สรุปเนื้อหาวิชา คณิตศาสตร์ ทุกบท ม.4-ม.6.pdf

* .4-5-6 * 2,000 * 14 (2541-2548) * .. .4-5-6

* .4-5-6 * 2,000 * 14 (2541-2548) * .. .4-5-6

Release 2.2 / / / / / / /

.. ()

http://math.reads.it [email protected]

2 Math E-Book Release 2.2

http://math.reads.it Release 2.0 13 2548 Release 2.1 28 2548 Release 2.2 14 2549 ( Release 2.0) 2548 O-NET & A-NET SCIENCE CENTER () 159

3/1 . 10200

.. . 0-2225-7160, 0-2221-5884

7-Eleven

O-NET / A-NET 3

.+

+

+

+

..2544 4 ( .4 .6) ()

14 (..41 ..48) (2532 2548, ) (O-NET / A-NET)

:]

Math E-Book Release 2.2 ( )

O-NET / A-NET 4

(1) () :] () ( !)

(2) ( )

() ..

( 2548) 2 ..

- . . ()

- - . - . () . . Science Center - -

( ) , , ,

- 3 !

- Thaiware.com, se-ed.net, f0nt.com ...

[email protected] http://math.reads.it :]


O-NET / A-NET 6

( O-NET)

1 () 2 ( 2.2 2.5) 3 ( 3.5) 5 ( 5.2 5.5) 7 ( 7.9) 8 ( 8.1) 13 ( 13.1 13.4 ) 16 ( 16.1 16.6) 17 ( 17.5 17.6 )

( A-NET)

2 13 16 ( 16.4) 17


O-NET / A-NET 7

a

1 11 1.1 12 1.2 - 15 1.3 21 2 31 2.1 32 2.2 36 2.3 39 2.4 44 2.5 48 58 3 59 3.1 60 3.2 63 3.3 65 3.4 67 3.5 69 82 4 83 4.1 : 84 4.2 : 86 4.3 : 92 4.4 : 94 4.5 : 96 4.6 : 99 4.7 : 102 5 119 5.1 120 5.2 121 5.3 124


O-NET / A-NET 8

5.4 127 5.5 131 fog 146 6 147 7 157 7.1 158 7.2 160 7.3 162 7.4 165 7.5 166 7.6 169 7.7 171 7.8 172 7.9 173 8 187 8.1 187 8.2 191 8.3 192 8.4 195 () 204 9 205 9.1 206 9.2 208 9.3 211 9.4 215 9.5 216 10 227 10.1 228 10.2 230 10.3 231 10.4 233 10.5 235 10.6 237 10.7 240 : 250


O-NET / A-NET 9

11 251 11.1 252 11.2 254 11.3 256 11.4 259 268 12 269 12.1 270 12.2 272 12.3 274 13 279 13.1 280 13.2 282 13.3 284 13.4 285 14 295 14.1 296 14.2 298 14.3 300 306 15 307 15.1 307 15.2 309 15.3 312 15.4 317 15.5 319 332 16 333 16.1 333 16.2 335 16.3 337 16.4 339 16.5 341 16.6 345 358


O-NET / A-NET 10

17 359 17.1 360 17.2 363 17.3 374 17.4 378 17.5 383 17.6 388 1 (14 ) 403

c | 2541 408 d | 2542 417 e | 2542 426 f | 2543 435 g | 2543 444 h | 2544 453 i | 2544 462 j | 2545 471 k | 2545 481 l | 2546 492 n | 2546 502 o | 2547 512 p | 2547 523 q | 2548 532

1 541 (17 ) () 1 | 2532 2541 542 2 | 2541 2548 573 : 1 ( 2 , 70 ) 588 2 (35 ) 606 : Math E-Book 616 657


O-NET / A-NET 11

{ s,e,t }

1 e

(Set) , 7, 360 , (Element Member)

A, B, C

{ } A , B 7, C 360 , D 7, E 3 33 A {= , , , , , , } (comma) B { 2, 1, 0, 1, 2}= B {0, 1, 1, 2, 2}= C {2, 3, 5}= D {2, 3, 5}= C D= (360 2 2 2 3 3 5= ) E {4, 5, 6, 7, ..., 32}= ...


O-NET / A-NET 12

(Finite Set) A n(A) n(A) 7= , n(B) 5= , n(C) 3= , n(E) 29= (Null Set Empty Set) { } n( ) 0 =

(Infinite Set) F 2, G 0 1 F {1, 0, 1, 2, 3, ...}= , n(F) G { | } G { x | 0 x 1}= < < x () 0 x 1< < ()

2 B , 3 C , 0.5 G 2.5 B , 4 C , 0 G

(Relative Universe) U U U { 2, 1, 0, 0.5, 7}= U H { x | x 0 }= > H {0, 0.5, 7}= =U H {0, 1, 2, 3, ...}=

U (R ) H { x | x 0 }= > H { x | x 0 }= >R

1.1

(Subset) B A B A ( B A B A) B A B A B A B A

A {m, p, r, w}= B B A 16 {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}

S oae! S

ao {a, b} A a A a b A

S ui o! S eo ie { , 0, 1, {2, 3},(4,5)} oo 5 a e, e 0, e 1, e {2,3}, aoaa (4,5) aoeae 1 oaaae 1

{(1, 2),(2, 1), {1, 2}, {2, 1}} oo 3 a oaa (1,2), oaa (2,1), ae {1,2} (oaa 1-2 a 2-1 o a e 1-2 ae 2-1 oeoaa oaaa)

eooaea e e a ooa ... ooe a a ia oaa..

O-NET / A-NET 13

( 16 16.3)

n r = n n!r (n r)! r !

x ! 1 2 3 ... x= 7 3 7 7 ! 1 2 3 4 5 6 7 353 4 ! 3 ! 1 2 3 4 1 2 3

= = =

1. A 2. A A 3. n n2 ... ( 42 16= ) 4. (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}= { , , , , , , , , }

. A {a}=

12 2= {a} P (A) { , {a}}=

. B {a, b}= 22 4= , {a} , {b} {a, b} P (B) { , {a}, {b}, {a, b}}=

. 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 = 02 1= P (D) { }=

S ui o! S o aaaa a { } ao ...

(e) eeo oe oaoae ( iea 0) aeaae { }

o oe oo ae oe o oo e a

eoaa eoe { } e eo o ... ooa o n( ) 0 = n({ }) 1 =

S oae! S

ao {a, b} P(A) {a, b} A ao a A a b A


O-NET / A-NET 14

E { , {0}, { }}= P(E) , { }, {{0}}, {{ }}, { , {0}}, { , { }}, {{0}, { }}, { , {0}, { }} { }

A, B A {1, 3, 5, 7}= B {1, 2, 3, 4, 5, 6, 7}= . X X P (A)

X P (A) X A X 42 16= 4 4 44 4 1 4 6 4 1 161 40 2 3 + + + + = + + + + =

. X X P (A) n(X) 2< X P (A) X A 16 ( .) n(X) 2< 4 44 1 4 6 1110 2

+ + = + + = ( )

. 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=

1.1 (1) A, B A B A B x A y B (1.1) {x} B (1.3) {A} {B} (1.2) {y} A (1.4) {A} {B}

(2) A {{ }, a, b, {a}, {a, b}}= (2.1) { } A (2.3) {{a}, b} A (2.2) { } A (2.4) {a, b} A {a, b} A

(3) (3.1) A B B C A C (3.2) A B B C A C

(3.3) A B B C A C

O-NET / A-NET 15

(4) A (4.1) { x | x A } {A}= = (4.3) { x | {x} A } {A} = (4.2) { x | x A } A = (4.4) { x | {x} } =

(5) (5.1) n(A) 5= A 32 (5.2) n(A) 5= A 32 (5.3) n(A) 5= A 32

(5.4) n(A) 5= A 32

(6) A 511 A 511 5

(7) (7.1) (7.5) P ( ) (7.2) (7.6) P ( ) (7.3) { } (7.7) { } P ( )

(7.4) { } (7.8) { } P ( )

(8) A { , a, {b}, {a, b}}= (8.1) P (A) (8.6) a P (A) (8.2) { } P (A) (8.7) {a} P (A) (8.3) P (A) (8.8) {b} P (A) (8.4) { } P (A) (8.9) {{b}} P (A) (8.5) { , a, {b}} P (A) (8.10) { , a, {b}} P (A)

(9) A { , 1, 2, 3, {1}, {1, 2}, {1, 2, 3}}= (9.1) { , {1}, {1, 2}} P (A) (9.3) {{1}, {2}, {3}} P (A) (9.2) { , {1}, {1, 2}} P (A) (9.4) {{1}, {2}, {3}} P (A)

(10) [Ent39] 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 }

1.2 -

(Venn-Euler Diagram) U ( ) A, B, C

S ui o! S ae A a B a oiao (eoa) aeoi i ooei.. ooiaoa..


O-NET / A-NET 16

U

A B

U

A B

U

B

A

A

C

B

2 3 1 0 4

5 7

9

11

U

U

A B

U

A B

U

B

A

U

A B

U

A B

U

B

A

U

A

U

A B

U

A B

U

B

A

A B A B A B

A {0, 1, 2, 3, 4}= B {1, 3, 5, 7, 9}= C {2, 3, 5, 7, 11}=

1. (Union : ) ... A B A B

A B B 2. (Intersection : ) ... A B A B AB ( ) A B A B A

3. (Complement : ' ) A' A cA A

4. (Difference Relative Complement : ) B A B A ... B A B A' = B A B A n(B A) n(B) n(A) n(B A) n(B) n(A B) =


O-NET / A-NET 17

A (B C) (A B) (A C)A (B C) (A B) (A C)A (B C) (A B) (A C)A (B C) (A B) (A C)

= = = =

(A B) ' A ' B '(A B) ' A ' B 'P (A) P (B) P (A B)P (A) P (B) P (A B)

= =

=

A B A cup B A B A cap B A, B A {1, 3, 5, 7}= B {1, 2, 3, 4, 5, 6, 7}= ( . . ) . Y A Y Y B (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 ( A 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=


O-NET / A-NET 18

C P(C)

2 4 60

C { } 0 {{ }, 0} { , {0}} {{ , {0}}}= { , , , , , } . n(P (C)) n(C) 6= 6n(P (C)) 2 64= =

. 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+ = ( )

1.2 (11) 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} =

(12) C ' B ' { x | 1 x 10 }= <

O-NET / A-NET 19

B { x | x= 3 } C { x | x 5 }= <

(13) [Ent38] A {0, 1}= B {0, {1}, {0, 1}}= (13.1) A P (B) (13.2) {1} P (A) P (B) (13.3) n(P (A B)) n(P (A B))

(14) (14.1) ' = U (14.7) A A ' = (14.2) ' = U (14.8) A A ' = U (14.3) A (A B) (14.9) A = U A A ' =U (14.4) B (A B) (14.10) A A = A =

(14.5) (A B) A (14.11) A A = (14.6) (A B) B (14.12) A B A B ' =

(15) (15.1) A B P (A) P (B) (15.2) A B = A = B = (15.3) A B = A = B = (15.4) A B = B C B = A ' C ' = U (15.5) A B = B C A C

(16) A, B (16.1) A B A B (16.5) x A x A B (16.2) A B B A (16.6) x A x A ' B '

(16.3) A B A B ' = (16.7) x A x A ' B ' (16.4) (A B) ' B ' A = (16.8) x A x (A ' B ') '

(17) (17.1) A (A B) (17.6) (A B) B (17.2) (A B) B (17.7) (A B) B (17.3) (A B) B (17.8) A (A B) (17.4) A (A B) (17.9) (A B) (B A ') (17.5) A (A B)

(18) (18.1) A C B C = A B= (18.2) A C B C = A B= (18.3) A C B C = A B= (18.4) A ' B '= A B=

(19) A B A = 3

O-NET / A-NET 20

(20) (20.1) [Ent21] (A B) (B A) (A B) (20.2) [A (A ' B)] [B (B ' A ')] (20.3) ( ) ( )[(A B) (B A)] A ' A ' [(A B) (B A)]

(20.4) ( )[(A B) ' (B C ')] [(D E) (C ' E ')] (A E ') '

(21) (21.1) (A B C) (A ' B C) (B ' C ') = U (21.2) (A B C D ') (A ' C) (B ' C) (C D) C = (21.3) P (A B) P (A B) (21.4) P (A B) P (B A) { } = (21.5) A B P (A B) P (A) P (B) =

(22) A {0, 1, 2, 3}= , B {{0}, 1, 2, {3}}= C {0, {1}, {2}, 3}= (22.1) P (A) P (B) P (C ') { , {1}, {2}, {1, 2}} = (22.2) P (A) P (B ') P (C) { , {0}, {3}, {0, 3}} = (22.3) P (A ') P (B) P (C) { , {0}} = (22.4) P (A) P (B ') P (C ') { } =

(23) n( ) 35=U , n(A) 22= , n(B) 18= n(A ' B ')

(24) 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)

(25) 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) '

(26) A { , 1, {1}}= A B ' = (26.1) n [ P (A) P (B) ] 8 = (26.3) P (A B) { } = (26.2) {1} P (A B) (26.4) P (B A) { } =

(27) [Ent36] A { , { }, 0, {0}, {1}, {0, 1}}= [ P (A) A ] [ A P (A) ]

(28) A (28.1) A B {1, 2, 3, 4, 5} = B {1, 3, 5}= (28.2) A B {1, 2, 3, ..., 15} = B {2, 4, 6, 8, 10}=


O-NET / A-NET 21

= + -

= + +

- - - +

e aa..

(29) A {1, 2, 3, 4, 5, 6, 7}= B {1, 2, 3}= X (29.1) B X A (29.2) X A B X

(30) B A n(A) 10= , n(B) 4= n(C) (30.1) C { S | B S A }= (30.2) C { S A | S B }=

(31) A {0, 2, 4, 6, 8}= B {0, 1, 2}= C {1, 2, 3}= D {0, 2, 3}= X X A (31.1) B C ' X (31.3) B D X (31.2) B C ' X (31.4) B D X

(32) {1, 2, 3, 4, ..., 8}=U A {1}= U B {2, 4, 6}= C {1, 7}= D (B ' C) D A

(33) { x | 2 x 6 }= <

O-NET / A-NET 22

U

Math Eng

Math Social

Thai

30 12 15 5 U ( )

1 5 5 12 12-5=7 15 15-5=10 30-5-7-10 = 8 ...

2 n(M) 12= , n(E) 15= , n(M E) 5 = n(M E) n(M E) 12 15 5 22 = + = 30 22 8 = ... 37 48 45 15 13 7 5

( ) U ( ) 1 5 5 15-5=10, 13-5=8, 7-5=2 37-10-5-2=20, 48-10-5-8=25, 45-2-5-8=30 5+10+8+2+20+25+30 = 100

2 n(M) 37= , n(S) 48= , n(T) 45= n(M S) 15 = , n(S T) 13 = , n(M T) 7 = n(M S T) 5 = n(M S T) n(M S T) 37 48 45 15 13 7 5 100 = + + + = () 100 ...

() ()


O-NET / A-NET 23

1.3 (37) 80 35 27 32

(38) [Ent33] 20 () 17 15

(39) [Ent34] 20 7

(40) 34 5 1 3

(41) [Ent26] 7 , 6 5

(42) 160 100 ( 30 35 ) 80 ( 55 ) 50 60

(43) 35 11 16 7 23 21

(44) 600 500 300 200 50 30 25 15

(45) 3 4 4 6 , 3 8 3 , 3 4 2 , 4 18 4 , , 3 4

(46) [Ent31] 100 50 , 35 , 25 , 32 , 20 , 10 ,


O-NET / A-NET 24

U

22 23

20

11

9

(47) [Ent38] 180 95 92 125 52 43 57

(48) [Ent39] 100 , , 1 6 . . 199 . 10 . 24

(49) 100 60 35 20 , 28 15 , 40 22 , 14 6 , 16 10 , 20 11 , 12 8

(50) 0 100 2 3 5

()

(1) (1.1) (1.4) (2) (2.1) (2.3) (3) (3.1) (4) (4.3) (5) (5.1) (5.4) (6) 9 , 126 (7) (7.1) (7.7) (8) (8.6), (8.8), (8.10) (9) (9.3) (10) 32, 6 (11) . (12) {1, 2, 4, 5, 6, 7, 8, 9, 10} (13) , , 16-2 (14) (15) (15.3) (15.5) (16) (16.3),(16.4),(16.6)

(17.1) A B (17.2) A B (17.3) (17.4) A B (17.5) A (17.6) A B (17.7) (17.8) A B (17.9) (18) (18.4) (19) A = B = A B = (20.1) A B (20.2) B (20.3) B ' (20.4) (A E) ' (21) (22) (22.3) (23) 13 (24) d, a (25) . (26) (26.4) (27) 61+3 (28.1) 8 (28.2) 32

(29.1) 16 (29.2) (8 1) 16 (30.1) 64 (30.2) (16 1) 64 (31.1) 16 (31.2) 16 (31.3) 8 (31.4) 24 (32) 16 (33) 4 (34) 56 (35) 31 (36) 13 (37) 66 (38) 6 (39) 11 (40) 13 (41) 9 (42) 130 (43) 6 (44) 385 (45) 26, 12, 24 (46) 13 (47) 20 (48) . (49) 22-13 (50) 26


O-NET / A-NET 25

B

A x

C

A

B

()

(1.1) x A x B (1.2) y B , y A () (1.3) {A} {B} A {B} {B} B (1.4) A B () {A} {B} (2.1) ( A 5 { } ) (2.2) { } A A (2.3) {{a}, b} A {a} A b A (2.4) {a, b} A () {a, b} A a A b A {a, b} A (3.1) () (3.2) B {A}, C {B}= = C {{A}}= ... A C (3.3) A C (A C) B A C (4.1) (4.2) () (4.3) {x} A x A (4.2) (4.4) {x} x x (5.1) 52 32= ... (5.2) 31 5(2 1) (5.3) P(A) 1 P(A) 32 ... (5.4) (6) n2 512= n 9= 5 9 9! 1265! 4! = ()

(7.1) (7.3) (7.2) 02 1= () () (7.4) (7.2) , (7.6) (7.5) P( ) ( 7.2) (7.7) { } P( ) { } ( 7.1) (7.8) { } P( ) P( ) ( 7.2) (8.1) P(A) A A ( , ) (8.2) { } P(A) { } A

A ( A ) (8.3) P(A) ! , (8.4) { } P(A) P(A) (8.1) (8.5) { , a, {b}} P(A) { , a, {b}} A A a A {b} A (8.6) P(A) (8.7) {a} P(A) {a} A a A (8.8) {b} P(A) {b} A b A (8.9) (8.8) {b} A (8.10) { , a, {b}} P(A) P(A) , a P(A) , {b} P(A)


O-NET / A-NET 26

B

A

C

B C

A

(9.1) { , {1}, {1, 2}} P(A) { , {1}, {1, 2}} A A {1} A {1, 2} A (9.2) { , {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 (9.3) {{1}, {2}, {3}} P(A) {{1}, {2}, {3}} A {1} A, {2} A, {3} A (9.4) {{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 (10) n(X) A () A P(S) ( A S ) () 1 A 7 A ( A 1 7) 2, 3, 4, 5, 6 A ... {2, 3, 4, 5, 6} ..

5n(X) 2 32= = n(Y) A A x 6 {1} {1, 2} {1, 3} {1, 4} {1, 5} {1, 2, 3} A 6 .. n(Y) 6= (11) A B, A C, B C A B C {3, 5} =

A C {0, 1, 2, 3, 5} = A C 4 B . A B {0} =

. B C {1} = .. {1, 4} . A C {1} = . B A {2, 4} = (12) {1, 2, 3, ..., 10} B {3, 6, 9}= =U C {1, 2, 3, 4, 5}= C ' B ' (C B) ' C B {3} = {1, 2, 4, 5, 6, 7, ..., 10}

(13.1) A P(B) A B ( 1 B ) (13.2) P(A) P(B) P(A B) =

P({0}) { , {0}}= = ( {1} P(A) P(B) ) (13.3) 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 = (14.1) (14.2) U (complement) (14.3) (14.6) (14.7) (14.11) A A = (14.8) (14.10) ... (14.12) (!) (15.1) () (15.2) A B = (15.3) A B A B = A B (15.4) A B = A B B C B = B C (B C ) = A ' C ' (A C) ' ' = = = U ( A C ) (15.5) A B = A B B C B C A C A C " A B B C A C " A C (16.1) A B= A B A B A B = = = (16.2) A B= A B B A = = (16.3) A B ' A (B ') ' A B = = (16.4) B ' A B ' A ' (B A) ' = = (16.5) x B

A

C

B

0 3 5 1

2 4


O-NET / A-NET 27

U

A B

(16.6) A ' B ' (A B) ' = x A x (A B) ' .. (16.7) x A x (A B) ' .. ( x B ) (16.8) x A x A B .. ( x A B ) (17) ( ) A B A B B A (17.1) A (A B) = A B (17.2) (A B) B = A B (17.3) (A B) B = (17.4) A (A B) A B = (17.5) A (A B) A = (17.6) (A B) B A B = (17.7) (A B) B = (17.8) A (A B) A B = (17.9) B A ' B A = (A B) (B A ') = (18.1) C = U A B (18.2) C = (18.3) C = U (18.4) (19) B = A B = () A = (20) 2 (17) (20.1) = = A B (20.2) ( ) ( ) = = B (20.3) ( ) ( ) = = B '

(20.4) 5 () A ' B ' B C =

D E ' C ' E =

A E ( (A E)) ' (A E) ' = (21.1) B C

[(A A ' ) B C] (B C) '= U

(B C) (B C) '= = U (21.2) C

C [(A B D ') A ' B ' D]= A, B, D

C [ (A B D ') (A B D ') ' ] C= =U

(21.3) (A B) (A B) , + P( ) P( ), + (21.4) A B B A P(A B) P(B A) (21.5) 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) (22) P( ) P( ) P( ) P( ) = , , ** (22.1) A B C ' {1, 2} = (22.2) A B ' C {0, 3} = (22.3) A ' B C = P(A ' B C) { } = (22.4) A B ' C ' = (23) n(A ' B ') n(A B) ' = n(A B) .. B A n(A B) n(A) 22 = = n(A B) ' 35 22 13 = = (24) 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 = =


O-NET / A-NET 28

U

A B

32 55

U

U

(25) 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 A B . {b, e} A B ' A B ' A B {b} = = (26) A B ' A B = = A B ( A B A = ) (26.1) 3n[P(A B)] n[P(A)] 2 8 = = = (26.2) {1} P(A B) {1} A B 1 A B (26.3) P(A B) P( ) { } = = (26.4) B A (27) , { }, {0} 6(2 3) (6 3) 61 3 64 + = + = (28.1) A {2, 4,= {1,3,5} }

32 8= (28.2) A {1, 3, 5, 7, 9, 11, 12, 13, 14, 15,= {2,4,6,8,10} } 52 32= (29.1) X {1, 2, 3,= {4,5,6,7} }

42 16= (29.2) X {= {1,2,3} , {4,5,6,7} } ... 3 4(2 1)(2 ) 112 = (30.1) n(C) = 6S 2 64= = (30.2) n(C) = S

4 6(2 1)(2 ) 960= = (31.1) {0} X {0, 2, 4, 6, 8} 42 16= (31.2) {0} X X {0, 2, 4, 6, 8} 31.1 5 42 2 16 = (31.3) {0, 2} X {0, 2, 4, 6, 8} 32 8= (31.4) {0, 2} X X {0, 2, 4, 6, 8} 31.3 5 32 2 24 = (32) {3, 5, 8} D {2, 3, 4, ..., 8} 42 16=

(33) { 2, 1, 0, 1, 2, ..., 6}= U A {0, 1, 4}= U B {0, 1, 2}=

2{0, 1} X {0, 1, 2, 4} n(C) 2 4 = = (34) x {a, b, c, d, e, f} {a, c, d} X X {= {a,c,d} , {b,e,f} 3 3} (2 1)(2 ) 56 = (35) n(A) 5=

1S {B | B A, n(B) 1}= = , 2S {B | B A, n(B) 2}= = , ...

5S {B | B A, n(B) 5}= = 1 2 3 4 5S S S S S S= =

A (n(B)=0) 5n(S) 2 1 31= = (36) 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= ) (37) 35 + 35= 27 + 27= 32 32= +2++ = 94 80 (+++) 14= ++ = 80 14 66 = ( .. n(A ' B ') n(A B) ' = = ++) (38) 20=+, 17=+, 15=+ 2(++)=20+17+15 ++=26 26 20 6= =


O-NET / A-NET 29

U

U

U

U

100 60

24 11

U

.

600 500

3 4

U

32 35

20

A

B

y xz ?

C

(39) +++=20 +=2(+)-7 =

+2+=20 .....(1) -+2=7 .....(2) (1)x2 - (2); 3+3=33 +=11 (40) (, ) +++=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 (41) +=7, +=6, +=5 2(++)=18 .. ++ 9= (42) ( )

30 = + + 50 30 20= = 35 = + + 60 35 25= = 55 = 80 55 25= = 3 +++2=120 +++2=70 +++=100 =20 ++=80 +++=60 =10 ++=50 80 50 130+ = ( .., )

(43) +=16 ....(1) +=7 +=21-7=14 ....(2) +2+=16+14=30 ++=24 =30-24= 6 (44) +=200 +=30 +2+=230 =15 ++=215 600 215 385= = ( 43 44 , 42) (45) 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 (46) 3 , , n(A B C) ' 100 n(A B C) = 3 (36) n(A B C) = 32 35 20 87+ + = 13 (47) n(A B C) =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 30 20 =


O-NET / A-NET 30

22 23 20

11

9

x y

x

8

y

4

(48) 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+ + = (49) (, , ) 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 22 13 9 =

(50) {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) n(A) n(B) n(C) n(A B) n(A C))+ + n(B C) n(A B C) +

** 0 ** ... 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 n(A B C) 51 34 21 17 11 = + + 7 4 75 + =

n( ) 101=U n(A ' B ' C ') 101 75 26 = =

S ui o! S o (50) o ee A oeo 6 a a B oeo 8 a A B ae eoa.. o a 6 a 8 a 48 a ... aa! ... eo 6 a 8 aai! a o ... o 24 a a


O-NET / A-NET 31

Rea+l

2 ai

(Natural Number) (Counting Number) 1,2,3,4,... {1,2,3,4, ...}=N

( ) (Fraction)

(Integer) {..., 3, 2, 1, 0, 1, 2, 3, ...}= I

(Rational Number) { a/b | a, b= Q I b 0 }

1.


O-NET / A-NET 32

C

R

'Q Q

mI

I

I- 0I +I N

Q I

2 (Imaginary Number) (Complex Number : C ) 11

(Irrational Number) 'Q 2 1.4142...= , 3 1.7321...= , 3.1416... 2. N I Q , , .. 'Q

(Real Number : R )

2.1 ( .) [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= = [1] (Identity) a a ... 0 1 [2] (Inverse) a a ... a a a 1/a 1a [3] (Closure Property) a, b a b + R R a, b a b R R [4] (Commutative Property) a b b a+ = + a b b a=

S eiei! S

1. oo e (oooe e ) aeoaeo 2. e eeaoi ( 8) e oae a (a 2.718..)


O-NET / A-NET 33

[5] (Associative Property) a (b c) (a b) c a b c+ + = + + = + + a (b c) (a b) c a b c= = [6] (Distributive Property) a (b c) a b a c+ = + (a b) c a c b c+ = + [7] ( +R ) a 0 a + R a + R () [1] a c b c a b+ = + = a c b c a b = = c 0 [2] 0 a a 0 0= = ( 1)a a = ( a)b a( b) a b = = ( a)( b) a b = ( a) a = * [3] a b 0 a 0= = b 0= [4] a b a ( b) = + 1a b a b = b 0 ( 10 ) [5] a (b c) a b a c = [6] 1a 0 a 0 [7] a acb bc=

[8] a d ac bdb c bc++ = a d adb c bc =

[9] 1a b

b a =

[10] a b ac bc= a acb c b=

a b adc d bc=

1. [7] [10] 2. a b c a b c = = b 0 ( 0 ) ( A, B, C, D) A. B. C. Q D. I

. N A B C D


O-NET / A-NET 34

. A 2 2 0 B 2 2 2 = C D ()

. { x | x 0 }< A B C D () 2

. {1.414, 22/7} A B () C (22/7 ) D

. { 1, 0, 1} A 1 1 2+ = B C D ()

. { 10 x | x } I { 10 x | x } {0, 10, 20, 30, ...} = I A B () C D ()

2.1

(1) (1.1) 0.343443444... (1.2) 0.112112112... (1.3) 2a a (1.4) 2a a

(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 ab c= b c=

S ui o! S

oooia o oeei aeo ioi..

oo aaia 1 oai (ao ooi i a i 1 ) ... a oaooa (ao a o i i oaa..)

O-NET / A-NET 35

(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 x y 1= x y A . 0 . . . 0

(7) 16 5+

6 5+

(8) . (a b) a c = . (b c) b a = . (a b) (c b) b = . (c a) (b a) b =

(9) . x y 3 x y (x y) = + + . x y 2(x y) 3 x y = + . 3 1x y x y x y = + .

1x y 2 x y x y = +

(10) [Ent24] a b 3ab (a b) = + + x (y z) (z y) x =

(11) A A a ba b 2+ = a ba b 2 =

(11.1) A (11.2) A

* a b c a a b c b b c a c c a b


O-NET / A-NET 36

2.2 x n n 1n n 1 1 0a x a x ... a x a+ + + + a () n p(x) p(c) x c 3 2p(x) 4x x 2x 6= + 3 2p( 1) 4( 1) ( 1) 2( 1) 6 3 = + = n n 1n n 1 1 0a x a x ... a x a 0+ + + + = (Remainder Theorem) p(x) x c p(c) 0 () x c p(x) p(x) x c p(c) = 0 (Factor Theorem) p(x) c p(c) = 0 x c ... x c p(x) n c x (k/m) p(x) .. k 0a m na ( k/m ) x c p(x) c k 0a m na ... c k/m () c 2x 2 (x 2)(x 2) = + 3 22x x 6x 1 + x 2 3 22x x 6x 1 + x 2

3 22(2) (2) 6(2) 1 7 + = ... ( ) 3 22x x 6x 1 + x 1+ 3 22x x 6x 1 + x 1+

3 22( 1) ( 1) 6( 1) 1 4 + = ... ( )


O-NET / A-NET 37

p(x) x 3 , x 1 12 , x 2 25 . p(x) x 3 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 p(x) x c ... x c 2p(x) 2x 7x 3 (2x 1)(x 3)= + + = + + p(x) x c c 1/2= c 3= p(c) 0= ()

22c 7c 3 (2c 1)(c 3) 0+ + = + + = c 1/2= c 3=

. p(x) x c 7 c 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= 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= (Synthetic Division) () ... x c () 4 3 2x 3x 4x x 6 + + x 2 1. ( 1, 3, 4, 1, 6 ) c ( 2) 2 1 3 4 1 6


O-NET / A-NET 38

2. ( 1) ... ( 2)

2 1 3 4 1 6

21

/2

3. ( 3 2 1 + = ) ( 2) ...

2 1 3 4 1 6

2 2 4 101 1 2 5 4

+

4. () ... 3 2x x 2x 5 + + 4 32x 7x 6 + x 1+ 22x 2x 5 11 0 ( 2x ) 3 23x 7x 4 + 4 () 1, 2, 4 3 () 1, 3 1, 2, 4, 1/3, 2/3, 4/3 ...

0 ...

3 23x 7x 4 (x 1)(x 2)(3x 2) + = + ( 2 )

2.2

(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

1 2 0 7 62 2 5

2 2 5 11

1 3 7 0 43 4 4

2 3 4 4 06 4

3 2 0


O-NET / A-NET 39

(14) 4 3 2x x 3x x 1 + 3 22x x 75x a+ + + x 5 a

(15) x 2 3 2 ax ax x 2b4 + + 21 x x ba +

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) ... 3x 7x 6 + , 3 23x 7x 4 + 4 3x 3x 6x 4 + (18) ... 3 2x 2x 5x 6 + 3 2x x 10x 8+ +

(19) 6 5 4 3 23x 2x 64x 96x 27x 98x 40 + + +

(20) 2 2 2 2x a b 2abx b 0+ + = (20.1) a (20.2) b (20.3) a (20.4) b

2.3 [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> + > + a b a c b c , c 0> > > a b a c b c , c 0> < < [4] a c b c a b+ > + > a c b c a b , c 0> > > a c b c a b , c 0> < < [5] (Trichotomy Property) a, b R a b= a b< a b> [6] a b a b () [7] a b c a b< < < b c< a b c a b< < < b c< a b c a b<

O-NET / A-NET 40

a b

(Interval) (Number Line) (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 }< ( , ) R

* 23 25 2x a x b< < - a 0> b 0> 2 2(a , b ) - a 0 0 ( 0 ) 2a 2b x ( 4, 3) x 1, 0, 1 ... 0 9, 16 ... 2x [0, 16) : x 2x () 2 a x b< < c y d< < xy ac, ad, bc, bd ... xy x ( 1, 3) y ( 5, 4) xy 5, 4, 15, 12 ... xy ( 15, 12) 1/5, 1/4, 3/5, 3/4 ... x / y ( 3/5, 3/4)

x y+ (a c, b d)+ + ( ) x y (a d, b c) y d y c < < ... x

O-NET / A-NET 41

-3 0 1 1 2 2 2 - + - + - + - +

-3 0 1 2

-3 0 1 1 2 2 2 - + - + - + - +

-3 0 1 1 2 2 2

(Equality) (Inequality) ( > < )

S oa oaa a 1. oao oo aa ooa eo 2. oao oo oaaeoeeo o ao o o a eo/o 3. aee ao ao eaa eaeooi (oae a o a eoo)

1. ( > ) 1 2 3

1 2

(x c )(x c )(x c )... 0(x d )(x d )...

> 2

3(x 3)(x 1) 0x(x 2)

+ >

2. 1 2 3 1 2c , c , c , d , d , ... ()

3. +, , +, + 4. + ( 1 2x d , d , ... )

5. () x [ 3, 0) {1} (2, ) * 2 ( 2) 2 x ( -1 /) (x+1)(3-x) > 0 (x+1)(x-3) < 0

S ui o! S

eooaoaaa a.. ee a aee 0 e (x-2)(x-3) = 0 a x = 2, 3 o .. eo (x-2)(x-3) < 0 ae x < 2, 3 e! o oe eaa a!

O-NET / A-NET 42

-1 3 4 - + - +

-2 3 5 - + - +

2x 2x 19 4x 4+ =

( x 4 0 x 4 ) 2x 2x 19 4(x 4)+ = ... 2x 2x 3 0 = (x 1)(x 3) 0+ = ... { 1, 3}

2x 2x 19 4x 4+

<

x 4 < 4 ...

2x 2x 19 4 0x 4+ <

2x 2x 3 0x 4 <

(x 1)(x 3) 0x 4+

<

( x 4 ) ... ( , 1] [3, 4) (

2B B 4AC

2A )

( )

+ + +

2(x 2)(x 5)(x 2x 2) 0x 3 <

(The Axiom of Completeness) (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

S ui o! S

eaao e oo oa aa.. a oo o e

x2+x-3 < 0 1 1 122

+

ee a

oo ,1 13 1 132 2

+

O-NET / A-NET 43

2.3 (21) (21.1) (a b)(b c)(c d) 0 > a b c d> > > (21.2) a b< n N n na b< (21.3) a 0> , b 0> a b a b ab2

+ >

(21.4) a 0> , b 0> a b 2 2b a 1 1a ba b+ > +

(22) a b c< < (22.1) a ba b2

+< < (22.3) 3 3 3a b c< <

(22.2) a b ca c3+ +< < (22.4) ab bc<

(23) 7 x 5 < < 3 y 6< < (23.1) 2x y (23.2) 2xy

(24) 6 x 2 < < 2 y 3< < (24.1) xy (24.3) x/y (24.2) x y

(25) 20 . 5 .

(26) A B 4 3x 2 13

O-NET / A-NET 44

(32) (32.1) x(x 1)(x 2) 0(x 1)(x 2)

(33) 3 2x x 4x 4 0 + >

(34) A 3 2x 2x 5x 6+ + B

2x 1 1x 5

O-NET / A-NET 45

-b b

[1] a 0> [2] a a= a b b a = [3] ab a b= nna a= [4] aab b= b 0

[5] 2 22a a a= = [6] a b a b+ +< a b a b >

* [7] n n nn n n

a ,a a ,==

=

( ) * [1] x b= 2 2x b= () x b= x b= () * [2] x b x b< x b> x b> x b< x b> 3 x 1 = 3 x 1 = 3 x 1 = 3 x 1 = ... x 2= x 4= ... {2, 2, 4, 4}

3 x 1 < 3 x 1 < ... 1 3 x 1 < < ... 3 4 x 2 < < 2 x 4< < [ 4, 2] [2, 4] 1. 2x 1 x 2 x 3+ = + ... 2 2

S ui o! S

oi a e x 2 x+ = ..

x 2 x+ = o x 2 x+ = eoi aoao a ( ao) a ooeoaa eao ai a ..

e o e x 2 x+ < !

x x 2 x +< < eao ... i oao i ao aa..

O-NET / A-NET 46

-1/2 2

-1/2 2

x < -1/2 -1/2 < x < 2 x > 2

-1/2 2

x < -1/2 -1/2 < x < 2 x > 2

x x 1/2< , 1/2 x 2 ( : ) 2. ... 3 3

( ) ( ) x 3 x 3 x 3 x 3

= + = +

=

-2x - 1 -x + 2

(2x 1) ( ) x 3 3x 1 x 3 x 2

+ = + = +

=

-x + 2

(2x 1) (x 2) x 3 x 3 x 3 0 0

+ = ++ = +

=

3. () () ( : 0 0= 3 0> 1 0= 3 0< ) x 3= x 2>

x { 3} [2, )

2.4 (41) (41.1) n + I n 1> nn a a= (41.2) a, b 0> a b a b =

(42) m (42.1) 4x 0.5 m+ < 3 2x 1 0.5 < < (42.2) x 2 5 mx

+ < x (2,6) (42.3) 2x 25 m < x 5 6+ <

(43) x 1 5 < y 2 4 < x y+

O-NET / A-NET 47

(44) (44.1) 2x 6 x 8 0 + = (44.2) x 1 x 1 2 + + = (44.3) [Ent30] x 4 x 3 1 + =

(45) A 2 3x 2 3 x+ = + B 2 3x 2 3x+ = + B A '

(46) 28(x 2) 14(x 2) 3 0+ + + =

(47) 2A { x | x 3x 3 2x 3 }= + + = +I 5 3xB { x | 2 }x 2= =+I

2 2a b+ a, b A B

(48) 2 3x( x ) x=

(49) (49.1) 2x 1 3x 2 < + (49.4) 3 xx 1 2 <

(49.2) 3 x 2 6< < (49.5) x 2x 1 <

(49.3) 1x 0x+ > 2x x 2 0 <

(50) A x 2 x 42+ + <

B x x 7< (A B) '

(51) 4x 5A { x | x 5 }2+= <

O-NET / A-NET 48

(54.3) x 7 5 5x 25 < < (54.4) x 1 x 3 x 5 + <

(54.5) 22

x 5x 4 1x x 2 + >

* (55) x 3 x 2 <

(56) x (56.1) (1 x )(1 x) + (56.2) (1 x )(1 x) +

2.5 *

[1] m n n m m (Dividend) n (Divisor) m, n n 0 n m m n q= q I [1.1] a b b c a c [1.2] a b a b< [1.3] a b a c a (bx cy)+ (Linear Combination) b c bx cy+ x, y I [2] m, n n 0 m n q r= + q , 0 r n <

O-NET / A-NET 49

[3.2] p mn p m p n [4] (Even Numbers) (Odd Numbers) 2 n n I 2 n 1+ n I [5] (... : the Greatest Common Divisor : GCD) (... : the Least Common Multiple : LCM) d ... a b d a d b n a n b n d ... a b (a, b) c ... a b a c b c a n b n c n ... a b [a, b] [5.1] ... ... (a, b) [a, b] a b = [5.2] ... (a, b) d= (a/d, b/d) 1= [5.3] ... ... a b a b 1 1a b q r= + 1 2 2b r q r= + 1 2 3 3r r q r= + 2 3 4 4r r q r= + ... ( 0) ... ( kr ) ... 138 182 (182) (138)1 (44) (138) (44)3 (6) (44) (6)7 ( ) (6) ( )3= + = + = + =2 2 ... 2 ( 2 ) (m, n) 1= m n (Relative Primes) ( m n ) ... ... ... ...

2.5

(57) 3 2(19) (288) 5

(58) ... 252 34 d 252 x 34 y= + x, y

(59) ... 504 38

(60) ... ... x 128 16 384 x

(61) [Ent37] x, y x y< (x, y) 9= , [x, y] 28215= x 3 x, y

(62) [Ent38] x, y 80 x 200< < x p q= p, q x, y ... 15015 y

O-NET / A-NET 50

()

(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) 4+3 (16) 155/9 (17) (x 1)(x 2) (18) (x 1)(x 2)(x 3)(x 2)(x 4) + + (19) 2(x 2)(x 4)(x 5)(3x 1)(x 1) + + + (20.1) {b, b} (20.2) {0} (20.3) {0, 2b} (20.4) { a 1, a 1} + (21) (21.1) (21.2) (22) (22.4) (23.1) ( 6, 46) (23.2) ( 252, 180) (24.1) ( 18, 4) (24.2) ( 9, 4) (24.3) ( 3, 2/3)

()

(1.1) (1.2) (1.3) a 2= (1.4) a 3= (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}

(25) [7.5, 10) . (26) 2, 4 (27) 2 (28) (29) ( 1 0 1 2) (0) + + + + (30) (31) a ( , 2) (3, ) (32.1) ( , 1) (0, 1) (32.2) [ 4, ) {1} (32.3) 11 (33) [ 2, 1] [2, ) (34) 5 (35.1) ( , 1) (1/3, 1) (35.2) (2, 8] (36) 0 (37) 5 (38.1) 7 (38.2) (38.3) 8 (38.4) (39) 0 (40) 5/2 (41) (42.1) 3.5 (42.2) 17/3 (42.3) 96 (43) [0, 12) (44.1) 2, 2, 4, 4 (44.2) [ 1, 1] (44.3) [3, 4] (45) [ 2/3, 0) (46) 8

(47) 90 (48) 1, 6 (49.1) ( 1/5, ) (49.2) ( 4, 1) (5, 8) (49.3) ( 1, 2) {0} (49.4) 3 21( 1, 3) [ , )2

+ (49.5) ( , 2] ( 1, 1) [2, ) (50) (2, ) (51) (52) 7 (53.1) 2, 3, 5 (53.2) 1, 1, 9 (54.1) ( , 3/7) (1, ) (54.2) ( , 4) (0, ) (54.3) (2, 4) (6, 12) (54.4) ( 1, 3) (54.5) ( , 1] [ 1/3, 3] {1, 2} (55) ( , 1/2) (5/2, ) (56.1) ( , 1) ( 1, 1) (56.2) (1, ) (57) 1 (58) 2 (252)(5) (34)( 37)= + (59) 2 ( 504)( 4) ( 38)(53)= + (60) 48 (61) 495, 513 (62) 105, 165


O-NET / A-NET 51

(5.1) A 2 2 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 (x, y 0) (7) a 1/a ... 16 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 (10) a * b 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= + + + + + + ... (11.1) A 5 7 62+ = 6 A )

a b b a2 2+ += ) ...

(11.2) A 3 3 4.52 = 4.5 A A

ab ba2 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) 2(1) 2a ( 2) a+ = + a 3=

(14) 4 3 2(5) (5) 3(5) (5) 1 + 3 22(5) (5) 75(5) a= + + + ... a 81=

(15) 0 3 2 a(2) a(2) (2) 2b 04 + + = .... (1)

21 (2) (2) b 0a + = .... (2) a 4, b 3 a b 7= = + = (16) 2(x 2x 3) (x 3)(x 1) = +

4 3 24 3 2

(3) a(3) b(3) 3(3) 4 0( 1) a( 1) b( 1) 3( 1) 4 0 + + + + = + + + + =

19 37a , b9 9 = =

2(x x 2) (x 2)(x 1)+ = +

3 23 2

( 2) 10( 2) c( 2) d 0(1) 10(1) c(1) d 0 + + + = + + + =

c 7, d 18= = 155a b c d 9

+ + + = (17) () 3(x 7x 6) (x 1)(x 2)(x 3) + = + 3 2(3x 7x 4) (x 1)(x 2)(3x 2) + = + 4 3(x 3x 6x 4) (x 1)(x 2)(x 2)(x 2) + = + ... 2(x 1)(x 2) x 3x 2= = + (18) 3 2(x 2x 5x 6) (x 1)(x 3)(x 2) + = + 3 2(x 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= + + (19) 2(3x 1)(x 2)(x 4)(x 5)(x 1)+ + + (20.1) 2 2a 0 x b 0= = (x b)(x b) 0 { b, b} + = (20.2) 2b 0 x 0 {0}= = (20.3) 2 2 2a 1 x b 2bx b 0= + + =

2x 2bx 0 x(x 2b) 0 {0, 2b} + = + = (20.4) 2 2b 1 x a 2ax 1 0= + + =

2(x a) 1 0 (x a 1)(x a 1) 0 + = + + + ={ a 1, a 1} +

(21.1) c b a> > c d> ( )( )( ) 0 + > (21.2) 2 1 < 2( 2) < 21 (21.3) ... (a b) / 2 ab+ >

2 2a b 2 ab a 2ab b 4ab + > + + > 2 2 2a 2ab b 0 (a b) 0 + > >

( a b ) Math E-Book Release 2.2 ( )

O-NET / A-NET 52

x

2 2h x+ h

+ - + 3 2 3 2 +

+ - + 1/2 2/3

+ - + 4 5/2

+ - + 3/2 7/3

+ - + 1 3.5 1 3.5 +

(21.4) ... 3 3

2 2b a b a

aba b+ +>

3 3b a ab(b a) + > + (b a) + 2 2(b ab a ) ab(b a) + > + 2 2 2b 2ab a 0 (b a) 0 + > >

( a b ) (22.1) (22.2) ... ( min maxx X x < < ) (22.3) ( 3x ) (22.4) b 0= ab bc= (23.1) 27 x 5 0 x 49 < <

2 27 7(x 2x 1) 0 (x 1) 02 2 + + > + > (x 1 3.5)(x 1 3.5) 0+ + + > 3.5 1.8 a 0, b 2= = . {0} {0, 2} . { 2} {0, 2}

O-NET / A-NET 53

+ - + 2 3

-1 0 1 2 2 - + - + - +

-2 1 1 3 + - + - +

-4 -2 3 - + - +

-4 -1 0 2 5 5 + - + - + - +

-2 1 2 - + - +

-3 -1 2 - + - +

-5 -3 -1 2

-1 1/3 1 - + - +

-1 0 2 2 8 - + - + - +

+ - + -2 5/2

+ - + -5 6

+ - + 5 2

+ - + -1/2 3

+ - + 1/2 2

-1/2 1/2 2 3

(31) 3 2(1) a (1) a 2 5+ > 2a a 6 0 (a 3)(a 2) 0 > + >

a ( , 2) (3, ) (32.1) ( , 1) (0, 1) (32.2) A B (A ' B ') ' A B [ 4, 1) (1, ) = = [ 4, ) {1} (32.3) | 3 | | 2 | | 0 | | 1 | | 5 | 11 + + + + = (33) (x 1)(x 2)(x 2) 0 + > [ 2, 1] [2, ) (34) 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 + + + = (35.1) 1 2 0x 1 3x 1 22

4x 4 x 4x 4 0(x 2) (x 1)+ + + >

2 2

2 2x 8x x 8x0 0(x 2) (x 1) (x 2) (x 1)

+ + +> <

2x(x 8) 0(x 2) (x 1)

+ <

x 1+ x 1 0 x 1+ > > 4 0x 2 > x 2 > (2, 8] (36) 2x 5A ; 0x 2

+ >

2x 1 2x 1 x 5B; 1 0 0x 5 x 5 <

O-NET / A-NET 54

-1 1

. . .

3 4

. . .

-2/3 0

. . .

-2/3

. .

(41.1) nn n n a ,a a ,

==

=

(41.2) a 0, b 1= = |a b| 1 , |a| |b| 1 = = (42.1) 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 + <

(42.2) 1 22 x 6 13 x< < < < 2 1 2 171 5 6x 3 x 3 < < < + <

x 2 2 25 1 5 6x x x + = + = +

17(5, )3 x 2 17| 5 |x 3 + <

(42.3) 6 x 5 6 11 x 1 < + < < < 2 20 x 121 25 x 25 96 < < < <

2| x 25 | 96 < (43) 5 x 1 5 4 x 6 < < < < 4 y 2 4 2 y 6 < < < < 6 x y 12 |x y| [0, 12) < + < + (44.1) 2|x| 6|x| 8 0 + = (|x| 4)(|x| 2) 0 = |x| 2= 4 {2, 2, 4, 4} (44.2) . x 1< x 1 x 1 2 2x 2 x 1 + = = =

. 1 x 1 x 1 x 1 2 2x 2 x 1 {1} + + = = = [ 1, 1] (44.3) . x 3< x 4 x 3 1 2x 6 x 3 + + = = =

. 3 x 4 x 4 x 3 1 2x 8 x 4 {4} + = = = [3, 4]

(45) A ; . x 2/3< 2 3x 2 3x 2 2 = =

. 2/3 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, ) = 2B A ' B A [ , 0)3 = = (46) 28 x 2 14 x 2 3 0+ + + =

(2 x 2 3)(4 x 2 1) 0 + + = 3x 2 2 + =

14

3 3 1 1x { 2 , 2 , 2 , 2 }2 2 4 4 + + 8 (47) ** A; 2

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; |5 3x| |2x 4| = + 2

2 2(5 3x) (2x 4) 0 + = (5 3x 2x 4)(5 3x 2x 4) 0 + + = (1 5x)(9 x) 0 B {9} = = () A B {0, 1, 2, 3, 9} =

2 2a 9, b 3 a b 90= = + =

O-NET / A-NET 55

(48) 2 x 0< x 0> x 0< ... x 0> ( x 0 00 )

221 xx 3 32( x) x x x = =

. 21 x 32 = 2x 6 x 6= =

. x 1= 1 1 {1, 6} ( log ) (49.1) 44, 45 . 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, ) (49.2) 6 x 2 3 < < 3 x 2 6< <

4 x 1 < < 5 x 8< < ( 4, 1) (5, 8) (49.3) 1x 0|x|+ >

. x 0< 21 x 1x 0 0x x > >

(x 1)(x 1) 0x + >

( 1, 0) (1, ) ( 1, 0) . x 0>

21 x 1x 0 0x x++ > >

() (0, ) (0, ) ( 1, 0) (0, ) 2x x 2 0 (x 2)(x 1) 0 < + < ( 1, 2) x ( 1, 0) (0, ) x ( 1, 2) x ( 1, 0) (0, 2)

(49.4) . x 1< 3 3x xx 1 2 x 1 + < <

23 3 x xx 0 0x 1 x 1+ + < <

2x x 3 0 x 1 0 x 1x 1+ + + > > + > ( 1, 1)

. x 1> 3 3x x 0x 1 2 x 3 < <

2 23 x 3x x 3x 30 0x 3 x 3 + 3 21 3 21(x )(x )2 2 0(x 3)

+ > ()

21 4 3 21 3 21[ , 3) [ , )2 2

+

3 21[1, 3) [ , )2+

3 21( 1, 3) [ , )2+

(49.5) . x 0< x x2 2 0x 1 x 1

< < x 2x 2 x 20 0x 1 x 1

+ + + + ( , 2] ( 1, ) ( , 2] ( 1, 0) . x 0>

x x2 2 0x 1 x 1 < < x 2x 2 x 20 0x 1 x 1 +

( , 1) [2, ) [0, 1) [2, ) ( , 2] ( 1, 1) [2, ) (50) A; . x 2< x 2 x 4 0 x 2 2x 8 02

+ + < < x 10 0 x 10 ( , 2) < < . x 2> x 2 x 4 0 x 2 2x 8 02+ + + + < x x 7 0 7< < 7B ( , )2= A B ( , 2] = ... (A B) ' (2, ) = (51) 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 < <

15 5A [ , ]4 4= (51.1) A ( a b A2

+ )

(51.2) 5 15 10( ) A4 4 4+ = (52) 2A ; 14 x 2 14 < <

212 x 16 < < 20 x 16 > 

| x 1 | 9 + > a 1, b 1, c 9= = = (54.1)

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( , ) (1, )7 (54.2) ()

2 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, ) (54.3) |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) (54.4) 4 . x 1< x 1 x 3 x 5 1 x ( 1, 1) + + < + <

. 1 x 3

22

x 5x 4 1x x 2 + <

O-NET / A-NET 57

. 22

x 5x 4 1x x 2 + >

2 22

x 5x 4 x x 2 0x x 2 + + >

( 6x 2) (3x 1)0 0(x 2)(x 1) (x 2)(x 1) ++ + > <

( , 2) [ 1/3, 1) .

22

x 5x 4 1x x 2 + <

2 22

x 5x 4 x x 2 0x x 2 + + + <

2(x 3)(x 1) 0(x 2)(x 1) ++ <

( 2, 1] (1, 3] . . ( , 1] [ 1/3, 3] { 2, 1} (55) . x 0< | x 3| |x 2| < 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 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> ( , 1/2) (5/2, ) (56.1) . x 0<

2(1 x)(1 x) 0 (x 1) 0+ + > + > x 1= () ( , 1) ( 1, 0) . x 0> (1 x)(1 x) 0 (x 1)(x 1) 0 + > + < ( 1, 1) [0, 1) ( , 1) ( 1, 1) (56.2) (1, ) ( x 1= 1 (1 |x|)(1 x) + )

(57) 3 29 8 9 4 6 = 1

3 2(4x 1) (58x 2) x 3 2( 1) ( 2) 4 = x 5 3 2(19) (288) 5 4 ... 4 5 1 (58) ... 252=34(7)+14 .....() 34=14(2)+6 .....() 14=6(2)+2 .....() 6=2(3) (... 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) (59) ... -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) 2 = (-504)(-4) + (-38)(53) (60) x 128 16 384 x 48 = = (61) ... 9 3 3= ... 28,215 3 3 5 11 57= x y 9 x 3 3 5 11 495= = y 3 3 57 513= = (62) ... 1=

x y 1 15,015 3 5 7 11 13 = = x 2 80 x 200< < x 13 7= x 13 11= y 3 5 11 165= = y 3 5 7 105= =

O-NET / A-NET 58

5 14 . 00 00

22 5 14 . 00 00

22 5 14 . 00 00

41

22 5 14 . 00 00

44 1 14

2 2 .2 5 14 . 00 00

442 1 14

8430

2 2 .2 5 14 . 00 00

442 1 14

8444 30 00

2 2 . 62 5 14 . 00 00

442 1 14

84446 30 00

26 763 24

2 2 . 6 72 5 14 . 00 00

442 1 14

84446 30 00

26 764527 3 24 00

3 16 89.... ....

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

eo .. (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 :]


O-NET / A-NET 59

~lgc

3

(Logic) ()

(Truth Value)

(Proposition Statement) , , 1 2,

, ,

p, q, r

2 (True; T) (False; F)

S ui o! S ao eoe a a e a ... e 1. uo eoaeeiiia ioe e a! 2. e ai oa e a ea ea e aaoaioe a (eaoi a aoe a ao 3.4 a..)


O-NET / A-NET 60

3.1

() (Connectives)... (and), (or), - (if-then), (if and only if) (not) ...

~ (Negation)

p q p q (p q ) 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

T T F F - T F -

(Truth Table)

2n n ... 1 2 , 2 4 (), 3 8

2 (Equivalent) ( ) () p (q r) (p q) (p r) p (q r) (p q) (p r) p q ~ p q ~ q ~ p p q (p q) (q p)

~(p q) ~ p ~ q ~(p q) ~ p ~ q ~(p q) p ~ q ~(p q) ~ p q p ~ q


O-NET / A-NET 61

3.1 (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)]

(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) [Ent25] 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


O-NET / A-NET 62

(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)

(10) (10.1) p q . (p q) (q ~ p) . (~ q ~ p) (~ q p) . (p ~ q) (q p) . (p ~ q) (~ p ~ q)

(10.2) [ ]((q ~ t) p) ((q ~ t) ~ p) r~ q . q ~ t p . (t q) p . t q r . (t q) r . (t r) p (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 . a 0> b 0> ab 0< . a 0> b 0> ab 0> . ab 0< a 0> b 0> . ab 0> a 0< b 0<

O-NET / A-NET 63

(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) [Ent34] p q . ~(~ p q) . ~ p q . ~(q ~ p) . q ~ p

(15) p q ~(p q) p (q r)

3.2

() (Tautology)

p q p q T T F F

T F T F

F F F T

. (r p) (p r) . (r ~ p) (p r)

p r (4 ) p r r p p r (r p) (p r) p r r ~ p p r (r ~ p) (p r) T T T T T T T T T T T F T F F T F F F T F T T T T F T T T T F F F T T F F T T T

. .


O-NET / A-NET 64

( p, q, r, s, ...)

, -

, + , + ( , + , + )

3.2

(16) (16.1) (p q) [(p q) r] (16.2) (p q) [(p q) r] (16.3) [Ent29] [(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) [Ent29] [ ](p q) r (p r)(p q)

(17) (17.1) [Ent23] ~(p ~ q) (p q) (17.2) [(~ p q) p] (p q) (17.3) [(p q) ~ p] (~ p q)

. (r p) (p r) . (r ~ p) (p r)

. (r p) (p r)

T FF T T F

- ... p r ... p r ... ( p , r )

. 2 T F F T ... ...

(r ~ p) (p r) p r ~ p r ...


O-NET / A-NET 65

(17.4) (p q) [(q ~ p) (p ~ q)] (17.5) [(p q) (p q)] [(~ p ~ q) (~ p ~ q)] (17.6) [p (q r)] [(p q) (p r)] (17.7) [Ent29] [p (q r)] [(p q) r] (17.8) [Ent29] [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)]

3.3

1 2 3 np , p , p , ..., p q (valid) (invalid)

1.

1 2 3 n(p p p ... p ) q

2. (1) p q p q

(2) p q ~ q ~ p p q ~ q ~ p

(3) p q q r p r

(4) p q r s p r q s


O-NET / A-NET 66

(5) p q p

(6) p p q

3.3

(22) [Ent39]

(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

(24) (24.1) 1. p (q r)

2. ~ s p 3. q

(25) 1.

2.

(26) 1.

2. 3. 4.

(22.2) p (r s) ~ p (r s)

(24.2) 1. ~ p q 2. q ~ r 3.

p

S ui o! S ooeu i o oea a e ao oeiu eua a! ... a oa a ioou (eaoaeu )


O-NET / A-NET 67

. .

. .

3.4

x 2 ( )

x ( ) x 3 x 2 (Open Sentence)

( x) P(x), Q(x), R(x) p, q, r

(Quantifier) x 2 x (For All x; x ) x (For Some x; x ) () P(x) x 2 x [P(x)] x ... x 2 x [P(x)] x ... x 2 { }1,2,3=U x [P(x)] , x [P(x)] { }3,4=U x [P(x)] , x [P(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)]

P(x) x 2 Q(x, y) x+y x y [P(x) Q(x, y)] x y ... x 2 x+y y x [P(x) Q(x, y)] y x ... x 2 x+y


O-NET / A-NET 68

() x y [P(x) Q(x, y)] x y [P(x) ~ Q(x, y)]

3.4 (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) , Q(x) 2P(x) x x= > , Q(x) x= 3 x 1 . x [P(x)] U (0, 1) . x [Q(x)] { }2, 3, 5, 8= U

(30) 3 2x (x 5x 1 4) x ( 0 x 2)x 1 + < < >

(31) N 2[ x (x 1 2) x(x 1 0)] x 0x

+

(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) > =

(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) + >

S ui o! S a o 32.2 a 32.3 i i eoaa a ea a some e a

o 32.2 eii oeoeoa a o 32.3 eoiee x eaa a oeaea...

ao 32.4 a 32.5 i eoa eaa all e o

uia oaaa all a a, some a o

O-NET / A-NET 69

(34) [Ent21] { }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) [ ]P(x)~ 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) [Ent39] 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) . 22x [x 5 0] y [ ]y + = <

22x [x 5 0] y [ ]y + > . [Ent38] x [x 6] x [x 8] < > x [x 6] x [x 8]

3.5

(Reasoning)

2 ( )

(Inductive Reasoning)

O-NET / A-NET 70

... , , 1. A {2, 4,6, 8, 10, ...}= 2 12, 14, 16, ... ()

2. 1 1= , 1 3 4+ = , 1 3 5 9+ + = , 1 3 5 7 16+ + + = , 1 3 5 7 9 25+ + + + = n 2n

3. 1, 3, 7, 15, 31, ... 2, 4, 8, 16 63 ( 32 )

4. 11 11 121 = , 111 111 12321 = , 1111 1111 1234321 = 11111 11111 123454321 =

5. 3 12 , 51 , 96 , 117 , 258 , 543 , 2930 , 5022 , 7839 3 3 3

1. ... ( )

4 (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

3. ... ( )


O-NET / A-NET 71

( ) (Deductive Reasoning)

... 2 10 2 10 1. (1) ... (2) 2. (1) ... (2) * 3. (1) ... (2) * (valid)

...

1. (1) ... (2) ( )

2. (1) ... (2) ( )

3. (1) ... (2) ( )

(-)

( ) ( ) ( )

S ui o! S oo a ai o .. euiaaeo oeu (eo ao 3.3) e eua ae iaaa eoeuuo eie i ae i ...

eeo eueuo eeu ea eoio a iaa!


O-NET / A-NET 72

.

.

A B

A B

C

.

.

( ) 1. (1)

(2) ... 2. (1)

(2) ... ( 2 ) 3. (1)

(2) ... 4. (1)

(2) ... 2 3

* !

3.5 (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, , , , a2 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) [ ..47] 125, 726, a, 40328, 362889


O-NET / A-NET 73

(41) 2 ( )

(41.1) 37 3 11137 6 22237 9 333

= = =

(41.5) 11 11 12111 12 13211 13 143

= = =

(41.2) 9 9 819 99 8919 999 8991

= = =

(41.6) 1089 1 10891089 2 21781089 3 3267

= = =

(41.3) 1 9 11 212 9 111 3123 9 1111 4

= = =

(41.7) 2 (3) 3 (3 1)2 (3) 2 (9) 3 (9 1)2 (3) 2 (9) 2 (27) 3 (27 1)

= + = + + =

(41.4) 9 9 7 889 98 6 8889 987 5 8888

+ = + = + =

(41.8) 3 4 2 (1 2 3)4 5 2 (1 2 3 4)5 6 2 (1 2 3 4 5)

= + + = + + + = + + + +

(42) -

(42.1) (42.2)

(42.3)

(42.4)

(42.5)

(42.6) (42.12)

(42.7) (42.13)

(42.8) (42.14)


O-NET / A-NET 74

(42.9) (42.15)

(42.10) (42.16)

(42.11) (42.17)

()

(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 19) 16.1, 16.2, 17.2, 17.7, 19.1 (20) . .

(21) (22) (23) (24.1) s r (24.2) r (25) (26) . (27) . (28) . (29) . . (30) (31) (32) 32.1, 32.4 (33) 33.1 (34) 34.2, 34.4 (35.1) (35.2) (36) . (37) . (38.1) x [P(x) Q(x)] (38.2) x [P(x) Q(x) ~R(x))] (38.3) [ ]P(x)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) (42.3), (42.5), (42.6), (42.9), (42.13)

O-NET / A-NET 75

()

(1) . T p p ... ( T T T, T F F ) F p F ... p p p ... p ~ p F ... () . ( ) T p T, F p p, p p p, p ~ p 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 p p, F p ~ p, p p T, p ~ p F ( 1 ) (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 (3.9) p r T, q s F ( q F, s F ) p q T p F r T r s 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 ) ... (3.13) ( -) ((p ~ q) ~ p) (p q) F T F T F p q ... p q


O-NET / A-NET 76

... (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 . (s r) (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 .

(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

p q . (14.2) F*F ... ( T,T,T,F ) p q ~(p q) ... . ~ (~ p q) ~(p q)

O-NET / A-NET 77

(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 (16.1) (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 (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 (17.1) ~(p ~ q) ~(~ p ~ q) p q

(17.2) (~ p q) p (~ p p) (q p) T

q p (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

(18.1) [(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 F T T T F q (19.1) (p ~ p) (q ~ q) F F T () () (19.2) [p T] [~ p F] p ~ p F


O-NET / A-NET 78

(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) (22.2) p (r s) ~ p r s (23.1) 1. p q

2. (p q) r3. ~ (s r)4. p

s

(23.2)

1. p ~ q2. r s3. r q4. ~ s

p (24.1)

1. p (q r)q (p r)

3. qp r

2. s p

s r (24.2)

1. ~ p q2. q ~ r

~ p ~ rr p

3.

p 3. r (25) 1. p ~ q2. q

~ p (26)

1. p q2. ~ p r3. s ~ r4. ~ q

? ~ s . ~ s p . ( ) . s p . ~ r ~ s ~ r ( r ) . p ~ q (27) . x 2x >0 x 0= . x 2= 8 4, 2 4> < . x 1= . x

1. p q

4. p q2.(p q) r r3. ~ s ~ r

r ~ s ~ s

2. r s

4. ~ sr

3. r qq

1. p ~ q~ p

1. p q

4. ~ q~ p

2. ~ p rr

3. s ~ r

~ s

1. p q2. q s

p s3. ~ s

~ p

O-NET / A-NET 79

(28) . x x 2 ... ( x 3= ) . x ... x 0.5 ... ( x 2= F F T ) . x ... x ... ( T T , ) . x ... x 227

... 227 x 1= x x (29) . 2x ... x x> (0, 1)...=U . x ... x ... 3 x 1 ... 2, 3, 5 , 8 ... 3 1 . . (30) 3x (x 5x 1 4) + < x 1= 7 4 <

2x(|x 1| 0 x 2) < > F T , T T T (31) 2x(x 1 ) x 2= 22 1 3 = x(x 1 0) + > ( + 1

0 ) 2x( 0)x < x 1=

21 x 0=

2x(x 1) T T T (32.3) x(x 1 0 + > 2x 1) x 0= T (32.4) 2x(x 0) > x 0= x(x 0) = x 1= F F F

(32.5) 2x(x 0 > x 0)= ( x 1, 0, 1= ) T (33.1) x y

2x y 2+ > x 1, y 1= = (33.2) x y x 1 y 0= = x 0 y 0= = x 1 y 0= = (33.3) x y x 0= y (33.4) x y x 0, y 0= = (34.1) x 1, y 1= = 2 0 (34.2) x y x 0, y 0 x 0, y 1 x 1, y 1= = = = = = (34.3) x y x 1= y 1= x 0= y 1= x 1= y 1= (34.4) x y (34.5) x y x 0, y 0= = x 1, y 1= = x 1, y 1= = (35.1) x y ( y x= ) (35.2) x y x 2, y 2= = , x 2, y 2= = (36) . x y(xy 1) = x 2= y I x y(xy y) =

x 1= xy y= y F T F . x y(xy 1) = x y(xy y) = ( .) F T F . x y(xy 1) = x 0= y R x y(xy y) = ()

F T F . x y(xy 1) = x + R y + R x y(xy y) = ()

T T T ... .

O-NET / A-NET 80

.

(37) . x 10, y 5= = 15 50> . x, y 0< . x y ( x y x> ) . x y y x> . (38.1) x [P (x) Q(x)] (38.2) x [P (x) Q(x) ~ R(x))] (38.3) [ ]P (x)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) . . x[x 6] x[x 8] < < (40.1) a 9= ( / 2 ) (40.2) a 22= ( 2 / 5 ) (40.3) a 5= ( ) (40.4) a 48= ( / 2) (40.5) a 5= ( 2)

a 3= () 3 13 1

(40.6) 5a 6= () (40.7) a 25= () (40.8) a 3 3 3 3 3= ( 3 5 ) (40.9) 7 5, 6, _, 8, 9

9612, 72 , _, 4032, 36288

72 7 504 = 504 8 4032 = 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) = + + + + + + (42.1,42.2) 2 (42.3) (42.4) (42.5) (42.6) (42.7) 2

O-NET / A-NET 81

(42.8) (42.9) (42.10) (, ) ! (42.11) (42.12)

(42.13) (42.14) 2 (42.15) (42.16) ( 42.10 , ) (42.17)


O-NET / A-NET 82

0 1

0

1

inv

0 and

0

1

1 or

0

1

1 nand

0

1

0 nor

0

1

1 xor

eo .. 0 1 ... 0 0 , False 1 5 ( 12 ), True (Gate) (1) INVERTER ( ) 0 1 1 0 (2) AND ( ) 1 1 (3) OR ( ) 0 0 (4) NAND NOR ( ) AND OR AND OR (5) XOR ( -) 1 0 1 (0 1 0) 0 False 1 True AND OR ( 1+1 1, 2 ..) , :]

A B A and B (AB) 1 1 0 0

1 0 1 0

1 0 0 0

A B A or B (A+B) 1 1 0 0

1 0 1 0

1 1 1 0

A not A (A ) 1 0

0 1

A nand B = AB = A B+ A nor B = A B+ = A B () A xor B A B


O-NET / A-NET 83

O

y

x

Q1 (+, +)

Q2 (, +)

Q3 (, )

Q4 (+, )

G (e,o)

4 eiiea

(Analytic Geometry) , , ,

(Plane)

(Coordinate) (Cartesian Coordinate) 2 (Origin; O) x y

xy 4 (Quadrant; Q) 1, 2, 3, 4

(Ordered Pair) +x +y (2, 4)


O-NET / A-NET 84

P (x1,y1)

Q (x2,y2)

2 22 1 2 1PQ (x x ) (y y )= +

P (x1,y1)

Q (x2,y2)

1 2 1 2x x y yR ( , )2 2+ +

P (x1,y1)

Q (x2,y2)

1 2 1 2mx nx my nyR ( , )m n m n+ ++ +

m

n

P (x1,y1) Q (x2,y2)

C

R (x3,y3)

1 2 3 1 2 3x x x y y yC ( , )3 3+ + + +

P (x1,y1)

Q (x2,y2) R (x3,y3)

S (x4,y4)

T (x5,y5)

( 10) ( 11)

4.1 :

P, Q P(x, y) Q (2, 4) Q (2, 4) [1] P Q PQ

(Pythagorean Theorem) [2] m:n [3]

( Centroid) 2 : 1

[4] () ( det 9)

1 12 23 34 45 51 1

x yx yx y1x y2x yx y

=

1 2 2 3 3 4 4 5 5 1 2 1 3 2 4 3 5 4 1 51 (x y x y x y x y x y x y x y x y x y x y )2= + + + +

S o! S 1. ae e 3 e e o 3 a 3 a

2. eu eo o ai ... eei o ae io o


O-NET / A-NET 85

O

y

x B (2,0)

D (2,4) C

P Q

4.1 (1) 1P (1, 7) 2P ( 5, 2) 1 2PP

(2) P, Q AB , CD A (2, 7) , B (6, 3) , C ( 2,5) , D (8, 1) PQ

(3) OBCD , P BC , PC PQ= PQC

(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 1P (1, 2) 2P (3,5)

(9) (1, 7) , (8,6) , (7, 1)

(10) 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 (4/3, 1) A

(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


O-NET / A-NET 86

P (x1,y1)

Q (x2,y2) 2 12 1

y ym tan x x= =

O

y

x a

b

P (x1,y1)

m

P (x1,y1)

Q (x2,y2)

. PQR 15

(16) A (1, 4) , B ( 3, 2) , C ( 1, 3) , D ( 4,5) , E ( 2, 7)

4.2 :

P Q P Q PQ L 1L , 2L [1] (Slope; m)

(Parallel; & ) (Perpendicular; ) -1

m > 0 () m < 0 () m = 0 x y m

[2] [2.1] 1 1(x , y ) m

11

y mx =

yx

1 1y m ( x ) = y x [2.2] 1 1(x , y ), 2 2(x , y )

(2.1) 2 11 1

2 1

y yy ( x )x x =

y x

[2.3] (Intercept)

Intercept Form 1a b+ =

x y

a, b x y x (a,0) y (0,b) a, b

O-NET / A-NET 87

1. (Common Form) A B C 0+ + =x y 2. 3 Slope-Intercept Form m c= +y x c y Slope-Point Form 1 1y m ( x ) = y x Intercept-Intercept Form 1a b+ =

x y

3. = A CB By x

Am B= Cc B=

P(1, 3) Q(5, 9)

. P Q PQ 9 3m 3/25 1

= = . 1L PQ PQ

1L PQ L1 2m 3= ( 1 )

PQ 1 5 3 9( , )2 2+ + ... (3,6)

Documents

สรุปเนื้อหาวิชา คณิตศาสตร์ ทุกบท ม.4-ม.6.pdf