( ) 1.12 A, B A {1, 3,
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E-BookRelease 2.5 1.5 ( A Y B ) A Y 1, 3, 5, 7 Y ( A Y ) {1, 3, 5,
7} 24 1 15 Y B 2, 4, 6 Y ( 1, 3, 5, 7 Y B ) 23 8 15 8 120 . Z {1,
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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 27. 27 [email protected] (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
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B)) (17) (17.1) U(17.7) A A (17.2) U (17.8) A A U (17.3) A (A B)
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(19.6) x A x AB (19.3) A B A B(19.7) x A x AB (19.4) (A B) B
A(19.8) x A x (A B ) 28. 28 Math E-Book Release 2.5(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 A B A
B S ( A B A B A B)(22) A B A 3 (23) (23.1) (A B) (B A) (A B) (23.2)
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{0, 3}}(25.3) P (A) P (B) P (C) {, {0}}(25.4) P (A) P (B) P (C )
{}(26) n (U) 35 , n (A) 22 , n (B) 18 n(A B) 29.
[email protected](27) n (A) a , n (B) b , n (C) c , n (D) dn (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)
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1, {1}} A B (29.1) n [ P (A) P (B) ] 8(29.3) P (A B) {} (29.2) {1}
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2.5(36) U { x I | 2 < x < 6 } I 2A {k | k U } B { k |k U }C
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100 2 3 5 36. 36 Math E-Book Release 2.5 ()(1) (1.2), (1.6),
(1.9),(20.1) A B(34.1) 16 (1.12), (1.15), (1.16) (20.2) A B(34.2)
16(2) (2.2), (2.5), (20.3) (34.3) 8 (2.6) (20.4) A B(34.4) 24(3)
(3.1), (3.3) (20.5) A(35) 16(4) (4.6), (4.8), (20.6) A B(36) 4
(4.10) (20.7) (37) 56(5) (5.3) (20.8) A B(38) 31(6) P(B) {, {},
{{0}},(20.9) (39) 13 {{}}, {, {0}}, {, {}}, (21) (21.4) (40) 50
{{0}, {}}, {, {0}, {}}} (22) A B (41) 11(6.1) (6.4) A B (42)
66(6.5) (23.1) A B(43) 6(6.6) (23.2) B(44) 11(7) (7.1), (7.7)
(23.3) B (45) 13(8) (8.1), (8.4) (23.4) (A E) (46) 9(9) 9 , 126
(24) (47) 130(10) 32, 6(25) (25.3) (48) 6(11) (11.1), (11.4) (26)
13 (49) 385(12) (12.1) (27) d, a (50) 26, 12, 24(13) (13.3) (28) .
(51) 15(14) . (29) (29.4) (52) 50, 95, 40,(15) {1, 2, 4, 5, 6, 7,
8, 9, 10} (30) 61+3=64 230, 115, 255(16) , , 16-2=14(31.1) 8(53) 7,
19(17) (31.2) 32 (54) 20(18) (18.3), (18.5) (32.1) 16 (55) 13(19)
(19.3), (19.4),(32.2) 112(56) . (19.6) (33.1) 64 (57) 22-13=9(33.2)
960(58) 26 37. [email protected] ()(1) 2 a {b, c}(3.1) {} A ( )
( A 5 {} ) 8 (1.2) (1.6) (3.2) {} A A ( A ) 4 (3.3) {{a}, b} A ,
{a} , {{b, c}} , {a, {b, c}}{a} A b A ( ) (3.4) .. {a, b} A 8 (1.9)
(1.12) (1.15) ( A ) (1.16) {a, b} A a A b A (1.1) {a, b} A(1.3) b (
{b, c} )(1.5) (1.7) 1.3(4.1) .. P(A) A(1.10) (1.11) (1.13) 1.3
(1.14) b c(4.2) .. {} P(A) {} A A A (2.1) (4.3) ! .. ( 5 (4.4) ..
{} P(A) P(A) ) 4.1 (2.2) {0} (4.5) .. {, a, {b}} P(A)(2.3) {1} {,
a, {b}} A () A a A {b} A(2.4) {0, 1} () (4.6) .. P(A) (2.5) (2.6)
.. 2.2(4.7) .. {a} P(A) {a} A(2.7) a A (4.8) .. {b} P(A) {b} A(2.8)
0 b A (4.9) .. 4.8 (2.9) 1 {b} A (4.10) .. {, a, {b}} P(A)(2.10) 0
1 P(A) a P(A) {b} P(A) ( 4.1, 4.6, 4.8) (2.11) 0 {1} (2.12) {0, 1}
38. 38Math E-Book Release 2.5(5.1) {, {1}, {1, 2}} P(A) (7.1) .. {,
{1}, {1, 2}} A (7.2) .. A {1} A {1, 2} A 20 1 ().. .. () (5.2) {,
{1}, {1, 2}} P(A) P(A) ( A ) {1} P(A) ( {1} A 1 A ) (7.3) .. {1, 2}
P(A) ( {1, 2} A 1 A, 2 A ) ( ).. (7.4) .. (7.2) (5.3) {{1}, {2},
{3}} P(A) {{1}, {2}, {3}} A(7.5) .. P() {1} A {2} A {3} A( 7.2 ) ..
(7.6) .. (5.4) {{1}, {2}, {3}} P(A)(7.7) .. {} P() {} {1} P(A), {2}
P(A), {3} P(A) {1} A, {2} A, {3} A( 7.1 ) 1 A, 2 A, 3 A .. (7.8) ..
{} P() P() ( 7.2 ) (6) P(B) { , {}, {{0}}, {{}}, {, {0}},{, {}},
{{0}, {}}, {, {0}, {}} }(6.1) B .. B (8.1) .. A B .. 2 n(A) 25 32 (
B )(8.2) ..(6.2) P(B) .. B () 31 B (8.3) .. P(A) 1 P(B) .. P(A) 32
(8.4) .. 8.3(6.3) {} B .. B {} () {} B .. B n(A) B (9) 2 512(6.4)
{} P(B) .. n(A) 9 {} B B A 5 9 9! {} P(B) .. P(B) 126 ()5! 4! 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 39. [email protected](10) X (13.1) (13.2) A
()(1) A P(S) .. A S(13.3) {x} A x A(2) 1 A 7 A (13.1) A .. 1 A 7
A(13.4) {x} x ( A x ) 5 2, 3, 4, 5, 6 {2,3,4,5,6} n(X) 25 32 (14) A
B, A C, B C Y A A B C {3, 5}( 32 X) B 0, 1, 2 1 6A 35.. 0 2{1},
{1,2}, {1,3}, {1,4}, {1,5}, {1,2,3} 6 n(Y) 6 C B 1 435 A C {0, 1,
2, 3, 5}A 0 2 A C (11.1) x A x B A C 4 B (11.2) x y B , B . B C {1,
4} y A ()(11.3) {A} {B} A {B} {B} B(11.4) A B(15) U {1, 2, 3, ...,
10} {A} {B} 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}(12.1) ()(12.2) .. B {A}, C {B} (16.1) A
P(B) A B C {{A}} A C A B 1 B(12.3) .. A, B, C A B, B C A(16.2) P(A)
P(B) P(A B) A C P({0}) {, {0}}B {1} {, {0}} C(16.3) A B {0, 1, {1},
{0, 1}} n(P(A B)) 24 16 A B {0} n(P(A B)) 21 2.. 16 2 14 40. 40
Math E-BookRelease 2.5(17.1) (17.2) .. U (19.5) .. A B x B
(complement) (19.6) x A x (A B) (17.3) (17.6) x A x A B(17.7)
(17.11) A A x A x (A B)..(17.8) (17.10) (19.7) x A x (A B) (17.12)
(!) .. x B (19.8) x A x A B 17.3 17.12 .. x B (20) (18.1) ()( )
(18.2) .. A B U(18.3) .. A B A B .. A B AB A B ABBA(18.4) A B A B B
C B (20.1) A (A B) B C A A B( B C ) (20.2) (A B) B C A B B (20.3)
(A B) B A C (A C) U .. ( A C ) (20.4) A (A B) A B(18.5) A B A
B(20.5) A (A B) A B C B C (20.6) (A B) B A B A C A C(20.7) (A B) B
.. (20.8) A (A B) A B A B B C (20.9) B A B A A C .. A C (A B) (B A
) A C B (21.1) .. C U (19.1) .. A BA B AB AB( A, B )(21.2) .. C
(19.2) .. A B(21.3) .. C U A B BA (21.4) (19.3) .. A B A (B ) A
B(19.4) .. B A B A (B A) (22) A B A, B (A B ) 41. 41
[email protected](23) 2 (25.2) .. A B C {0, 3} (20) (25.3) .. A B
C (23.1) A B P(A B C) {}(23.2) ( ) ( )(25.4) .. A B C B(23.3) ( ) (
) B(26) n(A B ) n(A B) (23.4) 5 n(A B) () .. B A A BB C n(A B) n(A)
22 n(A B) 35 22 13 D E C E A E(27) n(A) a, n(B) b n(A B) b ( (A E))
(A E) B A ( B A ).. D C B A n(A B C D) n(D) d (24.1) BC 2 n(A B C
D) n(A) a [(A A) B C] (B C)U (B C) (B C) U ..(28) P(C) C {a,
c}(24.2) C n(P(A)) 8 n(A) 3 C [(A B D ) A B D] n(P(B)) 16 n(B) 4 A,
B, D C A C BC [ (A B D ) (A B D ) ] C .. A {a, c, } B {a, c, , } U
{b, d, e} A B b A B (24.3) .. (A B) (A B) A {a, c, b} B {a, c, d,
e} P() P() . d A B ( B A) ..(24.4) .. A B B A . e C B ( B C) .. P(A
B) P(B A) . b A B .. . .. (A B) A B A B {b}(24.5) A B P(A) P(B)
P(A) P(B) P(B) ......(1) (29) A B A B A B A B B A B ( A B A ) P(A
B) P(B) .....(2).. (1) (2) (29.1) n[P(A B)] n[P(A)] 23 8..(29.2)
{1} P(A B) {1} A B 1 A B ( 1 A ) ..(25) (29.3) P(A B) P() {} ..P()
P() P() P( ) (29.4) B A (25.1) ..A B C {1, 2} 42. 42Math
E-BookRelease 2.5(30) A P(A) 3 32.2 33.2 , {} , {0} 34.2 34.4 ..
n(P(A) A) 26 3 61 n(A P(A)) 6 3 3 61 3 64 (35) {3, 5, 8} D {2, 3,
4, ..., 8} D 24 16 P(A) A A P(A) () (36) U {2, 1, 0, 1, 2, ..., 6}
A {0, 1, 4} ( k2 U )(31.1) A {2, 4, {1,3,5} } B {0, 1, 2} 23 8 {0,
1} X {0, 1, 2, 4}(31.2) A {1, 3, 5, 7, 9, 11, 12, 13, 14, 15,..
n(C) X {2,4,6,8,10} } 22 4 25 32 (37) X {a, b, c, d, e, f}(32.1) X
{1, 2, 3, {4,5,6,7} } {a, c, d} X 24 16 X { {a,c,d} ,(32.2) X {
{1,2,3} , {b,e,f} } {4,5,6,7} } (23 1) (23) 56 (23 1) (24) 112 (38)
n(A) 5(33.1) n(C) S 26 64S1 { B | B A, n(B) 1} ,(33.2) n(C) SS2 { B
| B A, n(B) 2 } , ... (24 1) (26) 960 S3 { B | B A, n(B) 3 }( 32 )
S S1 S2 S3 S4 S5 = { A }.. n(S) 25 1 31 (34.1) {0} X {0, 2, 4, 6,
8} X 24 16 (34.2) {0} X X {0, 2, 4, 6, 8}(39) .. X {0, 2, 4, 6, 8}
Un(A B ) n(A B) {0} X ( 34.1) 32 3255 25 24 16 n(B) 55 AB(34.3) {0,
2} X {0, 2, 4, 6, 8} n(A B ) n(A B) X 23 8 n(U) 100(34.4) {0, 2} X
X {0, 2, 4, 6, 8} .. 100 32 55 13.. X {0, 2, 4, 6, 8} n(A ) 40 {0,
2} X ( 34.3) 25 23 24 43. 43 [email protected](40) A (43) B n(A)
100 n(B) 150 .. 100 U n(A B) 300 100 200 () n(A B) n(A) n(B) n(A
B)20 , 17 , 15 200 100 150 n(A B) n(A B) 502( ) 20 17 15 52 50 26
() 26 ( ) 26 20 6 (41) .. 1 x 6(44) Ux 50 32 6 1 11 U32 20 , 11 2(
) 7 , (42) 35 U 2 20 .....(1) 35 2 7 .....(2) 27 27 2 (1 ) (2) ; 3
3 33 32 32 11 3 2 94 80 ( 80) 14 (45) n(A B ) n(A B) 80 14 66 U ..
n(A B) 80 32 48( , 48 35 27 n(A B) ) n(A B) 14 n(A B) 80 14 66 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 44. 44 Math E-Book Release 2.5(46) ( 43 )
(48) 24 16 .....(1) 7 11 21 7 14 .....(2) U 2 30() 24 .. 30 24 6 7
, 6 , 5 2( ) 18(49) U.. 9 200 600 30 . 500 2 230(47) 15 215 U 100
.. 600 215 385 48 49 60 , 47) 30 .. 50 30 20 35 .. 60 35 2555 .. 80
55 25(50) 3 2 120 2 70 100 20 4 !3 4 60 10( , 0 ).. 160 130 1. 6 8
3 ( 5) 2 4 18 4 14 ( ) ( ) 4 2 6 14 262. ( ) 4 2 6 12 3 4 26 2 24
45. 45 [email protected](51) A, B, C (53) A, B, C , , , , ..n(A)
140 , n(B) 195 , n(C) 155 , 42 n(A B) 50 , n(B C) 45 , 1 n(A C) 60
, n(A B C) 20 n(A B C) 42 3 .. 3 42 29 22 21 10 12 15 n(A B C)n(A B
C) 140 195 155 50 45 60 20 n(A B C) 7 355 7 355 B AB A 14 34 370 15
355 5 7 8 1 C 15 C 14 4 1 19 n[(A B C) ] n(U) n(A B C) 370 355
15(54) AB ? z x y (52) 3 C ( 180 95 92 125 52 43 57 x 1.14 1) x 20
B y n(A C) 20 43 20 23 A 50 30 120 z n(A B) 20 52 20 32 40 20 25..
95 20 23 32 20 1570C 50 (55) 3 , , n(A B C) 100 n(A B C) 70 25 95
.. U 3 3235 40 39 n(A B C)120 25 70 15 230 20 32 35 20 87 n(A) n(U)
n(A) 370 140 230 .. 100 87 1330 40 25 20 115 50 120 70 15 255 46.
46Math E-Book Release 2.5(56) (58) U {0, 1, 2, ..., 100} x, y x 20
y A {x|x 2 } 22 23 11B {x|x 3 }9C {x|x 5 }x y 20 23 22 11 9 100 n(A
B C ) n(A B C) (x 20 23 22) (y 20 23 11) 6 n(U) n(A B C) x 5, y 10
n(A B C) 3 70 , 64 n(A) 2 .. 51 65 n(B) 3 .. 34 . . 70 64 65 199
n(C) 5 .. 21 . ( 5 ) . 5 10 9 24 n(A B) 2 3 6 .. 17 n(A C) 2 5 (57)
3 (, 10 .. 11 , ) n(B C) 3 5 15 .. 7 n(A B C) 2 3 5 30 .. 4 xy n(A
B C) 51 34 21 17 11 7 4 758 4 n(U) 101 n(A B C ) 101 75 26 3 n(A B
C) 60 8 (58) 20 15 22 6 10 11 xS A 6.. x 22 B 8 A B()?40 4 15 13 18
8 6 9 y.. 6 8 .. y 13 48 xy 9 ! 6 8 ! ... 24 47. ( R2.9pre .4-5-6 )
(Natural Number) (Counting Number) 1, 2, 3, 4, ... N = {1,2, 3,
4,...} ( ) ( ) (Fraction) () (Integer) II {..., 3, 2, 1, 0, 1, 2,
3, ...} ( 0) (Rational Number) QaQ { b | a, b I b 0 } (N I ) ( I Q
) 48. 48 Math E-Book Release 2.5 1. 5 5.0 5 0.42 0.420 21 1.3333...
1.3 4 1 5 32. (Irrational Number) Q 2 1.41421.. 3 1.73205..
3.14159.. 3 2 1.25992.. () S e ( ) 2.71828.. 3. N I Q .. Q
(RealNumber) () R S 1. (, ) 3 2. 0( 0 ) ( ) 49. 49
[email protected] CRIm QQIQII-I0 I N Im C 1 (Imaginary Number)
(Complex Number) C 2.1 ( A, B, C, D) A B C Q D I . NA B C D . A 2 2
0 B 2 2 2 C D . {x | x < 0}A B C D () 2 50. 50Math E-Book
Release 2.5 . {1.414, 22/ 7} A B C 1.414 2 22/7 () 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 ae ea a a 0 0 a a .. 0 a 1 1 a a .. 1 (Inverse) a
a i ai ia e a (a) (a) a 0 .. a a 1 11 a (a) (a) a 1 .. a a a 1 ( a
a ) 0 0 1 2.2 x y x y 2 . a e a a e 2 a ..e 2 e a a e a 2 a ..e 2 2
51. [email protected] . a 2 a i 2 a i 2 2 .. i 4a ( 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 .. S ( a e e a ) .. ( ) .[1] (Reflexive Property)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 a c b c (, ) 52. 52Math E-Book Release 2.5[1]
0 1[2] a a a ( 0) 1/a[3] (Closure Property) a b a+b a b ab [4]
(Commutative Property) a b b a ab b a[5] (Associative Property) a
(b c) (a b) c ( a b c ) a (b c) (ab) c( abc)[6] (Distributive
Property) a (b c) ab a c (a b) c a c b c[7] a 0 a R a R [1] a b a
(b) ( )[2] a (b c) ab a c (a b) c a c b c () 0 0 [1] (b ) a b a b1(
) a b c c a b c [2] (a )a1 0[3] (a, b )1a b b a 53. 53
[email protected][4] (b, c ) a ac b bc[5] (b, c )a d ac bd a d ad
bcbcb c bc[6] (b, c, d )a/ba a ac a/b adc bcb/cbc/d bc 1. () a b a
c b c a c b c a b 2. () () 0 a b a c b c a b a/c b/c c 0 3. 0() a c
b c a b c 0 4. 2 a b a2 b2 a2 b2 a b a b .(1) (1.1) 0.343443444...
(1.2) 0.112112112... (1.3) a2 a (1.4) a2 a 54. 54 Math E-Book
Release 2.5(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 c a c b b a a(2.4) b, c 0 b c b c S .. 1 ( 1 ) ... ( ) (3) . .
2. . 4 (4) (4.1) (4.2) (4.3) 0 (4.4) 0 (5) A { x N | x Q } B N A
(5.1) A B (5.2) A B (6) A x A y A x y 1 xy A . 0. . . 01(7) 6 5 6 5
*a bc(8) aa bc. (a b) a c. (b c) b a bb ca. (a b) (c b) b. (c a) (b
a) b cc ab 55. 55 [email protected](9) . x y 3 x y (x y). x y 2 (x
y) 3 x y . x y 3 1. x y 2 x y 1xy xy xy(10) a b 3ab (a b) x (y z)
(z y) x (11) A A a b a b a b a b 2 2 (11.1) A (11.2) A .
(Polynomial) x p(x) p(x) anxn an 1xn 1 ... a1x a0 a n n (degree)
p(x) 4x3 x2 2x 6 p(c) x c p(x) p(x) 4x3 x2 2x 6 p(1) 4 (1)3 (1)2 2
(1) 6 7 p(2) 4 (2)3 (2)2 2 (2) 6 26 (Equality) ( x) ( ) p(x) anxn
an 1xn 1 ... a1x a0 0 a b c d ... 0 a 0 b 0 c 0 d 0 ...( 0 ) a, b,
c, d, 0 0 56. 56Math E-Book Release 2.5 () (b1x c1)(b2x c2)(b3x
c3)... 0 0 ccc x b x b x b ... 1 2 31 23 2.4 . x 6x 5 02 (x 5)(x 1)
0 x 5 0 x1 0 x 5x 1 ..{5, 1}. 6x2 13x 5 0 (2x 5)(3x 1 0 ) 2x 5 0 3x
1 0 x 5 2x 13.. {5 , 1 }23. 2x2 4x 1 0 ()1 2x2 4x 1.. 2(x2 2x) 12
2(x 2x + 1) 1 + 2( +1 +2 2 ) 2(x 1)2 1 .. 2 1/2 x 1 1 x 1 12 2{1 1
, 1 1 }22 B B2 4AC2 x 2A4 42 4(2)(1) x 1 1 .. {1 1 , 1 1 } 2(2)2 2
2 . , , 57. 57 [email protected] (Remainder Theorem) p(x) x c p(c)
p(x) 4x 2x x 3 x 3 32 p(3) 4(3) 2(3) (3) 3 96 x 2 3 2 p(2) 4(2)
2(2) (2) 3 3932 1 0 ( x ) 1. 1 . S2. 2.5 . 2x x 6x 1 x 2 3 2 2x x
6x 1 x 2 2(2) (2) 6(2) 1 13 2 3 2 . 2x3 x2 6x 1 x1 3 22x x 6x 1 x 1
2(1)3 (1)2 6(1) 1 4 2.6 p(x) x 3, x1 12, x 2 25 . p(x) x3 p(x) Ax2
Bx C 3 3 3 x 3 p(0) 3 A(0)2 B(0) C 3 x 1 12 p(1) 12 A(1)2 B(1) C 12
x 2 25 p(2) 25 A(2)2 B(2) C 25 A 2 , B 7 , C3 ... p(x) 2x2 7x 3
p(x) x 3 22(3) 7(3) 3 42 58. 58Math E-BookRelease 2.5. p(x) xc c 1
p(x) x c ... x c p(x) 2x2 7x 3 (2x 1)(x 3) p(x) x c c 1/2 c 3 2
p(x) xc ... p(c) 0() 2c 7c 3 (2c 1)(c 3) 0 2 c 1/2 c 3 . p(x) xc 7
c 1 p(x) x c 7 p(c) 7 2c2 7c 3 7 2c2 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 ) p(x) 7 2x2 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 2x3 x2 25x 12 p (3) 2(3)3 (3)2 25(3) 12 0
x3 2x3 x2 25x 12 x3 2x2 7x 4x 3 2x3 x2 25x 12 (x 3)(2x2 7x 4) 2x3
x2 25x 12 (x 3)(2x 1)(x 4).. 2x3 x2 25x 12 (x 3)(2x 1)(x 4) 59.
[email protected] 1. 3 3 2 2. 2x x 25x 12 0 3 x 3 1/2 4 k x
(a/b) p(x) a/b a a0 b an x k p(x) k 1. a a0 () b an ( x ) k a/b 2.
k ( a/b ) x k ( x (a/b) ) 2.8 2x3 x2 25x 12 () 12 1, 2, 3, 4, 6, 12
2 1, 2 k x k .. 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 12, 12, 1/2, 1/2,
3/2, 3/2 k p (k) 0 .. p (4) 2(4)3 (4)2 25(4) 12 0 x 4 2x3 x2 25x 12
x4 2x2 7x 3x4 2x3 x2 25x 12 (x 4)(2x2 7x 3) 2x3 x2 25x 12 (x 4)(2x
1)(x 3) .. 2x3 x2 25x 12 (x 4)(2x 1)(x 3) 3 60. 60 Math
E-BookRelease 2.5 k S x2 2 k 1, 1, 2, 2 (x 2)(x 2) ( k ) . () x c
() (Synthetic Division) x4 3x3 4x2 x 6 x 21. ( 1, 3, 4, 1, 6 ) c (
2) 21 34 1 6 2. ( 1) ( 2) 2 1 3 4 16 2 13. ( 3 2 1 ) ( 2) .. 2 13 4
16 2 2 4 10 1 12 5 44. .. x3 x2 2x 5 4 61. 61 [email protected]
2.9 2x3 7x 6 x1 1 20 7 6 2 2 5 .. 2x2 2x 5 112 2 5 11 0 S ( x2 0)
2.10 3x4 7x3 4x 3x 7x3 4x 04 x 3x4 7x3 4x (x)(3x3 7x2 4) 4 () 1, 2,
4 3 () 1, 3 1, 2, 4, 1/ 3, 2/ 3, 4/ 3 ...1 3 704 3 4 4 0 ... 2 3 4
4 0 64 3x3 7x2 4 (x 1)(x 2)(3x 2) 3 20 3x4 7x3 4x x (x 1)(x 2)(3x
2) .. 3x4 7x3 4x 0 {0, 1, 2, 2 }3 1. ( 2 ) 2. 62. 62 Math
E-BookRelease 2.5 .(12) 4x3 21x2 26x 17 x 4 a 3x3 13x2 11x 5 x 3 b
ba(13) x1 x2 2a x 2 xa a (14) x4 x3 3x2 x 1 2x3 x2 75x a x 5 a a1
2(15) x 2 x3 ax2 x 2b x x b4a ab(16) x2 2x 3 x4 ax3 bx2 3x 4 x2 x 2
x3 10x2 cx d a b c d (17) (17.1) x3 7x 6 0 (17.2) x3 4x2 x 6 0
(17.3) 6x3 11x2 4x 4 0 (17.4) x4 x3 11x2 5x 30 0 (17.5) 3x5 13x4
7x3 17x2 6x 0 (17.6) x 6 2x 5 14x 4 40x 3 11x 2 38x 24 0 (17.7) x 6
x 5 7x 4 9x 3 6x 2 28x 24 0(18) 3x6 2x5 64x4 96x3 27x2 98x 40 2/5 (
0) x 2/5S 5x 2 5 (19) ... x3 7x 6 , 3x3 7x2 4 x4 3x3 6x 4(20) ...
x3 2x2 5x 6 x3 x2 10x 8(21) x2 a2b2 2abx b2 0 (21.1) a (21.2) b
(21.3) a (21.4) b 63. [email protected]. (Inequality) ( >
< ) (, ) [1] a b b a Ra b a b R [2] (Transitive Property) a b b
c a c[3] a b a c b c ac bc c 0 a b ac bc c 0[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 ca b < c a b b < ca < b c a < b b ca
< b < c a < b b < c (Interval) 2 5 (2, 5) [2, 5] (2, 5]
[2, 5) 64. 64 Math E-BookRelease 2.5 (a, b) { x | a x b } a 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) () () S ( R ) A, B, C
, , , 2.11 A [1, 4] B (2, 3)AB AB (A B) A B [1, 3) -2 1 34A B (2,
4] -2 1 34 (A B) (, 2] (4, ) -2 1 34 65. 65 [email protected] 2.12
A [2, ) B (2, 3] AB BAA B {2} (3, ) BA -2 1 3 x a x b 2 a > 0 b
> 0 (a2 , b2) a 0 b 0 (b2 , a2) a 0 b > 0 0 ( 0 ) a2 b2 x (4,
3) x 1, 0, 1 0 9 16 x2 [0, 16) xx2 () 2.13 x 2. x (2, 5) x (4, 25)
2. x (5, 2) x (4, 25) 2. x (2, 5) x 0 0 4 25 .. x2 [0, 25) . x (5,
2) x2 [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) 66. 66Math E-Book Release 2.5 () a x
b c y d x y (ac, b d) ( ) x y (ad, 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 x2y x [0, 9] 2 2 x y 0, 0, 9, 45 .. x2y [9, 45].. x2y [0, 45] 1.
() a b a c b c a c b c a b 2. () () a b ( c 0 ) a c b c( c 0 ) a c
b c a b ( c 0 ) a/c b/c( c 0 ) a/c b/c a c b c ( c 0 ) a b ( c 0 )
a b 67. 67 [email protected]. () a b ( a, b 0 ) a2 b2 ( a, b 0 )
a2 b24. () 0 a b c d ... 0 () a, b, c,d, x2 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. x2 x 6 0 (2, 3)2. x2
x 6 > 0 (, 2] [3, ) x2 x 6 < 0 [2, 3] , , , ( 68. 68Math
E-Book Release 2.5) 1. 0 ( x ) ( 1 ) ()(x c1)(x c2)(x c3)... 2 (x
3)(x 31) > 0(x d1)(x d2)... x (x 2)2. x 0 ( c1, c2 , c3 , d1, d2
, ... ) 3 0 1 1 2 2 23. +, , +, , ... ( x ) + - + - + - + - +3 0 1
1 2 2 24. + ( x d1, d2 , ... )- + - + - + - + 3 0 1 1 2 2 25. () x
[3, 0) {1} (2, )3 0 12 0 S (x2)(x3) = 0 x = 2 3 .. (x2)(x3) < 0
x < 2 3 !!1. 2 ( 2) 2 69. 69 [email protected]. x B B 4AC 22A
2 (x 2)(x 5)(x 2x 2) < 0x3 + + 2 3 5 S () 1 1 12 x x 3 0 x 2 2 1
13 1 13 2, 2 2.16 x4 3x3 13x2 9x 30 0 (x 2)(x 5)(x2 3) 0 (x 2)(x
5)(x 3)(x 3) 0 + + +5 3 3 2 (5, 3) ( 3, 2) .. x4 3x3 13x2 9x 30 0+
+ +5 3 3 2 (, 5) ( 3, 3) (2, ) 2.17 x 2 2x 19 . x4 4 ( x 4 0 x4) x
2 2x 19 4(x 4) x 2 2x 3 0 (x 1)(x 3) 0 {1, 3} 70. 70 Math E-Book
Release 2.5x 2 2x 19. x4 0} x (x 5)2 2x 5(37) A > 0x22x 1 B 1x5
B Ax1(38) S 2x 2 a S a2 1 (39) A 2x 2 x 3 0 x 2 2x 2 B < 1 x 2
BA(40) 1212x 1(40.1) (40.3) x1 3x 1 x 2 21 x4 2(40.2)
>(40.4)> x1 x8 x 2x1(41) (41.1) { x | x2 7 } (41.3) (2, 6]
(3, 8] (41.2) { 1, 5, 7, 9 } [6, )(41.4) { x 2n | n I } 73.
[email protected](42) a A {x | x , n I } n11 b B {x | x , n I
}n ab(43) 2x2 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 a a > 0 a a a 0 5 5 3
(3) 3 2.18 . 2 2 ( 2 2 ) 2 2 . 3 (3 ) (3 ) 3 74. 74Math
E-BookRelease 2.5. 1 2 ( 1 2) 1 2. x4 x x () x 4 x > 4 x4 x 4 x
4 . 2x 1 2x 1 x > 1/2 2x 1 2x 1 x 1/2 . . x 0 2x 1 > 0 x >
1/2 [1] a > 0 [2] a aab ba [3] ab a bnan a aa(b 0) bb2 [4] a2 a
a2 [5] ab < a bab > a b a n [6] n nn an a n S a2 a a .. a2 a
(x 3)2 1 x 3 1 .. x 3 1 75. 75 [email protected]() 1 ( x k )[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 k k
1 S ( +, , +) 2.19 . 3x 2 4 3x 2 4 3x 2 4x 2x 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 1x 2 x 4.. {2, 2, 4, 4}. 3 x 0 ) [3] p(x)
q(x) q(x) > 0 () q(x) 0 1 2.21 2x 1 3x 2 (2x 1) (3x 2) 2 2 (2x
1)2 (3x 2)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}() 77. 77 [email protected] 2.22
3x 2 < 4x 1 (3x 2) 2< (4x 1)222 (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 2 < 1 x1 x 2 2 x 2 < x 1 1 2 2 1 .. S x 2 x x 2 x x 2 x
.. .. x 2 < x 1 x < x 2 < x .. 2 3 3( 1 2 ) 1. x 0 x 78.
78Math E-BookRelease 2.5 2x 1 x 2 x 3 2 2 1/2 2 x 1/2 , 1/2 < x
2 , x > 2 1/2 2 S .. 2. 3 x > 2 1/2 < x 2 x 1/2 1/2 2(2x
1) (x 2) x 3(2x 1) (x 2) x 3 (2x 1) (x 2) x 3x3 x3 3x 1 x 3 x3 x3 x
3x 20 0 0 0 3 > 0 1 0 3 < 0 3. 1/2 2x 3 x > 2 {3} [2, )
79. 79 [email protected] 1 2 1 2 x3 2.23 > 4 x1 2 x 0 x 1 2 ..
x > 1x3x3 > 4 4 > 0(x 1 2)x3 x 3 4x 123x 15 3(x 5) > 0
> 0 < 0x3x3x3 (3, 5] (3, 5] x 1 x3 x3 > 4 4 > 0(x 1 2
)x 1 x 3 4x 45x 7 5x 7 > 0 > 0 < 0 x 1x 1 x1 [7/5, 1)
[7/5, 1) .. () [7/5, 1 (3, 5]) .(44) (44.1) n I n 1 n an a(44.2) a,
b 0 ab a b(45) m (45.1) 4x 0.5 m 3 2x 1 0.5(45.2) x 2 5 m x (2, 6)
x (45.3) x2 25 m x5 6 80. 80 Math E-BookRelease 2.5(46) x1 5y 2 4
xy (47) (47.1) x2 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) 8 (x 2)2 14 (x 2) 3 0(50) A { x I | x2
3x 3 2x 3 }5 3x B {x I | 2} x2 a2 b2 a, b A B 2(51) (x )x x 3(52)
3(52.1) 2x 1 3x 2 (52.4)< x x1 2 x (52.2) 3 x 2 6 (52.5)< 2 x
1 1 (52.3) x 0x2 x 2 0 x x2(53) A x < 42 B x x7 (A B) 4x 5(54) A
{x R | x < 5} 2 (54.1) a, b A (a b)/2 A (54.2) a, b A ab A 1(55)
A { x R | x2 2 14 } B {x R | 1 0} x A B 81.
[email protected](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 5x 2x2 5x 4 (57.2) 2
(57.5) > 1x1x2 x 2 (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 n q ( q m/n ) 82. 82Math E-BookRelease 2.5 2.24 2 6 6
2(3)6 (24) 24 6(4)(4) (20) 20 (4)(5)3 0 0 3(0)5 12 (5 12 ) q 12
5(q) [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 a bn ) ac b a b c b ( an b a b ) (Linear
Combination) b c b x c y ( x y ) 1. 4. S1. a b a c a (b c) 3. a b a
bn 2. a b a (b c) 4. an 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. a bn a b(.. 4 62 ) 8. a b an b(..
2 6 ) 2.25 a4 (3x 2y) a (4x y) a 22 x 1 .. a4 (3x 2y) a (3x 2y) ..
a (3x 2y) a (4x y) a ((3x 2y) 2(4x y)) .. a 11 x a 22 x 83.
[email protected] 2 a4 (3x 2y) 3x 2y a4m .....(1) ( m ) a (4x y)
4x y an .....(2) ( n ) (1) 2 (2) ; 11 x a4m 2an 11 x a (a3m 2n) 2
22 x a (2a3m 4n) ( ) .. a 22 x m, n ( n 0 ) q, r m n q 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 0 2 ( 0) 0 S (Prime Numbers) p p 1, 0, 1 p 1, 1, p, p
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 84. 84Math E-BookRelease 2.5 1,
0, 1 (Composite Numbers) [1] p p mn p m p n( p )[2] () 1 16 2 2 2 2
2 4 210 2 3 5 7 5445 3 3 5 11 11 32 5 112 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) . 2431 .. 2, 3, 5, 7,
11, 13, , 43, 47 11 ( 13 17) 85. 85 [email protected] . 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 86. 86Math E-Book Release 2.5 ... ... ... ...
(Euclid) ... (Euclidean Algorithm) ... ... 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 (2)(6) (2) 3 ... 2 ( 2 ) ... ... ... (a, b)
[a, b] a b ... ... 2.27 ... ... 196 272 (276) (192) 1
(84).....(1)(192) (84)2 (24) .....(2)(84) (24) 3 (12) .....(3)(24)
(12)2 ... 192 276 12192 276 ... 192 276 4416 12 87.
[email protected] d ... a b d a b d a x b y x y 1 x y ... 2.28
... 196 272 12 12 196 272 = ......... (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)
12 196 272 12 192(10) 276(7) .(60) (19)3(288)2 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 88. 88 Math
E-BookRelease 2.5(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 p q 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 42
nq0 r0 , 0 r0 n n 2r0 r1 ,0 r1 r0 r0 2r1 q0 , r0 , r1 ... n 42 (71)
a b a 1998 b r 0 r 19981998 47 r r1 0 r1 r (r, r1) 6 . (a, b) 6.
(a, 1998) 6 . (b, r) 6 . (1998, r) 6 89. [email protected] ()(1)
(25.3) (3, 2/ 3)(48) [2/ 3, 0)(2) (2.3) (26) [7.5, 10) . (49) 8(3)
.(27) 2, 4 (50) 90(4) (4.1) (4.3) (28.1) [1, 2](51) 1, 6(5) (28.2)
(, 1] [1/2, )(52.1) (1/5, )(6) .(28.3) (, 2) (0, 1/6)(52.2) (4, 1)
(5, 8)(7) 6 5 1(29) 2(52.3) (1, 2) {0}(8) .(9) .(30) 2 3 21(52.4)
(1, 3) [, )(10) (31) 2(11.1) (32) (52.5) (, 2] (1, 1) [2, )(11.2)
(33) a (, 2) (3, )(53) (2, )(12) 1(34) [2, 1] [2, ) (54) (13) 3
(35) 5 (55) 7(14) 81(36.1) (, 1) (0, 1)(56.1) 2, 3, 5(15) 7 39/7
(36.2) [4, ) {1}(56.2) 1, 1, 9(16) 27 (36.3) 11 (57.1) (, 3) (1,
)(17.1) {1, 2, 3}(37) 0 7(17.2) {1, 2, 3} (38) 5(57.2) (, 4) (0,
)(17.3) {2, 1/2, 2/ 3}(39) 0(57.3) (2, 4) (6, 12)(40.1) (, 1) (1/
3, 1) (57.4) (1, 3)(17.4) {3, 2, 5, 5}(40.2) (8, 2] (1, 4]1(57.5)
((, 1] [ 3 , 3]) {2, 1}(17.5) {1, 0, 2, 3, 1/ 3}(40.3) (2, 5/2)
(58) (, 1/2) (5/2, )(17.6) {4, 1, 1, 2, 3}(40.4) (2, 8] (59.1) (,
1) (1, 1)(17.7) {3, 1, 2}(41.1) 7(59.2) (1, )(18) (x 2)(x 4)(x
5)(3x 1)(x 1)2(41.2) (19) (x 1)(x 2) (60) 1(41.3) 8(61) 2 (252)(5)
(34)(37)(20) (x 1)(x 2)(x 3)(x 2)(x 4)(41.4) (62) 2 (504)(4)
(38)(53)(21.1) {b, b}(42) 0(43) 5/2(63) 256(21.2) {0}(64) 251(21.3)
{0, 2b} (44) (45.1) 3.5(65) 48(21.4) {a 1, a 1} (66) 21(45.2)
17/3(22) (22.1) (22.2) (45.3) 96 (67) 3(23) (23.4) (46) [0, 12)(68)
105, 165(24.1) (6, 46) (69) 495, 513(47.1) 2, 2, 4, 4(24.2) (252,
180)(70) 200(47.2) [1, 1](25.1) (18, 4)(71) .(47.3) [3, 4](25.2)
(9, 4) 90. 90 Math E-Book Release 2.5 ()(1.1) (5) A { x | x x(1.2)
} {1, 4, 9, 16, 25, 36, ...}(1.3) a 2 A a2 a .. B N A (1.4) a 3 A
a2 a (5.1) A .. B (2.1) .. a 0 b .. 2 2 4 4 B (2.2) .. a 0 b 0( 0 )
(5.2) A (2.3) (.. 1 1 2 2 A b B a c ) .. 2 2 4 4 B (2.4) .. a 0 b c
(6). .. x 2 y xy 1 (3) . . .. x 0 () y xy 1 . .. xy 1 . ( 3 5 8 8
xy A (1 )) ( . .. x y 3 5 15 ) xy 1 y ( x, y 0). ( ( ) 0 3 4 1 )334
44 3(7) a 1/a .. . 1 6 5 4 65 4 .. . 1(4.1) .. (8). (a b) a b a b
.. (4.2) .. (a b) c a (b c) . (b c) b a b b ..(4.3) .. 0 . (a b) (c
b) b a b .. ( 0 . (c a) (b a) c b a .. 0 )a b(4.4) .. [ ] c a [ ]
bc (9) . x y y x.. x y 91. 91 [email protected](10) x (y z) x (3yz
y z) a 4, b 3 3x (3yz y z) x 3yz y z a 4/ 7, b 5(z y) x (3zy z y)
x.. a b 7 39/ 7 3(3zy z y) x 3zy z y x 2 a b (16) x 2x 4 3 3(x 3)(x
1) 2 x (y z) (z y) x x ax bx 3x 4( y, z x 3 x 1 ( 0) 4 32 x ) (3) 4
a(3) 3 b(3) 2 3(3) 4 0 (1 a(1 b(1 3(1 4 0 ) ))) a 19/9, b
37/9(11.1) A 57 x2 x 2 (x 2)(x 1) 6 6 A x3 10x2 cx d2 (2)3 10(2)2
c(2) d 0 32 ab ba (1) 10(1) c(1) d 0 .. 2 2 c 7, d 18(11.2) A .. a
b c d 27 33 4.5 4.5 A 2 ab ba(17.1) .. 2 2 1, 2, 3 () (x 1)(x 2)(x
3) 0 {1, 2, 3} (12) a 4(4)3 21(4)2 26(4) 17 7(17.2) b 3(3)3 13(3)2
11(3) 5 8 b a 8 7 1 1, 2, 3 () (x 1)(x 2)(x 3) 0 {1, 2, 3} (13) x 1
x2 2a (1)2 2a(17.3) x 2 x a (2) a2, 1/2, 2/3 ().. 1 2a 2 a a 3 ( 2
6x2 x 2 ).. (x 2)(2x 1)(3x 2) 0(14) {2, 1/2, 2/ 3} (5)4 (5)3 3(5)2
(5) 1 569(17.4) 2(5)3 (5)2 75(5) a 650 a .. 569 650 a a 81 3, 2 ()
1 0 5 x2 5 (x 3)(x 2)(x2 5) 0 (x 3)(x 2)(x 5)(x 5) 0(15) {3, 2, 5,
5} 0 (2)3 a(2)2 (a/ 4)(2) 2b 0 .... (1) (1/ a)(2)2 (2) b 0 .... (2)
92. 92Math E-Book Release 2.5(17.5) x (21.1) a 0 x2 b2 0 (x b)(x b)
0 .. {b, b}2, 3, 1, 1/3 ()(21.2) b 0.. x (x 2)(x 3)(x 1)(3x 1) 0 x2
0 .. {0} {1, 0, 2, 3, 1/ 3} (21.3) a 1 x2 b2 2bx b2 0(17.6) x2 2bx
0 x (x 2b) 01, 1, 1, 2, 3, 4 () .. {0, 2b} (x 1)2(x 1)(x 2)(x 3)(x
4) 0 (21.4) b 1 {4, 1, 1, 2, 3} x2 a2 2ax 1 0 (x a)2 1 0(17.7) (x a
1 a 1 0)(x )1, 2, 2, 3 () .. {a 1, a 1} 1 1 2 x2 x 2 (22.1) .. c b
a c d(x 1)(x 2)2(x 3)(x2 x 2) 0 ()()() 0 {3, 1, 2} (22.2) .. 2 1
(2)2 12 (22.3)(18) (a b)/2 ab2, 4, 5, 1/3 () a b 2 ab a2 2ab b2 4ab
3 0 3 3x2 3() a2 2ab b2 0 (a b)2 0 a b.. (x 2)(x 4)(x 5)(x 1/3)(3x2
3) (x 2)(x 4)(x 5)(3x 1)(x2 1)(22.4) 3 3 b 2 2a baab ab(19) ( b3 a3
ab (b a)) () x3 7x 6 (x 1 2)(x 3))(x (b a)(b2 ab a2) ab(b a)3x3 7x2
4 (x 1 2)(3x 2))(x( 0) 43x 3x 6x 4 (x 1 2)(x 2)(x 2))(x b2 2ab a2 0
(b a)2 0... a b ... (x 1)(x 2) x2 3x 2 (23.1) (23.2) (20) .. xmin X
xmaxx3 2x2 5x 6 (x 1 3)(x 2))(x(23.3) .. x3 x3 x2 10x 8 (x 1)(x
2)(x 4)( )... (23.4) .. b 0 ab bc ... (x 1)(x 2)(x 3)(x 2)(x 4) b
ab bc x5 17x3 12x2 52x 48 93. 93 [email protected](24.1) 7 x 5 0
< x2 49 (28.1) x2 x 2 < 0 3 y 6 6 y 3.. (x 2)(x 1) < 0.. 6
x2 y 46 + + (6, 46) 12(24.2) 3 y 6 9 y2 36 2 [1, 2] xy (7) 36 xy2 5
36(28.2) 2x2 x > 1.. 252 xy2 180 2x2 x 1 > 0 (252, 180).. (2x
1)(x 1) > 0 + +(25.1) xy 1 1/2 12, 18, 4, 6 .. (18, 4) (, 1]
[1/2, )(25.2) x y 8, 9, 4, 5 .. (9, 4)(28.3) .. x (x 2)(6x 1)
0(25.3) x/ y + + 3, 2, 1, 2/ 320 1/6.. (3, 2/ 3) (, 2) (0, 1/6)(26)
(29) x2 6x 7 ( h . h x2 2 2x .) h b b2 4acx 2a x h ..220 2x 2 h x 2
10 x 2h x2 (x2 6x 9) 2 < 0 (x 3)2 2 < 0 100 20 x x2 h2 x2 (x
3 2)(x 3 2) < 0x 100 h 5 h 2 2 2020 + + 3 23 2 0 20 < 50 h
0)(x+++ +1/2 2/32 12 0 [2, 1] [2, ) n 0 .. m n 2 (31). x2 (35) x3
2x2 5x 6 < 0 1 2x2 3x 20 < 0 .. (x 2)(x 1)(x 3) < 0.. (2x
5)(x 4) < 0 A + +++3 12 45/2 B (5, ) AB 4, 3, 2, ..., 2 53 12 13
.. . 4 3 1 0 1 2 5. x2 7 x 10 0 (x2 7 x 49) 409 0(36.1) + + +
3336361 0 1 2 2 (x 7)2 409 0 (, 1 (0, 1) )6 36 (x 7 409)(x 7 409) 0
66 66 (36.2)+ A + + 7 409 x 7 409 () 2 1 1 366 B+ + 27/6 x 13/64
23.. 4, 3, 2, ..., 2 (A B ) A B [4, 1) (1, ) 7 .. . ( [4, ) {1} )
(36.3) + + + + 4 1 0 2 5 52 2(32) 2x 4x 5 0 x 2x 2.5 0 (x2 2x 1 3.5
0 (x 1 2 3.5 0)) 3, 2, 0, 1, 5 (x 1 3.5)(x 1 3.5) 0 ++| 3| |2| |0|
|1| |5| 111 3.5 1 3.5 3.5 1.8 a 0, b 2 {0} {0, 2} {2} {0, 2} 95. 95
[email protected] 5 (40.1) (37) A; >0x 2 () ++ 12 0 2 5/2x13x
1 2x 12x 1 x 5 B;1 0 03x 1 2x 2 (x 1) x5x5 0 0 (x 1)(3x 1) (x 1)(3x
1)x 6 ++ 0x 5 5 6 + + 11/3 1 B A B A [2, 5/2)1 2 (2) 0 (, 1 ( , 1)
)3x 1x 1 2x 4 1x(38)2 0 0(40.2).. >0x2 x2x 1 x8 x 5 x5 0 0x2 x 2
x 8 x2 x x2 2x 8 >0 >0 (x 1 8) )(x (x 1 8) )(x++x2 2x 8(x
4)(x 2)5 20(x 2)2 x1(x 2)2 x1( ).. 4x 4 x2 4x 4>0 2x2 5x 2 >
0 (2x 1 2) > 0)(x(x 2)2(x 1) x2 8x x2 8x + + >0 0 x 2 (44.2)
a 2, b 3 x2 |a b| 1 |a| |b| 1..(2, 8] + (41.1) ( 7, 7) (45.1) 3 2x
1 0.5 7 2 2x 1.5 1 x 0.75(41.2) .. 4 4x 3 3.5 4x 0.5 3.5(41.3) (2,
8] |4x 0.5| 3.5 8 m |4x 0.5| m 3.5(41.4) {..., 6, 4, 2, 0, 2, 4,
...} (45.2) xx 2 5 1 2 5 2 x x 6 2 x 6 3 3 2 1 1 2 3 1 x x1(42) A {
2 , 2 , 4 , ... }13 5 2 6 17 x3 1 ( 1) m x 2 5 m 17 x 3 1 B { 1, 12
, 13 , ... } (45.3) 6 x 5 6 1 11 x 1.. 1 0 < x2 121 25 < x2
25 96.. a b 1 (1) 0 m |x2 25| m 96 97. 97 [email protected](46) 5
x 1 5 4 x 6 . 2/ 3 < x 0 .. 4 y 2 4 2 y 62 3x 2 3x 6x 0 x 0 6 x
y 12 . x >0.... |x y| [0, 12) 2 3x 2 3x 0 0 [0, ) ..A [0,
)(47.1) x2 |x|2 B; . .|x|2 6|x| 8 0 2/3 (|x| 4)(|x| 2) 0 . x 2/ 3
.. |x | 2 4 2 3x 2 3x 2 2 .. x 2, 2, 4, 4 . x > 2/ 3 .. 2 3x 2
3x 0 0 [2/ 3, )(47.2) .. B [2/ 3, ) B A B A [ 2 , 0)..3.11. x
1..(49) 8 x 2 2 14 x 2 3 0(x 1 (x 1 2 2x 2 x 1 ) ) .. (2 x 2 3)(4 x
2 1) 0. 1 < x 1..3 x 2 2 4 1(x 1 (x 1 2 2 2 )) [1, 1) 3 3 1 1 x
{ 2 2 , 2 2 , 2 4 , 2 4 }. x > 1..(x 1 (x 1 2 2x 2 x 1)) {1} 8..
[1, 1](47.3) (50) A; .. (x2 3x 3)2 (2x 3)2 0.. . (x2 3x 3 2x 3)(x2
3x 3 2x 3) 03 4 (x2 x)(x2 5x 6) 0. x 3.. x(x 1 2)(x 3) 0)(x(x 4) (x
3) 1 2x 6 x 3 A {0, 1, 2, 3}. 3< x 4..(x 4) (x 3) 1 1 1 [3, 4)
B; ( . x > 4.. 0 x 2 ) (x 4) (x 3) 1 2x 8 x 4 {4} .. |5 3x| |2x
4|.. [3, 4] (5 3x)2 (2x 4)2 0 (5 3x 2x 4)(5 3x 2x 4) 0(48) A; (1
5x)(9 x) 0. x B {9}.. 2/3 0.. A B {0, 1, 2, 3, 9}. x 2/ 3.. a 9, b
3 a2 b2 902 3x 2 3x 2 2 98. 98 Math E-BookRelease 2.5(51) (52.2) 2
x 0 x > 0 x 0 .. 3 x 2 6 6 x 2 3 3 x 2 6.. x > 0 4 x 1 5 x 8
( x)x x3 x x x321 2 2 (4, 1 (5, 8)). x 1 1 1 3 x 2 x 2 6. 0 0 00 ..
x 0 00 x 0 1. (52.3)x 0 |x| 2 1 x2 3 x2 6 x 6.. {1, 6} x 0 .. 1 x2
1 log ( ) x 0 0 xx(x 1)(x 1) 0 x(52.1) 1 (1, 0) (1, )( 47, 48 ..)
(1, 0) x 1/2 ..2x 1 3x 2 1 5x x 1/5 x >0.. (1/5, 1/2)1 x2 1x 0 0
x> 1/2 ..xx2x 1 3x 2 3 x x 3 () [1/2, ) (0, ) (1/5, ) (0, ) ..
(1, 0) (0, ) 2 ..x2 x 2 0( ) (x 2)(x 1 0 ) (1, 2).. (2x 1)2 (3x
2)2.. (2x 1 2 (3x 2)2 0 ) x (1, 0) (0, ) x (1, 2) (2x 1 3x 2)(2x 1
3x 2) 0 (x 3)(5x 1 0 (x 3)(5x 1 0 )) (1, 0) (0, 2) (, 3) (1/5, ) ..
3x 2 > 0 x > 2/3 (1/5, ) 99. 99 [email protected](52.4) x
> 2..x 1..x 2 x4 < 0 x 2 2x 8 < 0332 0x 1 x1 x1 0 x 1..
(1, 1) x x >1 ..52.1 ( 2) 33 0 x 3 x3x x 7 2x 7 x 7 (, 7) 2 2()x
> 7..(x 3 21)(x 3 21)x x7 0 7 2 2>07x 3.. B (, 2) 21 4. A B
(, 2] (A B) (2, ) [ 3 2 21 , 3) [ 3 2 21 , )[1, 3) [ 3 21 , )2 (54)
()2x |4x 5| |4x 5| < 10..(1, 3) [ 3 21 , )2 2x |4x 5| x 5/ 4
..(52.5) 2 2x 4x 5 6x 5 x 5/6 x > 0 x 0 (, 5/ 4) A x x > 5/4
.... 2x 4x 5 5 2x x 5/2A A (x2 x 2)2 2 2 2 2(3x 2) (4x 1) (3x 2)
(4x 1 0 ) (x25x 4 x2 x 2)(x25x 4 x2 x 2) > 0 (3x 2 4x 1)(3x 2 4x
1 0 ) (6x 2)(2x2 4x 6) > 0 (x 1)(7x 3) 0 (x 1)(7x 3) 0 2(3x 1
2(x 3)(x 1 < 0 )).. (, 3) (1, ) 1 (, 1] [ 3 , 3]7 .. 0 2 x x 2 0
x 2 1(57.2) ((, 1] [ 1 , 3]) {2, 1} 3 ()( x 1) (x 2)2 (2x 2) (x 2)2
(2x 2)2 0 (x 2 2x 2)(x 2 2x 2) 0 (x 4)(3x) 0 3(x 4)(x) 0.. (, 4)
(0, ) 101. [email protected] 2 (59.1) x2 5x 4x2 5x 4 x 02 >
1 2< 1 x x2x x 2 (1 x)(1 x) 0 (x 1)2 0 x2 5x 4 x 1 >1()x2 x 2
x2 5x 4 x2 x 2 (, 1) (1, 0) >0 x2 x 2 x > 0(6x 2)2(3x 1) (1
x)(1 x) 0 (x 1)(x 1) 0 >0 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) pq p*q(14.1) (p p) (q q) TTF . p q. p q TFF . p q. p q
FTFFFT(14.2) pq . ~ (~ p q). ~p q. ~ (q ~ p). q~p(15) p q ~ (p q) p
(q r) 113. [email protected] . (Tautology) 3.5 . (r p) (p r) p
r (4 )pr rpp r (r p) (p r)TT T T TTF T F FFT T T TFF F T T .. . (r
~ p) (p r) p r (4 )pr r ~pp r(r ~ p) (p r)TT T T TTF F F TFT T T
TFF T T T .. () 1. , - 114. 114Math E-BookRelease 2.5 2. , S - (
)3. ( ) 3.6 . (r p) (p r) ( 3.5) - (r p) (p r) T F F T T F p r p r
( p , r ) .. 115. 115 [email protected] . [(p q) (q r)] (p r) -
[(p q) (q r)] (p r) FT T F T FT T F F p r q .. 3.7 (r ~ p) (p r) (
3.5) 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 q) (p r) (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)] 116. 116Math E-Book Release
2.5(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)] . p1, p2 , p3
, ..., pn ( ) q (p1 p2 p3 ... pn) q 1. 2. (valid) (invalid) 117.
[email protected]. (p1 p2 p3 ... pn) q 3.8 1. 2. 3. 1. p q2. ~
p r3. ~ q r (p q) (~ p r) (~ q) r F TTT FT T T FF q .. 2. () () ()
p q p q p q p (~ q)~p(~ q) qr q (~ p) q(p) pr..() () p q ~ q ~ p p
q p p (q) p q 118. 118Math E-Book Release 2.5 3.9 1. p q 2. ~ p r
3. ~ q r 1. ~ q ~ p 3. ~ p [ ()] ~ p 2. r [ () ].. r S () .(22)
(22.1) 1. p q (22.2) p (r s) 2. q s ~ p (r 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. a2 2 a2 13.
a2 2 a 4. a2 1 a 119. 119 [email protected](24) (24.1) 1. p (q r)
(24.2) 1. ~ p q 2. ~ s p 2. q ~ r 3. q 3. p(25) 1. 2. (26) 1. 2. 3.
4. . . . . . x 2 ( ) x ( ) x 3 x 2 (Open Sentence) x P(x), Q(x),
R(x) ( Px , Qx , Rx ) , , -, P(x) Q(x) ~ P(x) Q(x) 120. 120 Math
E-Book Release 2.5 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 U {3, 4, 5}x [ P(x)] 3, 4, 5 2x [ P(x)] 3
2 () U {1, 2, 3}x [ P(x)] 1 2 ( x )x [ P(x)] 3 2 () U {0, 1, 2} x [
P(x)] x [ P(x)] x 2 x x (There Exists x) x [P(x)] x 2 All () Exist
() 3.11 U {2, 0, 1, 1.5, 2} . x [ x x ] x () 121. 121
[email protected] .x [ x x ] x 2 () . x [ x x ] x 0 1.5 () . x [ x
x 3 ] x 1 ( T F ) . x [ x x 3 ] x 2 . x [ x3 x ] x3 x 2 x . x [ x
x2 0.5 ] x 1.5 () x 0 ( 0.5) . x [ x x2 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) xy x y P(x, y) ,
xy x y P(x, y) xy yx x,y xy yx x,y 122. 122Math E-Book Release 2.5
xy , xy , yx , yx xy [...] x y ... x y [ ] x y yx [...] 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 () U {1, 2, 3} x y [ P(x) Q(x, y)] ..
(1,2), (2,1), (3,2) y x [ P(x) Q(x, y)] .. (1,1), (2,2), (3,3) [ ]
1. 2. x [P(x)] x [ ~ P(x)] xy [P(x) Q(x, y)] xy [P(x) ~ Q(x, y)] x
[ x x 4 ] x [ x x < 4 ] 1. 2. S x [ P(x)] x [ P(x)] x [ P(x)] x
[ ~ P(x)] 123. [email protected] .(27) U {2, 1, 0, 1, 2} . x [x
x2 0] . x [x3 x2 x x2 ] . 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) x x2 Q(x) x 3 x 1. x [P (x)] U (0, 1). x [Q (x)]
U 2, 3, 5, 8(30) x (x35x 1 4) x ( x21 0 x > 2)(31) N[x (x2 1 ) x
(x 1 > 0)] x 2 0 x (32) U {1, 0, 1}(32.1) x (x2 1) x (x2
1)(32.2) x (x 1 0) x (x2 1)(32.3) x (x 1 0 x2 1)(32.4) x (x2 0) x
(x 0)(32.5) x (x2 0 x 0) 32.2 32.3 some S 32.2 32.3 x 32.4 32.5 all
.. all , some 124. 124 Math E-Book Release 2.5(33) U {1, 0,
1}(33.1) xy (x2 y > 2)(33.2) xy (x2 y > 2)(33.3) xy (x2 y
> 2)(33.4) xy (x2 y > 2)(34) {1, 0, 1}(34.1) xy (x2 y y2
x)(34.2) xy (x2 y y2 x)(34.3) xy (x2 y y2 x)(34.4) xy (x2 y y2
x)(34.5) xy (x2 y y2 x)(35) (35.1) xy (x y y x) U {2, 0, 2}(35.2)
xy (x y 0) U {2, 2}(36) xy (xy 1) xy (xy y) . . . . (37) R . xy [x
y > xy]. xy [x y < 0] . xy [x < y] . xy [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) xy [(x y 5) (x y 1)](38.5) xy [x 0 y 0 xy 0](38.6)
xy (xy 0 x 0 y 0)(38.7) xy [(P (y) ~ R (x)) (~ Q (x) ~ P
(y))](38.8) xyz (x y z xy < z)(39) . x [x 5 0] y [22 ] x [x 5 0]
y [22 > ]yy. x [x 6] x [x > 8] x [x > 6] x [x 8] 125. 125
[email protected]. (Reasoning) 2 ( ) (Inductive Reasoning) , , , ,
(...) 3.13 . 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 n2 . 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 3 126. 126Math E-Book Release 2.5 100% 1. 20 4 3.14 () .
(n 1 2 2(n 1) ) n n 1, 2, 3, 4 4 1, 9 2, 16 4, 25 8 .. n 7, 8, 9,
... . n2 n 5 n n 1, 2, 3, 4 n2 n 5 5, 7, 11, 17 .. n 5 n2 n 5 25 2.
100 22 70 3. (, , ) (, , ) ( ) (Deductive Reasoning) 2 0 2 0 127.
127 [email protected] ( .) () 3.15 . (1) (2) . (1) (2) . (1) (2) (
) ( ) 1. (1) (2) ( ) 128. 128Math E-BookRelease 2.52. (1) (2) ( )3.
(1) (2) ( ) (-) () () ( ) ( )( ) 3.16 . (1) (2) . (). (1) .(2) ()
129. 129 [email protected] . (1) . (2) () . (1) (2) ().1. 2. (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) 1 ,2 , 3 , 4 ,a 2 3 4 5(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 130. 130 Math
E-BookRelease 2.5(41) 2 ( ) 37 3 111 11 11 121(41.1) 37 6 222
(41.5)11 12 132 37 9 333 11 13 143 9 9 81 1089 1 1089(41.2) 9 99
891 (41.6)1089 2 2178 9 999 8991 1089 3 3267 1 9 11 2 2 (3) 3 (3
1)(41.3) 12 9 111 3 (41.7)2 (3) 2 (9) 3 (9 1) 123 9 1111 4 2 (3) 2
(9) 2 (27) 3 (27 1) 9 9 7 88 3 4 2 (1 2 3)(41.4) 9 98 6 888 (41.8)4
5 2 (1 2 3 4) 9 987 5 8888 5 6 2 (1 2 3 4 5)(42) -(42.1) (42.2)
(43) -(43.1) (43.2) (43.3) 131. 131 [email protected](44) -(44.1)
(44.4) (44.2) (44.5) (44.3) (44.6) (45) -(45.1) (45.4) (45.2)
(45.5) (45.3) (45.6) 132. 132Math E-BookRelease 2.5 ()(1)p |F|p |F
(24.2) r(40.5) 5 3T |p|p|T (25) (40.6) 5/6p | T | T |~ p | T |~
p(26) . (40.7) 25p |~ p | T | F (27) . (28) . (40.8) 3 3 3 3 3(2)
(2.6) (2.10) (29) . . (40.9) 5047(3) (3.5), (3.7), (3.9), (30)
(41.1) 37 12 444 , (31) 37 15 555 (3.12) (3.11) T, T, F(32) , , , ,
(41.2) 9 9999 89991 ,(4) . . (33) , , , 9 99999 899991(5) (34) , ,
, , (41.3) 1234 9 11111 5 ,(6) . . (35.1) 12345 9 111111 6(7) . (p
q) (~ r ~ s) (35.2) (41.4) 9 9876 4 88888 , . p ~ q r(36) (37) .9
98765 3 888888 (38.1) x [P (x) Q (x)](8) .(9) . (41.5) 11 14 154
,(10.1) . (10.2) .(38.2) x [P (x) Q (x) ~ R (x))]11 15 165(10.3)
.(11) . (38.3) x [ P (x)] x [Q (x)] (41.6) 1089 4 4356 ,(12) (38.4)
xy [(x y 5) 1089 5 5445(13) (x y 1)] (41.7) 2 (3) 2 (9) 2
(27)(14.1) (14.2) . (38.5) xy [x < 0 y 0(15) 3:5 2 (81 3 (81 1 ,
) ) xy > 0](16) (16.1),2 (3) 2 (9) 2 (27) 2 (81) (38.6) xy (xy 0
x > 0 (16.2) 2 (243) 3 (243 1) y > 0)(17) (17.2), (41.8)
(17.7)(38.7) xy [P (y) ~ R (x) Q (x)] 6 7 2 (1 2 3 4 5 6) ,(18)
(38.8) xyz (x y < z 7 8 2 (1 2 3 4 5 6 7)(19) (19.1)xy z) (42)
(45)(20) . . (39) . . (21) (40.1) 9(43.1), (43.3), (44.1),(22)
(40.2) 22(44.4), (45.1)(23) (40.3) 5(24.1) s r (40.4) 48 133. 133
[email protected] ()(1) (2.1) [(p s) (p r)] (p s) T TT (2.2) [(q
s) r] [.....]T. T T T T ( F F ) (2.3) [(r q) (p q)] [.....] TF FT p
p .. (2.4) [(p q) (q r)] ~ s T( T T T, T F F ) TF p F .. (2.5) [(q
p) r] r Tp p p .. TT Tp ~ p F .. (2.6) [(p q) ~ r] [(~ p q) r] F
FFTTF. F (2.7) [(p ~ q) ~ r] [(p q) .....] F F F ( T T )T F .TFTp
T.. (2.8) (p q) ~ r [... ...]FF p p .. F ( F T T, F F F ) (2.9) [p
(q r)] [.....]Fp p p .. Fp ~ p T .. (2.10) [q (....)] [p (q ~ r)]F
FT F.T F. - F (2.11) [(~ p ....) (~ r ....)] [.....]T T F ( F T T T
T )T p p .. (3.1) (....) (p q) TFp T.. Tp T T .. Tp F ~ p .. (3.2)
(....) (p q) T ( T F F, F F T ) F .p p T .. - T (3.3) (~ r p)
(....) Tp ~ p ~ p .. - T ( T F F, F T T ) (3.4) r q T. r q T, F (p
q) (s p) (s q) T T T T TT p p .. (3.5) p T, q F, r TF p ~ p .. (~ q
(p r)) (~ r) Fp p T .. TT Fp ~ p 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) (....) TF 134. 134 Math
E-Book Release 2.5(3.9) p r T, q s F (4) s r F.. s T, r F q F, s F
pr T.. p T p q T p F p q T .. q T r T .. T F F . [(....) (q r)] (r
s) F (3.10) pq F p T, q FF F . [....] (~ r s) T (....) ~ qTT
T(3.11) p q T p T, q T p r F r F (5) p q T.. p q(3.12) p T, p ~ r T
r ~ s F .. r F, s T r F [(~ p r) ....] T F. ~ (....) ~ r T (3.13)
(3.14) p, q, r, sT ( . r (p ~ q) T p, q, r, s )FF . (....) (p q) T
.. T( .)(3.13) (6) p q, r ~s .. - . [.... (r ~ s)] [.... (~ r ~
s)]T TT ((p ~ q) ~ p) (p q) . [....] [(p ~ q) (r ~ s)]TF T TT F . .
T F p q .. p q (7) . ~ [(~ p ~ q) (r s)] , (p q) (~ r ~ s) . ~ [~
(p ~ q) ~ r] (p ~ q) r (3.14) - (8) p ,p ~ (r s) ~ p (~ r ~ s) q ,F
r TF .. (p q) r T T r s p q T r F .. r s . p~q F. ~pq F p ~ p . ~r
~q F. p ~r T.. . .. 135. 135 [email protected](9) . ~ (~ p ~ q) p
q ..(14.1). (~ p q) (q ~ p) .. p q p*p q*q (p*p)*(q*q). p (~ q p) ~
q pT T F FT q p ( .) T F F TF. F T T FFF F T TF pq(10.1) . . . (p
q) (q p) (14.2) F*F (10.2) {[(q ~ t) (p ~ p)] ~ q} rT [(q ~ t) ~ q
] r ( T, T, T, F ) [(q ~ q) (~ t ~ q)] r .. p q ~ (p q) T . ~(~ p
q) ~(p q) (~ t ~ q) r (t q) r(10.3)(15) [(q r) (q ~ r)] [(p s) (p ~
s) ] p q r q*r p*(q*r) [q (r ~ r)] [(p (s ~ s)] qpT T T FFF T T T F
FFT F T FFT F F TF(11) . . F T T FTF T F FT ab > 0, a < 0, b
< 0 F F T FTF F F TF.. . . : 3 : 5 (p q) r . (~ p ~ q) ~ r .. .
~ r (~ p ~ q) .. (16) - (16.1) (p q) [(p q) r](12) . p (q r) F . (~
q ~ r) ~ p T F T T T T F . ~ p q r .. . . . (16.2) (p q) [(p q)
r]FT F T T T T F(13) .~(p ~ r) ~ q ~ p r ~ q q (r ~ p) ..(16.3) [(p
q) (q r)] (p r). p (q r) ~ p (~ q r) F q (p r) ~ q (~ p r) ..T T FT
T F F T F. (p q) r ~ p ~ q r q (p ~ q) (p r) ~ p ~ q ~ p r .. ..
136. 136 Math E-Book Release 2.5(16.4) [(p r) (q r)] [(p q) r]
(17.5) (p q) (p q) F ~(p q) ~(p q) T TF T T T T T T F (~ p ~ q) (~
p ~ q) r .. .. (17.6) ~ p (q r)(16.5) [(p r) (q r)] [(p q) r] (~ p
q) (~ p r) F T T F (p q) (p r) T F .. F F F F F F p q (17.7) ~ p ~
q r p q ~(~ p q) r (p ~ q) r.. ..(16.6) [(p r) (q s) (p q)] (r s) F
(17.8) T T T F F F F F T T F F p , () q .. p q r TT T T T(16.7) (p
q) r (p q) (p r) TT F F F FTF T F F T TF TF F T TT F F T T T F FT T
F F q FT F T T.. FF T T TFF F F F (17) (18) (17.1) ~(p ~ q) ~(~ p ~
q) pq (18.1) [(p r) (q r)] (p q).. F FF(17.2) (~ p q) pF F (~ p p)
(q p) qp p q T r r .. (17.3) (p q) ~ p (18.2) [(~ p q) ~ p] (p q) F
(p ~ p) (q ~ p) FF FT F q ~p p q .. (17.4) (p q) (p q) (q p) (~ p
q) (~ q p) .. 137. [email protected](19.1) (p ~ p) (q ~ q) F F
T (22.1)1. p q (22.2)p (r s)() 2. q s ~pr s.. p s ~s ~p(19.2) [p T]
[~ p F] p ~ p F3. ~ s ~ p(19.3) p q ~ p ~ q ~(p q) (~p ~ q) ~ F
(23) (23.1) 1. p q 1. p q 2. (p q) r(20) p, q, r 4. p 3. ~ (s r) 4.
p q 2. (p q) rs. (p q) (~ p ~ q) rF3. ~ s ~ r TF F TF T r~s ~ s. (p
q) (~ p q) (23.2)1. p ~ q 2. r s. ~ ((p q) r) (~ (p q) ~ r)2. rs4.
~ sF 3. r qr TF 4. ~ s F F F F F T3. r q r p q.. 1. p ~ q ~p. (~ p
r) (~ q r) (~ p ~ q) r (p q) r . (~ p q) (~ p r)(24.1) 1. p (q
r)(24.2) 1. ~ p q q (p r)2. q ~ r ~ p (q r) p (q r) 3. q~p ~r pr r
p 2. s p 3... . . s r p 3. r(21) .. (25) 1. p ~ q 2. q ~p 138.
138Math E-BookRelease 2.5(26)1. p q 1. p q(29) . x x x2 2. ~ p r4.
~ q U (0, 1) 3. s ~ r ~p ( )4. ~ q2. ~ p r.. . ?r3. s ~ r. x ... x
~s... 3 x 1 2, 3, 5 , ~s 8 ... 3 1.. .. . . ~ s p .. ( ) . s p .. ~
p (30) x (x3 5x 1 4). ~ r ~ s .. x 1 7 4 r x(| x2 1| 0 x > 2) .
p ~ q .. ~ p ( 0) F T x (27) . x x2 0 x 0.. T T T. x 2 8 4 2 4 . x
1 . x (31) x (x2 1 ) x 2 22 1 3 x (x 1 > 0) 1 0(28) . x ... x x
(2 0) x 1 2 0 x 1 2 .. (T T) F F ... x 3 T F F. x ... x 0.5 (32.1)
x (x2 1) x 0, ... x 2 F F T x (x2 1) x 1.. T F F(32.2) x (x 1 0) x
0. x ... x x (x2 1 x 0 .. T T T ) (32.3) x (x 1 0 x2 1 )... x 0 ..
T T T (32.4) x (x2 0) x 0 x (x 0) x 1 .. F F F. x ... x (32.5) x
(x2 0 x 0) 22/7 ... 22/7 ( x 1, 0, 1 ) x 1 .. T T ~ F T 139. 139
[email protected](33.1) x, y x2 y > 2 (36) . U I.. x 1, y 1 xy
(xy 1) x 2 y (33.2) x y xy (xy y) .. x 1 y 0 x 1 xy y y x 0 y 0 ..
F T F x 1 y 0 . U I xy (xy 1) (33.3)x y xy (xy y) ( .).. x 0 y .. F
T F(33.4) x y . U R xy (xy 1) .. x 0, y 0 x 0 y 33.2 33.3 xy (xy y)
().. F T F . U R xy (xy 1) (34.1)x y x R y R .. x 1, y 1 2 0 xy (xy
y) () .. T T T .(34.2)x y .. x 0, y 0 x 0, y 1 x 1, y 1 (37). .. x
3, y 4 7 > 12(34.3) x y . .. x, y .. x 1 y 1 0 x 0 y 1 . .. x y
x 1 y 1 x y x . .. x y y x (34.4) x, y . .. x 0, y 0 34.2 34.3
(38.1) x [P (x) Q (x)](34.5) x y (38.2) x [P (x) Q (x) ~ R (x))]..
x 0 y 0 (38.3) x P (x) x [Q (x)] x 1 y 1 (38.4) xy [(x y 5) (x y
1)] x 1 y 1 (38.5) xy [x < 0 y 0 xy > 0] (38.6) xy (xy 0 x
> 0 y > 0) (38.7) xy [P (y) ~ R (x) Q (x)] (38.8) xyz (x y
< z xy z)(35.1) x y .. x y (35.2) x y .. x 2, y 2 x 2, y 2(39) .
. .. x [x 6] x [x 8] 140. 140Math E-Book Release 2.5(40.1) a 9 ( /
(42.1, 42.2) 2 )(40.2) a 22 ( 2 / 5 )(40.3) a 5 ( ) 2 (40.4) a 48 (
/ 2 ) () (40.5) a 5 ( 2) 3 1.. a 3 () 3 1(40.6) a 5() (43.1)6(40.7)
a 25 ()(40.8) a 3 3 3 3 3 ( 3 5 ) .(40.9) 7 5, 6, _, 8, 9 12, 72 ,
??? , 4032, 36288 ..6 9(43.2) 72 7 504 504 8 4032 .. a 5047 (41.1)
37 12 444 , () 37 15 555..(41.2) 9 9999 89991 , 9 99999 899991
(43.3)(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)(44.1) 3 (243 1)(41.8) 6 7 2 (1 2 3 4 5
6), 7 8 2 (1 2 3 4 5 6 7) .. 141. [email protected](44.2)(45.1)
2 ..() (45.2).. (44.3) 2 () () .. ..(44.4)(45.3) .. ()
..(44.5)(45.4) .. (, ) ! (45.5) .. () ( 45.4 .. , ) (44.6)(45.6) ()
.. () .. 142. .. 0 1 ... 0 0 , False 1 5 ( 12 ), True (Gate)
inv01(1) INVERTER ( )1 0 1 1 0and0 0(2) AND ( ) 1 1 1 or01(3) OR (
) 0 0 1nand 01(4) NAND NOR ( ) AND OR 1 AND OR nor 00(5) XOR ( -) 1
1 0 xor 1 (0 1 0) 01 0 False 1 True AND OR ( 1+1 1, 2 ..) Anot AA B
A and BA BA or B (A ) (AB) (A+B)101 11 1 11011 00 1 010 10 0 11 0
00 0 00 A nand B = AB = A BA nor B = A B = A B ()A xor B A B , :]
143. ( R2.9pre .4-5-6 ) , , (Analytic Geometry) (Plane)
(Coordinate) 2 2 (Cartesian Coordinate) 2 (Origin; O) y x y Q2 Q1
(, +) (+, +) 4 Ox (Quadrant; Q) 1, 2, 3, Q3 Q4 4 (, ) (+, ) +x +y
(OrderedPair) (0, 0) , (2, 4) 2 4 ( x 2 y 4 ) 144. 144Math E-Book
Release 2.5 1. 2 +x .. ( ) 2. 3 () (Space) 3 3 (x, y, z) .. ( ). :
P, Q P(x, y) Q(2, 4) Q (2, 4)[1] P Q PQ PQ () x ( y) Q (x2,y2)PQ
(x2 x1)2 (y2 y1)2 P (x1,y1) (Pythagorean Theorem) 3 ( ) ( ) 145.
[email protected][2] m:n Q (x2,y2) Q (x2,y2) m x1 x2 y1 y2 nR(
, ) 22 mx nx2 my1 ny2R( 1, )P (x1,y1)mnmnP (x1,y1) x y ( .) 4.1 P
(1, 2) Q (5, 4) . PQ PQ ((5) 1) (4 (2)) 36 36 6 2 22 . PQPQ (1(5) ,
(2) 4) (2,1)22 . R PR : RQ 2 : 3 3(1 2(5) 3(3)2(4)) R (,) ( 7 , 1)3
2 3 25 5[3] 3 ( Centroid) 1:2 x y R (x3,y3)xx x yy y C ( 1 2 3 , 1
2 3) 3 3 P (x1,y1) C Q (x2,y2) S( ) 146. 146 Math E-BookRelease
2.5[4] (*) ( ) ( det )T (x5,y5) x1y1P (x1,y1) x2y21 x3 y3 2 x4 y4 S
(x4,y4) x5y5 x1y1 Q (x2,y2)R (x3,y3)1 (x y x y x3y4 x4y5 x5y1 x2y1
x3y2 x4y3 x5y4 x1y5)2 1 2 2 3 4.2 ABCD (1, 6) , (2, 9) , (4, 1 , )
(3, 2) (1, 6) (4, 1) (2, 9) (3, 2)( ) 16411 29 2 3 2 16 1 [(1 36 4
18 (24) 2 (27) (2)] 54 ) 2 54 .(1) P1 (1, 7) P2 (4, 3) (1.1) P1 O
OP1(1.2) P2 O OP2(1.3) P1 P2 P1P2 147. [email protected](2)
A(2, 7) , B(6, 3) , C(2, 5) , D(8, 1) P Q AB CD PQy(3) OBCD D (2,4)
C P BC PC PQ PQC P QxOB (2,0)(4) ABC A(5, 3) , B(6, 1 , C(1, 8))(5)
ABC P(2, 1 , Q(5, 2) , R(2, 3) ) ABC (6) A(2, 8) , B(6, 12) , C(2,
4) P Q AB BC AP : PB 1 : 3 BQ : BC 3 : 4 PQ(7) . A(10, 5) , B(3, 2)
, C(6, 5) . D(1, 2) , E(3, 10) , F(4, 4) . A(2, 3) , B(6, 1) ,
C(10, 1) (8) P x P1 (1, 2) P2 (3, 5) (9) (1, 7) , (8, 6) , (7,
1)(10) 3 A(2, 1) , B(4, 3) , C(2, 5)(11) (m, n) (4, 5) , (4, 7) ,
(4, 1) m n (12) ABC B(6, 7) C(4, 3) P(4 , 1) A 3 148. 148 Math
E-BookRelease 2.5(13) P (13, 2) (13, 2)Q (6, 10) (0, 14) R (8, 4)
(16, 4) PQR(14) ABC PQR A(1, 3) , B(2, 0) , C(3, 5) , P(0, 0) ,
Q(8, 18) , R(12, 27)(15) P(3, 2) , Q(2, 3) , R(0, 4) . PQR 9 5 .
PQR 15 (16) A(1, 4) , B(3, 2) , C(1, 3) , D(4, 5) , E(2, 7) . : P Q
( P Q) PQ ( ) L L1 , L2 L ( L I 1) () S [1] (Slope; m) y x m ( x 1
y m ) m m > 0 () m < 0 () m = 0 x y m () 149.
[email protected] P () P Q Q (x2,y2) y2 y1 m ( tan ) x2 x1P
(x1,y1) (Parallel; //) (Perpendicular; ) 1[2] (Common Form) A x By
C 0( A, B, C ) x y 2 (x,y) 2 ( x y 0 y x ) [2.1] y m x c (Slope
& Intercept Form) m c y 1 (x1, y1) m m c m (x1, y1) P (x1,y1) 2
(x1, y1) (x2 , y2)Q (x2,y2)( c) P (x1,y1) 2 2 m c 150. 150 Math
E-BookRelease 2.5 4.3 P (1, 3) Q (5, 9). P Q m 5 19 3 3/2PQ. P Q 1
PQ 32 3 P Q y 2 x c 33 (3) 2 (1) c c 2 ( P Q ) 3 3 y 2 x 2 2y 3x 3
.. 3x 2y 3 02 PQ y m x c P (3) m(1) c Q (9) m(5) c33 3 m 2 c 2 .. y
2 x 2 3 y Ax CBB A (c) C (m) BB3 3 3x 4y 5 0 (4) 4 A/B x y S A x By
C 0 y y1[2.2] m (Slope & Point Form) x x1 (x,y) m 1 (x1, y1) m
y y1 m (x x1) 2 (x1, y1) (x2 , y2) y y1 y2 y1 (x x1) x2 x1 151.
[email protected] 4.4 P (1, 3) Q (5, 9) L1 PQ PQ PQ3 ()2 L1 PQ
mL1 2 ( 1 )3 PQ (15 , 3 9) (3, 6) 2 2 L1 y y1 m (x x1) (y 6) 2 (x
3) 3 3y 18 2x 6 .. 2x 3y 24 0x y[2.3] 1 (Intercept Form)a b a, b
(Intercept) x y ( x (a,0) y (0,b)) a, b y b xOa S 2 1 2 4.5 L5 y
(0, 1/ 3) x 1/2 L6 (1, 2) L5 . L5L6 x 1/2 x (1/2, 0) 1/ 3 0 L5 ..
mL5 0 (1/2) 2/3 L6 L5 mL6 3/2 152. 152 Math E-BookRelease 2.5. L5
xy 1 .. 2x 3y 11/21/33L6 (y 2) (x 1) .. 3x 2y 12 (1/13, 5/13) [3]
(x1, y1) A x By C 0 0 0 P (x1,y1)A x1 B y1 Cd d A2 B2Ax+By+C=0 [4]
Ax+By+C1=0C2 C1dd A2 B2Ax+By+C2=0 A x By C 0 S A, B 153.
[email protected] 4.6 L : 2x 3y 24 01. S (2, 5) L 1 d 2(2) 3(5)
24 13 13 SL122 3213 . L12 13 L1 () 2x 3y C 0 24 C C 2 13 22 32 26
24 C C 2 50 2x 3y 2 0 2x 3y 50 0 . L2 : 2x y 6 0 L12 13 (x1, y1)2x1
3y1 24 2 13 22 32 (x1, y1) 2x y 6 0 2x1 y1 6 0 x1 8 5 x1 8 y1 22 ..
x1 5 y1 4 (8, 22) (5, 4) . L2 . . L1 2 13 4.7 L3 3x y 2 3L4 3x 3y
18 . L3 A/B 3/1 3 . L4 x L4 3/ 3 3 (y:x) 3 x 60 154. 154 Math
E-Book Release 2.5. L3 L4 L3 L4 ( . .) A, B L4 3 3x y 6 3 6 3 2 3 4
3 dL3L4 2 32( 3) 122L3 L4 2 3 3. L3 , x, y 0 (0,0) x y x y x ( y 0)
2 y ( x 0) 2 3 1 2 2 2 3 23 [5] (*) L1 : A1x B1y C1 0 L2 : A2x B2y
C2 0 m1 m2 ( 180 )m1m2 m1 m2 tan () 1 m1m2 (x,y) L1 L2 A1x B1y
C1A2x B2y C2 A2 B211A2 B2 22Ans (Ans) 155. [email protected]
[6] (Projection) ( ) P L Q P1P2 L Q1 Q2 P (x1,y1) P2 (x2,y2)L:
Ax+By+C=0Q P1 (x1,y1) Q2 L: Ax+By+C=0 Q1 P L P L P P (x1, y1) y x (
45 x) Q (x12 y1 , x12 y1) .(17) A(1, 2) , B(2, k) , C(3, 4) k(18) y
Q(1, y) PR P(2, 6) R(4, 2)(19) AB x y 4 3 AB 3 A B(20) A(4, 5) ,
B(1, 2) ,C(2, 8) , D(2, 4) AB CD 156. 156Math E-BookRelease 2.5(21)
D ABCD A(4, 1) , B(5, 4) , C(1, 2)(22) (k, 7) (3, 2) (3, 2) (1, 4)
k (23) A(1, 5) B(3, 6) C(m, 4) D(1, m) m (24) C(5, 6) L (3, 1)
L(25) ABC A(1, 7) , B(8, 6) , C(7, 1)(26) (7) (27) (3, 0) (0,
2)(28) L (2, 5) (1, 3) x y (29) (6, 8) x 3x 4y 12(30) ABCD A(1, 2)
, B(2, 1) , C(3, 6) , D(2, 5) P P (31) 2x 3y 10 0 x y 1 2x y 5(32)
x 3x 4y 5 0 y (33) L 2x 3y 5 0 (1, 5) L x (34) M 3x 3y 5 7 N 2x 5y
7 4 L M y N 157. 157 [email protected](35) L1 (2, 2) (2, 0) L2 L1
(2, 0) L3 x 4/3 y 4(36) L1 2x 3y 6 0 L2 (2, 3) L1 L3 (2 , 1) L1 3
L2 L3 (37) A(3, k) 1 4 L A x L(38) L A(1, 2) B(2, 1) PQR L, x, y.
PQR 6 3 2 . PQR 4.5 (39) ABC A(2, 5) , B(4, 8) , C(2, 3) x y (40) x
y (1, 3) (41) (2, 4) 9 x (42) L 0.5, C(3, 0) , y A AB L y B x C BC
(43) ABC B , A (3, 5) , C (4, 4) AB 3/2 (44) 2x 3y 6 4x 6y 25 (45)
C Ax 2y C 0 3x 4y 5 0 158. 158Math E-BookRelease 2.5(46) L1 L2 4 L
12x 5y 15 0 L1 L1 , L2 x L1 L2(47) A(2, 1) , B(5, 4) , C(2, 3) A
BC(48) L 5x 12y 3 k P(3, 2) 4 k (49) 2x 4y 15 3x 4y 10 3 (50) 5x y
0 2x 3y 1 0(51) L1 ( 3, 2) (0, 1) L2 (2, 3) (1, 4)L1 L2(52) L1 (2,
3) (1, 0) L2 O L1 C L1 L 2 30 CO(53) 3x 4y 1 0 4x 3y 6 0(54) A (2,
1) x B (5, 6) y AB(55) A(1, 0) B(5, 8) P AB Q B x 1 PQ PQ(56) (2,
1) xy 0(57) (0, 7) 4x 5y 6 159. [email protected] . : (Conic
Section) (Circle) (Ellipse) (Parabola) (Hyperbola) 1. ()2. ()3.
()4. 2 () x y , , [1] (Translate Shift) y f(x) y k f(x h) h, k h k
h , h , k , k (0, 0) (h, k) 160. 160 Math E-BookRelease 2.5yy 2y=xy
= (x-3)2xO (3,0) x y y y+1 = x2 y+1 = (x-3)2 xx (0,-1) (3,-1)[2] ()
(Scale) y f(x) m y f(nx) m, n 1 n m ( m, n 1 ) h, k yyy = x2 3y =
x2 xx O 1 3yyy = (2x)2y/4 = x2 xx 1 2 4 161. 161 [email protected]
() y 2y = (x-3)2-2 2(y+1)=(x-3)2 x (3,-1) 1 2[3] () [2] (Flip) h, k
x y yyy=x 2-(y+1) = (x-3)2 x O x (3,-1) . : (Center; C) (Radius; r)
( .) (x, y) C (0, 0) r x2 y2 r2 C (h, k) (x h)2 (y k)2 r2 162.
162Math E-BookRelease 2.5 (x h)2 (y k)2 r2r C(h, k) r C (h,k) x2 y2
Dx Ey F 0 4.8 (1, 2) (2, 1)2 2Ax By Dx Ey F 01 (1, 2) (2, 1) 12 32
10 (x h)2 (y k)2 r2 (x 1)2 (y 2)2 ( 10)2 x2 2x 1 y2 4y 4 10 x2 y2
2x 4y 5 0 4.9 x y 2x 4y 10 0 22 x y (x 2x) (y 4y) 1022 () 2 2 (x 2x
+ 1) (y 4y + 4) 10 + 1 + 4 (x 1)2 (y 2)2 15 (1, 2) 15 x2 y2 Dx Ey F
0 (h, k) (D/2, E/2) 2 (1, 2) D 2 E 4 x2 y2 2x 4y F 0 (2, 1) F (2)2
(1)2 2(2) 4(1) F 0 F 5 x2 y2 2x 4y 5 0 163. [email protected] 3
D, E, F( h, k, r ) 3 3 x2 y2 Dx Ey F 0 r 2 h, k (h, k) r () (x h)2
(y k)2 r2 ( ) P (x1, y1) Q (*) (0 ) Q d P (x1,y1) C d x2 y2 Dx1 Ey1
F11d(x1 h)2 (y1 k)2 r2 .(58) (58.1) (x 4)(y 3) 1 xy 1 (58.2) y x 1
2 y x (58.3) x y 2x 4y 5 9 x2 y2 k22(59) (59.1) (3, 4) (1, 1)(59.2)
(1, 1) (2, 2)(59.3) y 2x (1, 1) 164. 164Math E-Book Release
2.5(59.4) (6, 3) , (2, 3) (2, 7)(59.5) (1, 5) x y 2x 2y 8 0 x2 y2
3x 3y 8 02 2(60) (0, 1) 3x2 3y2 11x 15y 9(61) (61.1) x2 y2 8 (2, 2)
(61.2) x2 y2 17 4( ) (61.3) x2 y2 16 (1, 8)( )(62) ()(62.1) 2 (x
2)2 (y 1)2 1 (x 6)2 (y 2)2 4 1(62.2) 1 , y x 2 x2 y2 4x 2y 1
0(62.3) 2x 3y 21 0 , 3x 2y 6 0 , 2x 3y 9 0(63) k x2 y2 6x 8y k 0
(64) k 0 y kx x2 y2 14x 49 k2(65) O C x2 4x 2 (y2 8y 9) OC OC (66)
4/3 x2 y2 4x 2y 4 A B D(1, 2) ABD (67) P(x, y) (x 1)2 (1 y)(1 y)
A(1, 0) 165. 165 [email protected] . : (Focus; F) (Directrix; )
(Axis) (Vertex) () V (0, 0) c x 4 c y y ( c , 2 c ) y2 4 c x x (c ,
c ) V (h, k) (x h)2 4 c (y k) (y k)2 4 c (x h) () 2c (x h)2 4 c (y
k)F(h,k+c) V(h, k)c c V (h,k) 4c cDirectrix : y=k-c x2 Dx Ey F
0Axis : x=h ()(y k)2 4 c (x h) 2c V(h, k)c c c Axis :VF (h+c,k) 4c
y=k (h,k)Directrix : y2 Dx Ey F 0x=h-c 166. 166 Math E-BookRelease
2.5 S (Latus Rectum) , , . 3 D, E, F ( h, k, c ) 2 4.9 (1, 2) (2,
1) Ax2 By2 Dx Ey F 0 y (x h)2 4 c(y k) (h, k) (1, 2) (x 1)2 4 c(y
2) c (2, 1) x, y (2 1 2 4 c(1 2)) 4 c 1/ 3 (x 1)2 (1/ 3)(y 2) 3(x2
2x 1) y 2 3x2 6x y 1 0 4.10 x 2x 2y 3 0 2 y ()2 x y (x2 2x) 2y 3
(x2 2x + 1) 2y 3 + 1 (x 1)2 2y 4 (x 1)2 2(y 2) (x 1 2 4(0.5)(y 2)
)1 F 1 (1, 2) 0.5 (1, 2 0.5) (1, 1.5) 0.5 V(1,2) 2 y 2 0.5 2.5
Directrix( 2y 5 0 ) 1:4 (c:4c)S (0,0) (1,2) () 167. 167
[email protected] .(68) (68.1) (2, 3) (5, 3)(68.2) O (3, 6)(68.3)
O (4, 6) x (68.4) (2, 3) (8, 2.1) x(68.5) (5, 2) (3, 0) y(68.6) (2,
2) x 2 0(68.7) (1, 3) , (9, 1) , (51, 2) x(68.8) (2, 3) , (3, 18) ,
(0, 3)(69) P(3, 6) 2x2 3y 0 (70) (70.1) x2 12y 0(70.2) y2 10y 12x
61 0(70.3) (4, 2) x 1 0 1 7(70.4) x (0, 3) (0, 6)(71) P(x, y)
(71.1) y 4 (2, 8) (71.2) x 4 (3, 1) 5 (72) 4y (x 1)2 13 x (73) 2x y
8y2 8x (74) (1, 6) y2 4x 4y 8(75) y 5 x2 6x 6 2y y2(76) x y x2 y2
6x 0 x (77) y2 4y 8x 20 P , P, (78) 6 8 168. 168 Math E-Book
Release 2.5. : ( F1, F2 ) (2a) (Major Axis) ( V1V2 ) 2a (Minor
Axis) ( B1B2 ) 2b a b a b , 1 a b a b x y . C (0, 0) 2a 2b x b 1 (
x)y 22ay b22x 1 ( y) a ()B1 (h,k+b) (x h)2 (y k)2 1 a2 b2ab C(h, k)
2a 2b V2 F2 CF1 V12 2c (h,k) (h+c,k) (h+a,k) c a bB2 Ax2 By2 Dx Ey
F 0V1 (h,k+a) () (y k)2 (x h)2 F1 1 2ab2(h,k+c) b C(h, k) B2 C
(h,k) B1 (h+b,k) 2a 2bc c a2 b2aF2 V2 Ax2 By2 Dx Ey F 0 169. 169
[email protected] C (h, k) (x a2h) (y b2k) 1 (y a2k) (x b2h) 1 2 2
2 2 x y S a c a b a c a2 b2 b c a 2a a, b, c ( r) (h,k) S.. c 2a()
4 4 4 h, k 4.11 (2, 1) (2, 4) 2 2 (2, 4) Ax By Dx Ey F 0 x y (y k)2
(x h)2 a2 b2 1 a (4) (1) 5 c (4) (1) 3 .. b 52 32 4 (y 1 2 (x 2)2 )
(h, k) (2, 1) a, b 52 42 1 16(y 1 2 25(x 2)2 400 16(y2 2y 1) 25(x2
4x 4) 400) 25x2 16y2 100x 32y 284 0 170. 170 Math E-BookRelease 2.5
4.12 7x 16y 28x 96y 60 0 22 x y 1 22 2 2 (7x 28x) (16y 96y) 60 7
(x2 4x) 16(y2 6y) 60 7 (x2 4x + 4) 16(y2 6y + 9) 60 + 28 + 144( 7 4
28 , 16 9 144 ) 7 (x 2)2 16(y 3)2 112 112 .. (x16 (y 3) 12)22 7 B1
(2, 3) a 4, b 7 c 16 7 3 V24 7 (2 4, 3) (2 3, 3) F2 3 C(2,3) F1 V1
(2, 3 7) B2 .(79) (79.1) (3, 1) y 8 6 (79.2) (0, 8) (0, 5)(79.3)
(4, 2) (2, 2) 4 (79.4) (2, 1) (2, 4) (6, 1)(79.5) (2, 1) (2, 4) c :
a 2 :5(80) (80.1) 4x2 9y2 36(80.2) 9x2 5y2 54x 50y 26 0(80.3) 5x2
9y2 10x 40(81) P(x, y) (81.1) (4, 0) (4, 0) 12 (81.2) (2, 7) (2, 1)
10 (82) 6 10 (82.1) (82.2) x 171. 171 [email protected](83) 4x2
9y2 48x 72y 144 0 3x 4y 5(84) 45 x x2 3y2 4x 2 0 (85) F1 F2 kx2 4y2
4y 8 y B x F1F2 B 347 k (86) 4 1 80 . . : ( F1, F2 ) (2a)
(Transversal Axis) V1V2 2a (Conjugate Axis) B1B2 2b ( a b )
(Asymptote) 2 ( a b ) ( ) C (0, 0) 2a 2b x b 1 y 2 2 a x (-) y b 1
22xa y (-) C (h, k) (x a2h) (y b2k) 1 (y a2k) (x b2h) 1 2 2 2 2
172. 172 Math E-BookRelease 2.5 ()B1 (h,k+b)(x h)2 (y k)2 1 a2 b2
cb C(h, k) F2 V2 C V1F1 2a 2ba (h,k) (h+a,k) (h+c,k) c a2
b2B2Asymptote : Ax2 By2 Dx Ey F 0Asymptotea(y-k)=b(x-h) F1 (h,k+c)
()Asymptote : (y k)2 (x h)2 1b(y-k)=a(x-h)V1 (h,k+a)a2b2b C(h, k)
B2C (h,k) B1 (h+b,k) 2a 2b a c a2 b2c Asymptote V2 F2 Ax2 By2 Dx Ey
F 0 a bS () a 4.13 (2, 1)(2, 4)2 2 (2, 4) Ax By Dx Ey F 0 x y (ya2
(xb2 1k)2 h)2 a (4) (1 3 )c (4) (1 5 ).. b 52 32 4 (y 1 2 )(x 2)2
(h, k) (2, 1) a, b 1 32 42 16(y 1 2 9(x 2)2 144 ) 16y2 9x2 32y 36x
164 0 173. [email protected] x 1 9x2 16y2 36x 32y 164 0 4.14 x
5y 10y 25 0 2 2 x 5(y 2y) 25 ()22 x x 2 2 x 5(y 2y + 1) 25 - 5 ()
x2 5(y 1)2 20 20 (x) (y 1) 12 2 204 (0, 1) a 20, b 2 c 20 4 24 (
20, 1) ( 24, 1) (0, 1 2) 2 22 a a c b S c c a b2 2 2 caacb b [1]
(Eccentricity; e) e c / a e 0 1 e 1 e 1 c a b 0 [2] (*) 2 ( ) 2 2 2
b /a 2 a2 /c 174. 174 Math E-BookRelease 2.5 Ax 2 By 2 Cxy Dx Ey F
0 C 0 4 (rotate) - a b () 2 (Rectangular Hyperbola) ( xy) xy k k 0
45 Asymptote (Q1 ,Q3)xy kk 0F1 C (0, 0) Asymptote V1C (0,0) V1 ( k,
k) V2 V2 ( k, k) F2 F1 ( 2k, 2k)F2 ( 2k, 2k)Asymptote (Q2 ,Q4)xy kk
0F1 C (0, 0)V1AsymptoteC (0,0) V1 ( k, k)V2V2 ( k, k) F2F1 ( 2k,
2k)F2 ( 2k, 2k) 175. [email protected] .(87) (87.1) (3, 1) (2,
1) 6 (87.2) (1, 6) (1, 4) 6 (87.3) (0, 4) (0, 4) (3, 0)(88) (88.1)
9x2 4y2 36(88.2) 9x2 16y2 18x 64y 199 0(88.3) 6x2 y2 36x 2y 59
0(88.4) 6x2 10y2 12x 40y 94 0(89) P(x, y) P (3, 0) (3, 0) 4 (90)
P(x, y) 4x 3y 11 4x 3y 5 144/25(91) (91.1) xy 4 (91.2) xy 2x y
3(92) 9x2 18x 16y2 64y 199 3x 4y 8 (93) F1 6x2 10y2 12x 40y 94 0 4
F1(94) 9 (x 1)2 4 (y 2)2 36 x 8 (95) E 6x2 5y2
