math e-book release2.2.04

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Math E-BookRelease 2.2.04 (..2547 ) http://math.reads.it thaiware.com Release 2.2 14 2549 Release 2.2.01-03 - 2549 Release 2.2.04 26 2550

( Release 2.0) 2548 3 ( Release 2.0) 2550 O-NET & A-NET SCIENCE CENTER () 159

()

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3/1 . 10200 . 0-2225-7160, 0-2221-5884 ..

7-Eleven

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

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

Math E-Book Release 2.2.04

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(1) () :] () ( !) (2) ( ) ( ) .. 2 .. - . . () - - . - . () . . Science Center - - ( ) - 3 .. ! - Thaiware.com, Se-ed.net, f0nt.com ...

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(eoo O-NET / A-NET) 2549 (Central University Admissions System) 4 [10%] 1. GPAX . 2. GPA 4-5 [20%] 3. O-NET (Ordinary National Educational Test) [35%-40%] 5 ( 2 ) ... O-NET .6 4. A-NET (Advanced National Educational Test) [30%-35%] ( 3 ) O-NET ( 2 3 ) ... A-NET 3

ae

(1) O-NET A-NET 1 (2) (Short Answer) (3) 1 (O-NET) 2 (A-NET)

| GPA 4% | O-NET 7% | A-NET 20% | GPA 4% | O-NET 8% | A-NET 10% | GPA 5% | O-NET 7% | A-NET 10% | GPA 4% | O-NET 7% | A-NET 10% | GPA 5% () | O-NET 20% | GPA 5% | O-NET 14% | GPA 5% | O-NET 8% | GPA 4% | O-NET 8% | GPA 4% | O-NET 7% | GPA | O-NET 7% | GPA 5% () | O-NET 7-10% | A-NET

(NIETS) http://www.ntthailand.com


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( 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


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1 1.1 1.2 - 1.3

a 1112 15 21

2 2.1 2.2 2.3 2.4 2.5

3132 36 39 44 48

58

3 3.1 3.2 3.3 3.4 3.5

5960 63 65 67 69

82

4 4.1 : 4.2 : 4.3 : 4.4 : 4.5 : 4.6 : 4.7 :

8384 86 92 94 96 99 102

5 5.1 5.2 5.3

119120 121 124


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5.4 5.5

127 131

fog

146

6 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

147 157158 160 162 165 166 169 171 172 173

8 8.1 8.2 8.3 8.4

187187 191 192 195

()

204

9 9.1 9.2 9.3 9.4 9.5

205206 208 211 215 216

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7

227228 230 231 233 235 237 240

:

250


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11 11.1 11.2 11.3 11.4

251252 254 256 259

268

12 12.1 12.2 12.3

269270 272 274

13 13.1 13.2 13.3 13.4

279280 282 284 285

14 14.1 14.2 14.3

295296 298 300

306

15 15.1 15.2 15.3 15.4 15.5

307307 309 312 317 319

332

16 16.1 16.2 16.3 16.4 16.5 16.6

333333 335 337 339 341 345

358


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17 17.1 17.2 17.3 17.4 17.5 17.6

359360 363 374 378 383 388

1 (14 ) | | | | | | | | | | | | | | 2541 2542 2542 2543 2543 2544 2544 2545 2545 2546 2546 2547 2547 2548

403408 417 426 435 444 453 462 471 481 492 502 512 523 532

1 (17 )() 1 | 2532 2541 2 | 2541 2548

541

542 573 588 606

: 1 ( 2 , 70 ) 2 (35 )

: Math E-Book Math E-Book Release 2.2.04

616 657

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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

12S uio! S eoie {, 0, 1, {2, 3},(4, 5)}

(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}

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 oa e 1-2 ae 2-1 oeoaa oaaa) eooaea eeaooa ... oo eaa ia oaa..

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

1.1 (Subset) B A B A ( B A B A) B A B A B A B A A B B {m}{m, p} {m, p, r} {m, p, r, w} {p} {m, r} = {m, p, r, w}

A

16 {w} {p, r} {p, w} {r, w}

{r} {m, w}

S oae! S ao {a, b} A a A a b A

{m, p, w} {m, r, w}

{p, r, w}


( )

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13

1. A 2. A A 3. n 2 n ... ( 2 4 = 16 ) 4. (Proper Subset) 2 n 1 ( ) ( A A A A ) ... (Power Set) A P(A) S oae! S A n P(A) 2 n A = {m, p, r, w} ao {a, b} P(A) {a, b} A P (A) = { , {m}, {p}, {r}, {w}, {m, p}, {m, r}, ..., {m, p, r, w} } ( 16 16.3) n r

ao a A a b A

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

x ! = 1 2 3 ... x 7 3 7 3 = 7 ! = 1 2 3 4 5 6 7 = 35 4! 3! 12 3 4 12 3

. A = {a} 2 = 2 {a} P (A) = {, {a}} . B = {a, b} 2 = 4 , {a} , {b} {a, b} P (B) = {, {a}, {b}, {a, b}} . C = {2, 3, 5} 2 = 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 = 2 = 1 P (D) = {}1 2 3 0

o aaaa a {} ao ... (e) eeooe oaoae (iea 0) aeaae { }

S uio! S

ooeo o aeoeo ooea eoaa eoe {} eeoo ... ooa o n() = 0 n({}) = 1


( )

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14

E = {, {0}, {}} P(E) {, {}, {{0}}, {{}}, {, {0}}, {, {}}, {{0}, {}}, {, {0}, {}}} A, B A = {1, 3, 5, 7} . X X P (A) X P (A) X A X 2 = 16 4

B = {1, 2, 3, 4, 5, 6, 7}

4 4 4 4 4 0 + 1 + 2 + 3 + 4 = 1 + 4 + 6 + 4 + 1 = 16

. X X P (A) n (X) < 2 X P (A) X A 16 ( .) n (X) < 2 4 4 4 0 + 1 + 2 = 1 + 4 + 6 = 11

( ) . Y A Y Y B A Y 1, 3, 5, 7 Y ... Y B 2, 4, 6 Y ( 1, 3, 5, 7 Y B ) ... 2, 4, 6 Y {2, 4, 6} 2 = 8 3

1.1(1) A, B (1.1) {x} B (1.2) {y} A (2) A = {{}, a, b, {a}, {a, b}}

A B

A B

x A

y B

(1.3) (1.4)

{A} {B} {A} {B}

(2.1) {} A (2.2) {} A

(2.3) {{a}, b} 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 (4) A (4.1) { x | x = A } = {A} (4.2) { x | x A } = A

15(4.3) (4.4){ x | {x} A } = {A} { 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) (7.2) (7.3) {} (7.4) {} (8) A = {, a, {b}, {a, b}}

(7.5) (7.6) (7.7) (7.8)

P () P (){} P () {} P ()

(8.1) (8.2) (8.3) (8.4) (8.5) (9)

P (A)

(8.6) a P (A) {} P (A) (8.7) {a} P (A) P (A) (8.8) {b} P (A) {} P (A) (8.9) {{b}} P (A) {, a, {b}} P (A) (8.10) {, a, {b}}

P (A)

A = {, 1, 2, 3, {1}, {1, 2}, {1, 2, 3}}

(9.1) {, {1}, {1, 2}} P (A) (9.2) {, {1}, {1, 2}} P (A)

(9.3) {{1}, {2}, {3}} P (A) (9.4) {{1}, {2}, {3}} P (A)n (Y)

(10) [Ent39] S = {1, 2, 3, 4, 5, 6, 7} n (X) X = { A P (S) | 1 A 7 A } Y = { A X | A 6 }

1.2 - (Venn-Euler Diagram) U ( ) A, B, C S uio! S ae A a B a oiao (eoa) aeo ii ooe i.. ooiaoa..


( )

O-NET / A-NETU U

16U

A A B AUA

B

BA B

A B

A B 04 1 9 B 2 3 57 11 C

A = {0, 1, 2, 3, 4}

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

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

A A B A B B A B B

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

A A B A BU

B A B A

A B

3. (Complement : ' ) A' A A c A 4. (Difference Relative Complement : ) B A B A ... B A = B A' B A B A U U U

A

A A B A B B

n (B A) n (B) n (A)

n (B A) = n (B) n (A B)


( )

O-NET / A-NET 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)

17 (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} 2 1 = 15 ... Y B 2, 4, 6 Y ( 1, 3, 5, 7 Y B ) ... 2 = 8 ... 15 8 = 120 4 3

. Z {1, 2, 3} Z Z A . ... {1, 2, 3} Z 1, 3 Z ( A Z = ) 2 Z ( Z A ) ... 2 1 = 3 ... Z A 5, 7 Z ( 1, 3 Z A ) ... 2 = 4 ... 3 4 = 12 2 2

. Z {1, 2, 3} Z = Z A {1, 2, 3} Z = 1, 2, 3 Z Z A 5, 7 Z ( Z = Z A ) ... 2 = 4 2


( )

O-NET / A-NET

18

C = {, {}, 0, {{}, 0}, {, {0}}, {{, {0}}}} . n (P (C)) n (C) = 6 n (P (C)) = 2 = 646

.

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 (P (C) 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))] . . () ( )

2 C

4

60 P(C)

60 + 2 = 62

1.2(11) .A B = {0, 1, 2, 3, 4, 5} A C = {0, 3, 5} A B = {1, 3, 5} B C = {2, 3, 5} A C = {0, 1, 2, 3, 5} A B ' = {0}C' B'

. B C ' = {1} . A C ' = {1}

.

B A ' = {2, 4}

(12)

U = { x I | 1 < x < 10 }

I = ( )


O-NET / A-NETB = {x | x

19C = {x | x < 5}

3 }

(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 (14.2) U ' = (14.3) A (A B) (14.4) B (A B) (14.5) (A B) A (14.6) (A B) B (14.7) A A ' = (14.8) A A ' = U (14.9) A U = U A = A ' (14.10) A = A A = (14.11) A A = (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) (16.6) (16.2) A B B A (16.7) (16.3) A B = A B ' (16.8) (16.4) (A B) ' = B ' A (17) (17.1) A (A B) (17.2) (A B) B (17.3) (A B) B (17.4) A (A B) (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 x A x A x A x A

x A B x A 'B' x A 'B' x (A ' B ') '

(17.6) (A B) B (17.7) (A B) B (17.8) A (A B) (17.9) (A B) (B A ')

3


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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) = (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)

C

(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 (U) = 35 , 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) A, B, C

n (A B C D)

(25) C A,

P (C) = {, {a}, {c}, C} , n (P (A)) = 8 , n (P (B)) = 16 ,

C B , {b, d, e} A B

b A B '

e (C B ') ' {b, e} (A ' B) '

. .

d (A B ') ' b (A ' B ') '

. .=

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

(26.3) 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 (29) A = {1, 2, 3, 4, 5, 6, 7} X (29.1) B X A (29.2) X A B X (30) B A

21B = {1, 2, 3}

n (A) = 10 , n (B) = 4

n (C)

(30.1) (30.2)

C = {S |B S A} C = {S A | S B }D = {0, 2, 3}

(31) A = {0, 2, 4, 6, 8} B = {0, 1, 2} C = {1, 2, 3} X X A (31.1) B C ' X (31.3) B D (31.4) B D (31.2) B C ' X (32) DU = {1, 2, 3, 4, ..., 8}A = U {1}

X X

C = {1, 7} (B ' C) DU = { x I | 2 < x < 6 }

B = {2, 4, 6}

A

(33) 2

I =

A = {k | k U }

B = {

k |k U}

C = {x | A B x

x A B}

(34) A = {a, b, c, d, f} B = {a, c, d, e} X X A B A B X (35) A = {1, 3, 5, 7, 9} Sk = { B A | n (B) = n (S) S = S1 S2 S3 S4 S5 (36) A, B U n (U) = n (A B ') = 32 n (A ' B ') k}

100 , n (A ') = 40 , n (B) = 55 ,

1.3 -

2 3

n (A B) = n (A) + n (B) n (A B) n (A B C) = n (A) + n (B) + n (C) n (A B) n (A C) n (B C) + n (A B C)

eaa..

= = +

+ +

-

- -

-

+


( )

O-NET / A-NET

22

30 12 15 5 U ( ) U

Math

Eng

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 ( ) Social 1 5 5 Math 15-5=10, 13-5=8, 7-5=2 37-10-5-2=20, 48-10-5-8=25, 45-2-5-8=30 Thai 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 + 451513 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

(47) [Ent38] 180 95 92 125 52 43 57 (48) [Ent39] 100 U , , 1 6 . . 199 . 10 . 2420 22 23 11 9

(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

() (1.1) x A x B A (1.2) x y B , 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 A B C (4.1) (4.2) () (4.3) {x} A x A (4.2) (4.4) {x} x x (5.1) 25 = 32 ... (5.2) 31 (25 1) (5.3) P(A) 1 P(A) 32 ... (5.4) (6) 2n = 512 n = 9 5 9 9! = 126 ()5! 4!

(7.1) (7.3) (7.2) 20 = 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(9.1){, {1}, {1, 2}} A{, {1}, {1, 2}} P(A)

26 (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}} n(P(A B)) = 24 = 16

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)

A B = {0}

n(P(A B)) = 21 = 2

16 2 = 14 (14.1) (14.2) U (complement) (14.3) (14.6) (14.7) (14.11) A A = {1} P(A), {2} P(A), {3} P(A) {1} A, {2} A, {3} A (14.8) (14.10) ... (14.12) ( !) 1 A, 2 A, 3 A (10) n(X) (15.1) () (15.2) A B = A () A P(S) ( A S ) () 1 A 7 A ( A 1 (15.3) A B A B = A B 7) 2, 3, 4, 5, 6 (15.4) A B = A B A ... {2, 3, 4, 5, 6} .. B C = B B C (B C = ) n(X) = 25 = 32 A n(Y) A A X 6 C B A ' C ' = (A C) ' = ' = U {1} {1, 2} {1, 3} {1, 4} {1, 5} {1, 2, 3} ( A C ) A 6 .. n(Y) = 6 (15.5) A B = A B (11) A B, A C, B C B C B C A B C = {3, 5} A C A C B A C = {0, 1, 2, 3, 5} 1 4 A C " A B B C 35 A A A C " 0 2 C 4 B A C B . A B = {0} C (16.1) A = B . B C = {1} .. {1, 4} A B = A B = A = B (16.2) A = B . A C = {1} A B = BA = . B A = {2, 4} (16.3) (12) U = {1, 2, 3, ..., 10} B = {3, 6, 9} A B ' = A (B ') ' = A B C = {1, 2, 3, 4, 5} C ' B ' (16.4) B ' A = B ' A ' = (B A) ' (C B) ' C B = {3} (16.5) x B {1, 2, 4, 5, 6, 7, ..., 10}


( )

O-NET / A-NET(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) ( )U

27

(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

A

B

BA =

(17.1)AB

AB AB A (A B)

(21.2)

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

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

(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'

(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(25) P(C)

28C = {a, c}

U = {2, 1, 0, 1, 2, ..., 6}

(33)

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) n[P(A B)] = n[P(A)] = 23 = 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} (26 3) + (6 3) = 61 + 3 = 64 (28.1) A = {2, 4, {1,3,5} } 23 = 8 (28.2) A = {1, 3, 5, 7, 9, 11, 12, 13, 14, 15, {2,4,6,8,10} } 25 = 32 (29.1) X = {1, 2, 3, {4,5,6,7} } 24 = 16 (29.2) X = { {1,2,3} , {4,5,6,7} } ... (23 1)(24) = 112 (30.1) n(C) = S = 26 = 64 (30.2) n(C) = S= (24 1 6) = 960 )(2

n(P(A)) = 8

A = {0, 1, 4} B = {0, 1, 2}

U

{0, 1} X {0, 1, 2, 4} n(C) = 22 = 4

(34)X = {

x {a, b, c, d, e, f}

{a,c,d} , {b,e,f} } (23 1)(23) = 56 (35) n(A) = 5 S1 = {B | B A, n(B) = 1} , S2 = {B | B A, n(B) = 2} , ... S5 = {B | B A, n(B) = 5} S = S1 S2 S3 S4 S5 = A (n(B)=0) n(S) = 25 1 = 31 (36) Un(A B ') = n(A B) = 32 =

{a, c, d} X

(31.1) {0} X {0, 2, 4, 6, 8} 24 = 16 (31.2) {0} X X {0, 2, 4, 6, 8} 31.1 25 24 = 16 (31.3) {0, 2} X {0, 2, 4, 6, 8} 23 = 8 (31.4) {0, 2} X X {0, 2, 4, 6, 8} 31.3 25 23 = 24 (32) {3, 5, 8} D {2, 3, 4, ..., 8} 24 = 16

32 55 n(B) = + = 55 A B n(A ' B ') n(A B) ' = n(U) = 100 = +++ = 100 32 55 = 13 ( .. n(A ') = 40 ) (37) 35 U + = 35 27 + = 27 32 = 32 +2++ = 94 80 (+++) = 14 ++ = 80 14 = 66 ( .. n(A ' B ') = n(A B) ' = ++) (38) U 20=+, 17=+, 15=+ 2(++)=20+17+15 ++=26 = 26 20 = 6


( )

O-NET / A-NET(39) +++=20 +=2(+)-7 = U

29

(43) +=16 ....(1) +=7 +=21-7=14 ....(2) 24 +2+=16+14=30 ++=24 11 +2+=20 .....(1) =30-24= 6 -+2=7 .....(2) (44) +=200 U (1)x2 - (2); 3+3=33 +=11 +=30 600 (40) +2+=230 =15 500 . ++=215 = 600 215 = 385 U (, ( 43 44 ) , 42) +++=34 .....(1) =5 .....(2) (45) =++1 .....(3) ++=3 .....(4) (2), (3) (1) (4) 4 ! +5+(+5+1)+=34 +5+=3 ( , = 0 ) =7, =9 3 4 = ++1 = 7+5+1 = 13 = 6 , ++ = 8 + = 3 = 5 .. (41) = 2 , + = 4 + = 18 4 = 14 .. = (+)+++(+) +=7, +=6, +=5 = 4 + 2 + 6 + 14 = 26 U = (+)++ = 4 + 2 + 6 = 12 2(++)=18 .. ++ = 9 3 4 = (42) U = 26 2 = 24 100 (46) 3 , , ( n(A B C) ' = 100 n(A B C) 60 ) U 3 32 (36) 35 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 ( .. , )

n(A B C) =

32 + 35 + 20 = 87

20

B C

13 A (47) ? z n(A B C) =180 y x =95+92+125-52-43-57+x x = 20 y = n(A C) 20 =43-20=23 z = n(A B) 20 =52-20=32 95 20 23 32 = 20


( )

O-NET / A-NET(48) x 20 y x+y+20+23+22+11+9=100 22 23 11 (x+20+23+22)=(y+20+23+11)+6 9 x = 5, y = 10 70 , 64 , 65 . . 70 + 64 + 65 = 199 . 5 . 5 + 10 + 9 = 24 (49) (, , )

30(50) U = {0, 1, 2, ..., 100}

2 } B = { x | x 3 } C = { x | x 5 } n(A ' B ' C ') n(A B C) ' .. n(U) n(A B C) n(A B C) A = {x|x

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 y x n(A B) 2 3 6 17 ... n(A C) 2 5 4 8 10 11 ... n(B C) 3 5 15 7 ... n(A B C) 2 3 5 n(A B C) = 60 8 30 4 =20+15+22-6-10-11+x ... x=22 n(A B C) = 51 + 34 + 21 17 11 ( ) 7 + 4 = 75 40-4 = 15+13+18-8-6-9+y ... y=13 n(U) = 101 22 13 = 9 n(A ' B ' C ') = 101 75 = 26S uio! S

o (50) oee A oeo 6 a a B oeo 8 a A B aeeoa.. o a 6 a 8 a 48 a ... aa! ... eo 6 a 8 aai! ao ... o 24 a a


( )

O-NET / A-NET

31

a+l Re

2 ai

(Natural Number) (Counting Number) 1,2,3,4,... N = {1,2, 3, 4, ...} ( ) (Fraction) (Integer)I = {..., 3, 2, 1, 0, 1, 2, 3, ...}

(Rational Number) Q = { a/b | a, b I b 0 } 1.


( )

O-NET / A-NET

32

S eiei! S 1. ooa (oooe a) aeoaeo 2. e eeaoi ( 8) eoaea (a 2.718..)

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

2.

(Real Number : R ) 2 (Imaginary Number) (Complex Number : C ) 11Q'

CR QIII0

Im

QII+ N

2.1 ( .) [1] (Reflexive Property) [2] (Symmetric Property) [3] (Transitive Property) [4] a = aa = b b = a

a = b

b = c a = c

a = b a+c = b+c a = b ac = bc

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

[4] (Commutative Property)

a+b = b+a ab = ba


( )

O-NET / A-NET

33a (b c) = (a b) c = a b c

a + (b + c) = (a + b) + c = a + b + c a (b + c) = a b + a c (a + b) c = a c + b c

[5] (Associative Property) [6] (Distributive Property)+

[7] ( R ) a 0 a R+ a R+ () [1] [2] * [3] [4] [5] [6] [7] [8] [9] [10]

a+c = b+c a = b ac = bc a = b 0a = a0 = 0 (a) b = a (b) = a b (a)(b) = a b ab = 0 a =0 (a) = a

c0

(1)a = a

b0

b=0

a b = a + (b)

a b = a b1

(

01 )

a (b c) = a b a c

a1 0 a ac = b bc a d ac + bd + = b c bc

a = b a b ab a = c bc

1

a d ad = b c bc

a ac = bc b

ab ad = cd 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 I } = {0, 10, 20, 30, ...} A B () C D ()

2.1(1) (1.1) 0.343443444... (1.2) 0.112112112... (1.3) a 2 a (1.4) a 2 a (2) a, b, c R

S uio! S oooia ooeei aeo ioi.. ooaa ia 1 o ai (ao oo i i ai 1 ) ... a o aooa (aoa oii oaa..)

(2.1) a b = a b = 1 (2.2) a b = 0 a = 0 b = 0 (2.3) b 0 a = c a = cb

(2.4)

b, c 0

b a a = b c

b = c


( )

O-NET / A-NET

35

(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 . 0 . (7) 6+ 5 1 6+ 5xy A

. . 0 * a b c a a b c b b c a c c a b

(8) . (a b) a = c . (b c) b = a . (c a) (b a) = b . (a b) (c b) = b (9) . x y = 3 x y + (x + y) . x y = 2 (x + y) 3 x y 3 1 . x y = . x y = 2 x y + 1 xy x+y xy

(10) [Ent24] 2

a b = 3ab + (a + b)

x (y z) = (z y) x

A

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

(11.1) A (11.2) A


( )

O-NET / A-NET

36

2.2 x anxn + an 1xn 1 + ... + a1x + a0 a () n p (x) p (c) x c p (x) = 4x3 x2 2x + 6 p (1) = 4 (1)3 (1)2 2 (1) + 6 = 3 anxn + an 1xn 1 + ... + a1x + a0 = 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 a0 m an ( k/m ) x c p(x) c k a0 m an ... c k/m () c x2 2 = (x 2)(x + 2)

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

3

2

2x x 6x + 1 x + 1 2x x 6x + 1 x + 1 2(1) (1) 6(1 + 1 = 4 ... ( ) )3 2 3 2

3

2


( )

O-NET / A-NET

37

p (x) x 3 , x 1 12 , x 2 25 . p (x) x 3 p (x) Ax + Bx + C 3 3 3 x 3 p (0) = 3 A(0) + B(0) + C = 3 x 1 12 p (1) = 12 A(1) + B(1) + C = 12 x 2 25 p (2) = 25 A(2) + B(2) + C = 25 A = 2 , B = 7 , C = 3 ... p (x) = 2x + 7x + 3 p (x) x 3 2(3) + 7(3) + 3 = 422 2 2 2 2 2

. p (x) x c c p (x) x c ... x c p (x) = 2x + 7x + 3 = (2x + 1)(x + 3) p (x) x c c = 1/2 c = 3 p (c) = 0 () 2c + 7c + 3 = (2c + 1 + 3) = 0 c = 1/2 c = 3 )(c2 2

. p (x) x c 7 c p (x) x c 7 ... p (c) = 7 2c + 7c + 3 = 7 2c + 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 ) p (x) 7 = 2x + 7x 4 = (2x 1)(x + 4) c = 1/2 c = 4 2 2 2

(Synthetic Division) () ... x c () x4 3x3 + 4x2 + x 6 x 2 1. ( 1, 3, 4, 1, 6 ) c ( 2) 2 13

4

1

6


( )

O-NET / A-NET

38

2. ( 1) ... ( 2) 2 1 13 2

4

1

6

3. ( 3 + 2 = 1 ) ( 2) ... 2 13 4 1 2 2 4 1 6 + 10

1

2

5

4

4. () ... x3 x2 + 2x + 5 4

2x 7x + 6 x + 1 1 2 0 7 2 2 2 2 5 2x 2x 5 11 0 ( x ) 2

3

6 5 11

2

3x 7x + 4 4 () 1, 2, 4 3 () 1, 3 1, 2, 4, 1/3, 2/3, 4/3 ... 1 3 0 ... 2 3 3x3 7x2 + 4 = (x 1 2)(3x + 2) )(x

3

2

7 0 4 3 4 4 4 4 6 4

0

3

2

0

(

2

)

2.2(12) 4x3 21x2 + 26x 17 x 4 a 3x3 + 13x2 + 11x + 5 x + 3 b b a (13) x1

x2 + 2a

x +2

x+ a

a


( )

O-NET / A-NET (14) (15) a+b

392x3 + x2 + 75x + a

x 5

x4 x3 + 3x2 x 1

a

x 2

x3 ax2 +

a x + 2b 4

1 2 x + x b a

(16) x2 2x 3 x4 + ax3 + bx2 + 3x + 4 x2 + x 2 x3 + 10x2 + cx + d a + b + c + d (17) ... (18) ... x3 7x + 6 , 3x3 7x2 + 4 x3 2x2 5x + 6

x4 3x3 + 6x 4

x3 + x2 10x + 8

(19)

3x6 2x5 64x4 + 96x3 27x2 + 98x + 40

S uio! S aeaee e 2/3 (e e) aaoo x-2/3 aa ... oei e 3x-2 a 3 eoo aa!

(20) x2 + a2b2 + 2abx b2 = 0 (20.1) a (20.2) b (20.3) a (20.4) b

2.3 [1] [2] (Transitive Property) [3] [4] [5] (Trichotomy Property) a, b R a [6] [7] a b+c a > b ac > bc , a > b ac < bc , a+c > b+c a > b ac > bc a > b , ac > bc a 0 c0 c b

a b a

b () b () b < c b < c

b < < ) S oaoaaa

1. oaooo aaooa eo 2. oaooo oaaeoeeo oaoo oaeo/o 3. aee aoao eaa eaeooi (oaeaoaeoo)

1. ( 2 (x c1)(x c2)(x c3)... > 0 (x + 3)(x 31) > 0(x d1)(x d2)... x (x 2)

>)

2. c1, c2 , c3 , d1, d2 , ... () -3 0 1 1 2 2 2 3. +, , +, - + - + - + - + + -3 0 1 1 2 2 2 4. + - + - + - + - + -3 0 1 1 2 2 2 ( x d1, d2 , ... ) 5. () x [3, 0) {1} (2, ) -3 0 1 2

* 2 ( 2) 2 x ( -1 /) (x+1)(3-x) > 0 (x+1)(x-3) < 0 S uio! Seooaoaa aa.. eeaaee 0 e (x-2)(x-3) = 0 a x = 2, 3 o ..eo (x-2)(x-3) < 0 ae x < 2, 3 e! ooeeaaa!


( )

O-NET / A-NET

42x 2+ 2x 19 = 4 x42

2

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

x44

x 2+ 2x 19 < 4 x4

x + 2x 19 4 < 0 x42

0 a > (21.2) a 0 , b > 0 a b (21.4) (22) a 0, b > 0

b > c > da+b > ab 2 b a 1 1 + 2 > + a b a2 b

a b

(22.1) (22.2) (23) (24) (23.1) (24.1) (24.2)

a+b a < < b 2 a+b+c < c a < 3

(22.3) (22.4)a3 < b3 < c3

ab < bc

7 < x < 5 x2 y

3 < y < 6

(23.2) xy2 (24.3) x/ y

6 < x < 2xy

2 < y < 3

xy

(25) 20 . 5 . (26) A B 4 < 3x 2 < 13 11 x < 4x + 1 < 2x + 7 A B ' (27) m n x2 + 6x + 7 < 0 m n (28) . . 20 3x 2x2 > 0 3x + 7x 30 < 02

137

(29) m 21 + 5x 6x2 > 0 n 3x2 1 > 1 + x 3x2

m+n

(30) a b 2x2 + 4x 5 > 0 (30.2) {a + b} {a, b} (30.1) {ab} {a, b} (31) x3 + a2x a 2 a x1

5


( )

O-NET / A-NET (32) (32.1)

44x (x 1 2) )(x < 0 (x + 1)(x 2)

(32.2) (A ' B ') ' A (x + 4)(x 3)(x + 2)3 > 0 (32.3) (33) x3 x2 4x + 4 > 0{x |

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

B

(x + 4)(x + 1)(x 2)3 > 0}' x (x 5)2

(34) A x3 + 2x2 < 5x + 6 A B (35) (35.1) 1 < 2x1 3x 1

B = (5, )

(35.2) [Ent29]

4 > x 2

2 x+1

(36) A

2x 5 > 0 x+2

B

2x 1 < 1 x+5

(37) [Ent38] S a2 + 1 x1 > 2 x +2

B A'

a S

(2, 6] [3, 8) { x = 2n | n I }

(38) (38.1) { x | x2 < 7 } (38.3) (38.4) (38.2) { 1, 5, 7, 9 } [6, ) (39) a b A = {x | x =

n , n I+ } n+1

B = {x | x =

1 , n I } n

a+b

(40) 2x2 5x + 2 < 5

2.4 (Absolute Value Modulus) a a a a 0 a b a b a a = a ,a < 0 ,a > 0


( )

O-NET / A-NET

45

[1] a > 0 a = a [2] ab = a b [3] [4] a b = b2

ab = baan = an

a

= a2

b 0

a2 [5] [6]

= a

a+b < a + b

ab >

a b

* [7] n n

n

a an = a

n = n =

( b 0) * [1] x = b x2 = b2 () x = b x = b () * [2] x b x b x > b x b -b

b

3 x = 1 3 x = 1 3 x = 1 ... x = 2 x = 4 ... {2, 2, 4, 4}

3 x

= 1

S uio! S

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

oia e x + 2 = x .. x + 2 = x o x + 2 = x eoiaoao a (ao) ao oeoaa eao ai a.. eo e x + 2 < x ! x < x + 2 < x eao ... ioa oiaoaa..

1. 2x + 1 x 2 = x + 3 ... 2 2


( )

O-NET / A-NET

46

x x < 1/2 , 1/2 < x < 2 , x > 2 ( : ) -1/2 2 2. ... 3 3 x>2 -1/2 < x < 2 x < -1/2 -1/2(-2x - 1) (-x + 2) = x + 3 x3 = x+3 x = 3 (2x + 1) (-x + 2) = x + 3 3x 1 = x + 3 x = 2

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

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

-1/2

2

x > 2

x {3} [2, )

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

n

an = a

ab = a b

(42) m (42.1) 4x + 0.5 < m 3 < 2x 1 < (42.2) (42.3) (43) x 2 +5 < m x

0.5

x (2, 6)

x2 25 < m

x+5 < 6

x1 < 5

y 2 < 4

x+y


( )

O-NET / A-NET (44) (44.1) x2 6 x + 8 = 0 (44.2) x 1 + x + 1 = 2 (44.3) [Ent30] x 4 + x 3

47

= 1

(45) A 2 + 3x = 2 + 3 x B 2 + 3x = 2 + 3x (46) (47) A = { x I | x2 + 3x + 3 = 2x + 3 } a2 + b2

B A'

8 (x + 2)2 14 (x + 2) + 3 = 0

B = {x I |

5 3x = 2} x+2 A B

a, b ( x )x = x 32

(48) (49) (49.1) 2x 1 < 3x + 2 (49.2) (49.3)3 < x 2 < 6 x + 1 > 0 x

(49.4) (49.5)

3 < x x1 2x x 1

< 2

x2 x 2 < 0x+2 + x < 4 2

(50) A B (51) A = {x R | x < 2 4x + 5

x < x7

(A B) '

< 5}

(51.1) a, b A (a + b)/2 A (51.2) a, b A a+b A

(52)

A = { x R | x2 2 < 14 }A B'

B = {x R |

1 1 > 0} x

c ax + b > c

(53) a, b, c (53.1) 4 < x < 1 ax + b < (53.2) x < 10 x > 8 (54) (54.1) 3x + 2 < 4x + 1 (54.2) [Ent41]x 2 < 2 x+1


( )

O-NET / A-NET (54.3) (54.4) (54.5)x 7 < 5 < 5x 25 x 1 + x 3 < x 5

48

x2 5x 4 > 1 x2 + x 2x 3 < x 2

* (55)

(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 I1. a b a a c a (b c)S i! S

3. a b a bn

2. a b a (b c) 4. an b a b * aouo aaoeaiaa! aoiuo! 3. a bn a b 1. a (b c) a b a a c 2. a (b c) a b 4. a b an b

[2] m, n n 0 m = n q + r q I , 0 < r < n q, r q (Quotient) r (Remainder) [3] (Prime Numbers) p 0, 1, 1 p 1, 1, p, p 2, 3, 5, 7, 11, ... ... 0, 1, 1 (Composite Numbers) [3.1] 1


( )

O-NET / A-NET [3.2]

49p 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 b = r1q 2 + r2 r1 = r2q 3 + r3 r2 = r3q 4 + r4 ... a = b q 1 + r1 ( 0) ... ( rk ) ... 138 182 (182) = (138) 1 + (44)

(138) = (44) 3 + (6)

(44) = (6) 7 + (2)

(6) = (2) 3

... 2 ( 2 ) (m, n) = 1 m n (Relative Primes) ( m n ) ... ... ... ...

2.5(57) (19)3(288)2

5 d = 252 x + 34 y

(58) ... 252 34 x, y

(59) ... 504 38 (60) ... ... x 128 16 384 x (61) [Ent37] x, y x < y x 3 x, y (x, y) = 9 , [x, y] = 28215

(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) (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)2

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

() (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 4 = 1) 4 3

3 3 +( ) = 0 4 4

. ( 4 , 4 ) (4.1) ( ) (4.2) (a b) c a (b c) (4.3) ( 0 ) ... ( 0 0 ) a (4.4) [ ] c a [b ] (5) A = {x | x x } = {1, 4, 9, 16, 25, 36, ...} A .. B = N - A = { A} ( )b c


O-NET / A-NET(5.1) A 2 2 B 2 2 = 4 4 B ... (5.2) A 1+ 1 = 2 2 A

51(14) (5)4 (5)3 + 3(5)2 (5) 1= 2(5)3 + (5)2 + 75(5) + a

... a = 81 (15) 0 a (2)3 a(2)2 + (2) + 2b = 0 .... (1) a = 4, b = 3 a + b (16) (x2 2x 3) = (x 3)(x + 1) 19 37 ,b = 9 9

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 ... 1 6 + 56+ 5

B

4 1 2 (2) + (2) b = 0 a

.... (2)= 7

(3)4 + a(3)3 + b(3)2 + 3(3) + 4 = 0 ( 1 4 a( 1 3 b( 1 2 3( 1 4 0 ) + ) + ) + )+ =

a =

(x2 + x 2) = (x + 2)(x 1 )

(2)3 + 10(2)2 + c(2) + d = 0 3 ) )2 ) (1 + 10(1 + c(1 + d = 0

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

c = 7, d = 18 a + b + c + d = 155 9 (17) () (x3 7x + 6) = (x 1)(x 2)(x + 3) (3x3 7x2 + 4) = (x 1)(x 2)(3x + 2) (x 3x + 6x 4) = (x 1)(x 2)(x 2)(x + ... = (x 1)(x 2) = x2 3x + 2 (18) (x3 2x2 5x + 6) = (x 1)(x 3)(x + 2) (x3 + x2 10x + 8) = (x 1)(x 2)(x + 4) ... = (x 1)(x 2)(x 3)(x + 2)(x + 4)4 3

2)

= x5 17x3 + 12x2 + 52x 48

= 3(3zy + z + y)x + 3zy + z + y + x

... (11.1) A 5+7 = 6 6 A )2

(19) (3x + 1)(x 2)(x 4)(x + 5)(x2 + 1) (20.1) a = 0 x2 b2 = 0 (x b)(x + b) = 0 {b, b}

(11.2)33 2

(20.4) b = 1 x2 + a2 + 2ax 1 = 0 (x + a)2 1 = 0 (x + a 1 + a + 1 = 0 )(x ) = 4.5 4.5 A A {a + 1, a 1}A ab ba = 2 2

a+b b+a = 2 2

) ...

(20.2) (20.3)2

b = 0 x2 = 0 {0}

x + 2bx = 0 x(x + 2b) = 0 {0, 2b}

a = 1 x2 + b2 + 2bx b2 = 0

...

(12) a = 4(4)3 21(4)2 + 26(4) 17 = 7 b = 3(3)3 + 13(3)2 + 11(3) + 5 = 8 b a = 8 7 = 1 (13) (1)2 + 2a = (2) + a a = 3

(21.1) c > b > a c > d ()()(+) > 0 (21.2) 2 < 1 (2)2 < 12 (21.3) ... (a + b) / 2 > a + b > 2 ab2 2 2 2

ab

a + 2ab + b > 4aba b)

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

(


( )

O-NET / A-NET(21.4) ... b3 + a3 > ab(b + a) b3 + a3 b+a > ab a2b2

52(27)x2 + 6x + 7 < 02

(b + a)(b2 ab + a2) > ab(b + a) b2 2ab + a2 > 0 (b a)2 > 0

b b 4ac 2a (x + 3 2)(x + 3 +

(x2 + 6x + 9) 2 < 0 (x + 3)2 2 < 0

( a b ) (22.1) (22.2) ... ( xmin < X < xmax ) (22.3) ( x3 ) (22.4) b = 0 ab = bc (23.1) 7 < x < 5 0 < x2 < 49 3 < y < 6 6 < y < 3 6 < x2 y < 46 (6, 46) (23.2) 9 < y2 < 36 252 < xy2 < 180 (252, 180) (24.1) xy 12, 18, 4, 6 (18, 4) (24.2) x y 8, 9, 4, 5 (9, 4) (24.3) x 3, 2, 1, 2 / 3 y

+

2) < 03 2

-

3 + 2

+

3 2 < x < 3 + 2 m=-3+1=-2 n=-3-1=-4 (28) .mn = 2 (2x 5)(x + 4) < 0

+

4

5/2

+

. x2 + 7 x 10 < 0 3 7 49 409 7 409 (x + x + ) < 0 (x + )2 0 )

+h2 5 < 20 4 n = 0

1/2 2/ 3

+5 > 0 2

2

m+n = 2

2x [7.5, 10) . (26) A ; 6 < 3x < 15 2 < x < 5 A = [2, 5)B ; 11 x < 4x + 1 10 < 5x x > 2

15 h2 0 ) 2 2

3.5)(x + 1 +

+

3.5) > 0

-

+1 + 3.5

1 3.5

4x + 1 < 2x + 7 2x < 6 x < 3 B = (2, 3] A B ' = A B = {2} (3, 5) A B ' 2 4

3.5 1.8 . {0} {0, 2} . {2} {0, 2}

a = 0, b = 2


( )

O-NET / A-NET(31)(1 3 + a2(1 a 2 > 5 ) ) a2 a 6 > 0 (a 3)(a + 2) > 0

53

+

a (, 2) (3, ) (32.1) - + - + - + (, 1) (0, 1) + (32.2) A B (32.3)-1 0 1 2 2

2

3

+

x2 + 8x x2 8x >0 1 4 > 0 x > 2 (2, 8] (36) A ; 2x 5 > 0x+2 x2

+

(A ' B ') ' = A B = [4, 1 (1, ) ) [4, ) {1}

+

B;

-2

-

5/2

+ +

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

(33)

| 3 | + | 2 | + | 0 | + | 1 | + | 5 | = 11 (x 1 )(x 2)(x + 2) > 0

B A' = B A

2x 1 2x 1 x 5 1< 0 < 0 x+5 x+5 x6 < 0 x+5 -5 6

+

[2, 5/2)

-

+

2

+

2 + (2) = 0 x 1 x 1 2x 4 2 > 0 (37)x +2 x+2 x 5 x+5 > 0 < 0 x+2 x +2

> 0

-2

1

[2, 1] [2, ) (34) x3 + 2x2 5x 6 < 0 (x 2)(x + 1 )(x + 3) < 0

+ 2 2

B = (5, )AB

A

-5

-3 -1 -3 -1

+

+

4 3 1 + 0 + 1 + 2 = 5 (35.1) 1 2 < 0x 1 3x 1 3x 1 2x + 2 (x + 1 ) < 0 < 0 (x 1 )(3x 1 ) (x 1 )(3x 1 )

a = 2 a2 + 1 = 5 (38.1) x ( 7, 7) 7 (38.2) (38.3) 8 (38.4) {..., 6, 4, 2, 0, 2, 4, ...} (39) A = { 1 , 2 , 3 , ...} a = 1 a + b = 1 1 = 0 (40) 2x2 5x + 2 < 5 2x2 5x 3 < 0 (2x + 1 )(x 3) < 02 3 4 1 1 B = {1, , , ...} 2 3

5

-

2

+

b = 1

-1

+1/3

1

+

(, 1) ( 1 , 1) -1/2 3 3 (35.2) 2 2x 5x + 2 > 0 (2x 1)(x 2) > 0 + + ( < ) 1/2 2 16 4 4 1 > >0 2 2 x +1 x+1 (x 2) (x 2)4x + 4 x2 + 4x 4 >0 (x 2)2(x + 1 )

+

-

+

-1/2 1/2

2

3

1 5 3 + ( ) = 2 2


( )

O-NET / A-NET(41.1) n

54(45) . < 1.5A;

.

a , n = an = a , n =

. -2/3

. 0

(41.2) a = 0, b = 1 |a b| = 1 , |a| |b| = 1 (42.1) 3 < 2x 1 < 0.5 2 < 2x 1 < x < 0.75

x < 2/ 3

2 3x = 2 3x 2 = 2

4 < 4x < 3 3.5 < 4x + 0.5 < 3.5 | 4x + 0.5 | < 3.5 1 2 < < 1 3 x 2 1 2 17 1 < < 5 < + 6 < x 3 x 3 x2 2 2 +5 = 1 +5 = +6 x x x 17 (5, ) 3 x 2 17 | + 5| < x 3

. . . .

2/ 3 < x < 0

2 + 3x = 2 3x 6x = 0 x = 0

(42.2)

2 < x < 6

x > 0 2 + 3x = 2 + 3x 0 = 0 [0, ) B;

A = [0, )

. . -2/3x < 2/ 3x > 2/3

2 , 0) 3

2 + 3x = 2 + 3x 2 = 2

(42.3)

0 < x2 < 121 25 < x2 25 < 96 | x2 25 | < 96

6 < x + 5 < 6 11 < x < 1

2 + 3x = 2 + 3x 0 = 0 [2/ 3, ) B = [2/ 3, )

(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) |x|2 6|x| + 8 = 0 (|x | 4)(| x | 2) = 0 |x | = 2 4 {2, 2, 4, 4} (44.2) . . -1 1 .

(46)

B A ' = B A = [2

8 x + 2 14 x + 2 + 3 = 0 3 2

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

x {2 +

. .

x < 1 x + 1 x 1 = 2 2x = 2 x = 1

8 (47) ** A; 2 (x2 + 3x + 3)2 (2x + 3)2 = 0 (x2 + x)(x2 + 5x + 6) = 0

3 3 1 1 , 2 , 2 + , 2 } 2 2 4 4

1 4

1 < x < 1 x + 1 + x + 1 = 2 2 = 2 [1, 1) x > 1 x 1 + x + 1 = 2 2x = 2 x = 1 {1}

(x2 + 3x + 3 2x 3)(x2 + 3x + 3 + 2x + 3) = 0

.

A = {0, 1, 2, 3}

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

[1, 1] (44.3) . . 3 4

.

B; |5 3x| = |2x + 4| ( 0 x -2 ) 2 (5 3x 2x 4)(5 3x + 2x + 4) = 0 (1 5x)(9 x) = 0 B = {9}(5 3x)2 (2x + 4)2 = 0

. .

x < 3 x + 4 x + 3 = 1 2x = 6 x = 3 3 < x < 4 x + 4 + x 3 = 1 1 = 1 [3, 4)

.

x > 4 x 4 + x 3 = 1 2x = 8 x = 4 {4}

() A B = {0, 1, 2, 3, 9} a = 9, b = 3 a2 + b2 = 90

[3, 4]


( )

O-NET / A-NET

55

(48) 2 x < 0 (49.4) . x < 1 3 3 x > 0 x < 0 1 x +1 (1, 1 )

.

. x = 1 1 1 {1, 6} ( log ) (49.1) 44, 45 . x < 1/2 2x + 1 < 3x + 2 1 < 5x x > 1/5

x > 1 3 3 1/2

[1, 3) [ 3 + 21 , ) (1/5, ) 2 [ 2 3 + 21 47 .. (1, 3) [ 2 , ) > 0 ( x > -2/3) ] (49.5) A x (49.2) A A 0 |x|

. x

x < 0

(1, 0) (1, )

1 x2 1 > 0 > 0 x x (x 1 )(x + 1 ) > 0 x x >0

. x+

x2 + 1 > 0 x

(1, 0)

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

1 > 0 x

A (, 1) [2, ) A 0 A [0, 1) [2, ) x (, 2] (1, 1) [2, ) (, 2] (1, 1 [2, ) ) [ 2 x > 0 x < 0 ] (50) A; . x < 2 x 2 + x 4 < 0 x 2 + 2x 8 < 0 2 x 10 < 0 x < 10 (, 2)

A 2A + 2 A1

0

A2 A 1

0

.

x > 2 x +2 + x 4 < 0 x + 2 + 2x 8 < 0 2 3x 6 < 0 x < 2 [2, 2]

A = (, 2] [ x 2 47 ]


( )

O-NET / A-NETB; . . 7 7 x < x + 7 2x < 7 x < (, ) 2 2 x < 7

56 (53.2) 10 + 8 = 1

x > 7 x < x 7 0 < 7 7 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

A B = (, 2]

x + 1 > 9 | x + 1 | > 9 a = 1, b = 1, c = 9 (54.1) (3x + 2)2 < (4x + 1 2 (3x + 2)2 (4x + 1 2 < 0 ) ) (3x + 2 4x 1 )(3x + 2 + 4x + 1) < 0

x + 1 < 9

2

(x + 1 )(7x + 3) < 0 (x 1 )(7x + 3) > 0

. x > 5/4

(, 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 5 , ] 4 4

(, 3) (1, ) 7 (54.2) () ( x -1) (x 2)2 < (2x + 2)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

(51.1) ( a + b A )2 A

A = [

(51.2) (52)

A ; 14 < x2 2 < 14

5 15 10 + ( ) = A 4 4 4

12 < x2 < 16 4 < x < 4B;

0 < x2 < 16 A = (4, 4)

1 1 x x1 1> 0 > 0 < 0 x x x B = (0, 1 )

(, 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 < 4 x > 6 . . (2, 4) (6, 12) (54.4) 4 . x < 1 x + 1 x + 3 < x + 5 1 < x (1, 1 )

.

A B ' = A B = (4, 0] [1, 4)

1 < x < 3 x 1 x + 3 < x + 5 x < 3 [1, 3) x 1 + x 3 < x + 5 x < 3 x 1 + x 3 < x 5 x < 1 x >5 3< x < 5

7 (53.1) 4 + 1 = 3 2 2 4 + 3 < x + 3 < 1 + 3 | 2x + 3 | < 5 2 2 2 5 3 5 < x+ < 5 < 2x + 3 < 5 2 2 2

. .

a = 2, b = 3, c = 5

(1, 3) (54.5) 2 54.1, 54.2 .. x -2 1 .. x2 5x 4 x2 5x 4 > 1 < 1 2 2 x +x2 x +x2


( )

O-NET / A-NET. x2 5x 4 > 1 x2 + x 2 x2 5x 4 x2 x + 2 >0 x2 + x 2 (6x 2) (3x + 1 ) >0 5/2 =28(-4)+(-38)(-3) ... () (, 1/2) (5/2, ) 2 =(-504+(-38)(-14))(-4)+(-38)(-3) 2 = (-504)(-4) + (-38)(53) (56.1) . x < 0 2 (60) x 128 = 16 384 x = 48 (1 + x)(1 + x) > 0 (x + 1) > 0 (61) ... 9 = 3 3 x = 1 ... 28,215 = 3 3 5 11 57 () x y 9 (, 1) (1, 0) x = 3 3 5 11 = 495 . x > 0 (1 x)(1 + x) > 0 (x 1 )(x + 1 < 0 )

y = 3 3 57 = 513

(1, 1) [0, 1) (, 1) (1, 1) (56.2) (1, ) ( x = 1 1 (1 |x|)(1 + x) )

(62) ... x y = 1 15,015 = 3 5 7 11 13

= 1

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

eo

..(1) 514 514 2 14 5 ( ) 2 ( 00 00 )

(1) (2)2

5 14 . 00 002 5 14 . 00 002 5 14 . 00 00 4 1 2 5 14 . 00 00 4 1 14

(3)2

(2) ( 5) ( 5) 2 2 ... 2

(3) 2 2 4 ... 4 5 1 (4) () 2 4 ... ( 14) 114

(4)2 4

(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)

(5)

(6) (4) (5) (22) 44 ( 00) 30 3000

(6)

2 2 . 5 14 . 00 00 4 42 1 14 84 30 2 2

(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 :]

2 2 . 5 14 . 00 00 4 42 1 14 84 44 30 002 2 . 6 5 14 . 00 00 4 42 1 14 84 446 30 00 26 76 3 24 2 2

(7)

(8)

2 2 . 6 7 5 14 . 00 00 4 42 1 14 84 446 30 00 26 76 4527 3 24 00 3 16 89 .... ....


( )

O-NET / A-NET

59

~

lgc3

(Logic) () (Truth Value) (Proposition Statement) , , 1 2, , , S uio! S aoeoea aea ... e 1. uo eoaeeiiia ioe ea! 2. eai oaea eaea e aaoaioe a (eaoia aoe aao 3.4 a..)

p, q, r 2 (True; T) (False; F)


( )

O-NET / A-NET

60

3.1 () (Connectives)... (and), (or), - (if-then), (if and only if) (not) ... ~ (Negation) pT T FF

qT F TF

p q (p q )TF FF

p q (p q )T T T F

p q (p q )FT TT

p q (p q )T F FT

p (~p )F F TT

T T F F - T F - (Truth Table) 2n n ... 1 2 , 2 4 (), 3 8 2 (Equivalent) ( ) () p (q r) p (q r)

(p q) (p r)

(p q) (p r)

~ (p q) ~ (p q)

~p ~q ~p ~q p~q

pq pq

~p q

~q~p

~ (p q) ~ (p q)

(p q) (q p)

~p q

p~q


( )

O-NET / A-NET

61

3.1(1) p T p

Tp

T p

T p

Fp

Fp

Fp pT

Fp

pp p ~p

pp p ~p

pp p ~p

p F

pp 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 p (3.2) (p ~ q) (p q) (3.3) (~ r p) (~ (r s) (r ~ q)) p, q r, s p, r, r q (3.4) (p q) (s p) (s q) p q , q r (3.5) (~ q (p r)) (~ r) q n (3.6) n [(m q) ~ s] p q , q r (3.7) (p r) q (p q) (r s) , q s (3.8) (q p) (r s) (p r) (q s) , p q (3.9) [Ent25] r s (p ~ r) (p q) (3.10) (p r) ~ 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)] ~ p (~ r ~ s)


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(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)] . r (p ~ q) . (s r) (p q) . ~ (p s) ~ r (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 p) ~ q p . .~ (p ~ q)

~p q

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

(10) (10.1) p q . (~ q ~ p) (~ q p) . (p q) (q ~ p) . (p ~ q) (~ p ~ q) . (p ~ q) (q p) (10.2) [((q ~ t) p) ((q ~ t) ~ p)] ~ q r . (t q) p . q ~ t 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 b < 0 . a > 0 b > 0 ab < 0 . a > 0 b > 0 ab > 0 . ab < 0 a > 0 b > 0 . ab > 0 a < 0 b < 0ab > 0


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(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)p T q T p q F

T FF

F TF

F FT

(15) p q ~ (p q)

p (q r)

3.2 ( ) (Tautology)

. (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 F F T F T F T T T F T F T T T F T T T T F F T F T F T F T T T F T T T T T T

. .


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

. (r p) (p r)(r p) (p r) T F F T T F

. (r ~ p) (p r)

.

- ... p r ... p r ... ( p , r ) . 2 T F F T ... ... , - ( )

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

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


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(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 p1, p2 , p3 , ..., pn q (valid) (invalid) 1. (p1 p2 p3 ... pn) q 2. (1) p q (2) p qp ~ q

(3)

pq

qr

(4)

pq rspr q s

q

~ p

pr

p q ~q~p


( )

O-NET / A-NET (5) pqp

66(6) p pq

S uio! S ooeuio oea aea o oeiueuaa! ... aoaaioou (ea oaeu)

3.3(22) [Ent39] (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 = 1 3. a2 = 2 a 4. a2 1 a (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.


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. . . .

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 U = {1,2,3} x [P (x)] , x [P (x)] U = {3,4} 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 xy [P (x) Q (x, y)] x y ... x 2 x+y yx [P (x) Q (x, y)] y x ... x 2 x+y


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

3.4(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 [Q (x) ~ P (22/7)] . x [P (x) ~ Q ()] (29) P (x) , Q (x) P (x) = x > x2 , Q (x) = 3 x 1 . x [P (x)] U (0, 1) . x [Q (x)] U = {2, 3, 5, 8} (30) x

x (x3+5x 1 < 4) x ( x2 1 < 0 x > 2)

(31) N 2 [x (x2 1 ) x (x + 1 > 0)] x < 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) (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)

S uio! S ao 32.2 a 32.3 iieoaaa ea a some e a o 32.2 eii oeoeoa a o 32.3 eoiee x eaaa oeaea... ao 32.4 a 32.5 ieoa eaa all e o uiaoaaa all a a, some a o


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= {1, 0, 1}

(34) [Ent21] (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) (35.2) xy (x + y = 0) U = {2, 0, 2} U = {2, 2}

(36) xy (xy = 1) xy (xy = y) . . . . (37) . xy [x + y > xy] . xy [x < y]R+

. xy [x + y < 0] . 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) [Ent39] 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 < ] y x [x + 5 0] y [ 22 > ] y

. [Ent38]

x [x < 6] x [x > 8]

x [x > 6] x [x < 8]

3.5 (Reasoning) 2 ( ) (Inductive Reasoning)


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... , ,

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 n2

3. 1, 3, 7, 15, 31, ... 63 ( 32 ) 4. 11 11 = 121 , 111 111 = 12321 , 11111 11111 = 1234543211111 1111 = 1234321

2, 4, 8, 16

5. 3 12 , 51 , 96 , 117 , 258 , 543 , 2930 , 5022 , 7839 3 3 3 1. ... ( ) 4 (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. ... ( )


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( ) (Deductive Reasoning) ... 2 10 2 10

1. (1) ... (2) 2. (1) ... (2) * 3. (1) ... (2) * (valid) ... S uio! S

ooa aio.. euiaaeo oeu (eoao 3.3) e eua aeiaaa eoeuuo eiei aei ... eeoeueuo eeuea eoioaiaa!

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

( )


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72

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

.

.

2. (1) (2) ... ( 2 ) 3. (1) (2) ... 4. (1) (2) ... A 2 B .

. A

B

C 3

* !

3.5(40) a (40.1) 1, 3, 5, 7, a (40.2) 2, 7, 12, 17, a (40.3) 1, 2, 3, 4, a (40.4) 3, 6, 12, 24, a (40.9) [ ..47] (40.5) (40.6) (40.7) (40.8)3, 1, 1, 3, a

1 2 3 4 , , , ,a 2 3 4 5

1, 4, 9, 16, a3, 3 3, 3 3 3 , 3 3 3 3 , a

125, 726, a, 40328, 362889


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73

(41) 2 ( ) (41.1)37 3 = 111 37 6 = 222 9 9 = 81 37 9 = 333 9 99 = 891 1 9 = 11 2

(41.5)

11 11 = 121

11 12 = 132

11 13 = 143 1089 1 = 1089 1089 2 = 2178 2 (3) = 3 (3 1)

(41.2)

9 999 = 8991 12 9 = 111 3 9 9 + 7 = 88

(41.6)

1089 3 = 3267 2 (3) + 2 (9) = 3 (9 1 ) 3 4 = 2 (1 + 2 + 3)

(41.3)

123 9 = 1111 4 9 98 + 6 = 888

(41.7)

2 (3) + 2 (9) + 2 (27) = 3 (27 1 ) 4 5 = 2 (1 + 2 + 3 + 4)

(41.4)

9 987 + 5 = 8888

(41.8)

5 6 = 2 (1 + 2 + 3 + 4 + 5)

(42) - (42.1) (42.2)

(42.3) (42.4) (42.5) (42.6) (42.7) (42.8) (42.12) (42.13) (42.14)


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74

(42.9) (42.15) (42.10) (42.16) (42.11) (42.17)

()(40.1) 9 (40.2) 22 (40.3) 5 (21) (40.4) 48 (40.5) 5 3 (22) T|p|p|T (40.6) 5/6 (40.7) 25 (23) p | T | T |~ p | T |~ p (24.1) s r (24.2) r (40.8) 3 3 3 3 3 (40.9) 5047 p |~ p | T | F (25) (26) . (41.1) 37 12 = 444 , 37 15 = 555 (2) 2.6 2.10 (27) . (28) . (41.2) 9 9999 = 89991 , (29) . . 9 99999 = 899991 (3) 3.5, 3.7, 3.9, (30) (31) (41.3) 1234 9 = 11111 5 , 3.12 (32) 32.1, 32.4 12345 9 = 111111 6 (3.11) T, T, F (4) . . (41.4) 9 9876 + 4 = 88888 , (33) 33.1 (5) 9 98765 + 3 = 888888 (34) 34.2, 34.4 (6) . . (41.5) 11 14 = 154 , 11 15 = 165 (35.1) (7) . (p q) (~ r ~ s) (35.2) (36) . (37) . (41.6) 1089 4 = 4356 , . p ~ q r 1089 5 = 5445 (38.1) x [P (x) Q (x)] (41.7) 2 (3) + 2 (9) + 2 (27) (8) . (9) . (38.2) x [P (x) Q (x) ~ R (x))] (10.1) . (10.2) . + 2 (81 = 3 (81 1 , ) ) (38.3) x [P (x)] x [Q (x)] (10.3) . (11) . 2 (3) + 2 (9) + 2 (27) + 2 (81) (38.4) xy [(x + y = 5) (x y 1)] (12) + 2 (243) = 3 (243 1) (38.5) xy [x 0 y = 0 xy 0] (41.8) 6 7 = 2 (1 + 2 + 3 + (13) (14.1) . (14.2) . (38.6) xy (xy > 0 x > 0 y > 0) 4 + 5 + 6) , 7 8 = 2 (1 + (15) 3:5 (38.7) xy [P (y) ~ R (x) Q (x)] 2 + 3 + 4 + 5 + 6 + 7) (16 19) (38.8) xyz (x + y < z xy > z) (42) 16.1, 16.2, 17.2, 17.7, 19.1 (42.3), (42.5), (42.6), (39) . . (20) . . (42.9), (42.13) (1)p|F|p|F


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() (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

(3.1) (3.2) (3.3)

(....) (p q) T

(....) (p q) T

T

T

F

(~ r p) (....) T q Tq

T

.

(3.4) r

T

. - (

T p p, F p T, p T T, p F ~ p T F F, F F T

T T (3.5) p T (3.6) (3.7) (3.8)

(p q) (s p) (s q) T T, q F, r T

T

r

T

)

(~ q (p r) (~ r) F )n F

p p T, p ~ p ~ p

T

. T p p, F p ~ p,

q F

F

n [....] T

(....) q F

p p T, p ~ p F

q F, s F, r T, 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

(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 p q ... p q ( )

T

F


O-NET / A-NET

76

... (9) . ~ (~ p ~ q) p q ... . (~ p q) (q ~ p) ... (3.14) . p (~ q p) ~ q p q p ... ( -) [p ~ (r s)] ~ p (~ r ~ s) . . F T F (10.1) . . (p q) (q p) T T r s (10.2) {[(q ~ t) (p ~ p)] ~ q} r ... r s [(q ~ t) ~ q] r T [p ~ p] [(q ~ q) (~ t ~ q)] r ... T (~ t ~ q) r (t q) r . (4) s r F s T, r F (10.3) [(q r) (q ~ r)] [(p s) (p ~ s) ] p r T p T [q (r ~ r) (p (s ~ s)] q p p q T q T F T . [(....) (q r)] (r s) F . F F (11) . . . [....] (~ r s) T ab > 0, a < 0, b < 0 ... T . . (5) p q T p q (p q) r r ~ s F r F, s T . (~ p ~ q) ~ r [(~ p r) ....] T . ~ r (~ p ~ q) F (12) . p (q r) . (~ q ~ r) ~ p . ~ (....) ~ r T T . ~ p q r . . . r (p ~ q) T . F F (13) . ~ (p ~ r) ~ q ~ p r ~ q . (s r) (p q) T q (r ~ p) T (6) p q , r ~ s . p (q r) ~ p (~ q r) . [.... (r ~ s)] [.... (~ r ~ s)] T q (p r) ~ q ( ~ p r) T T . (p q) r ~ p ~ q r . (p ~ q) (p r) ~ p ~ q ~ p r . [....] [(p ~ q) (r ~ s)] T (14.1) T T (7) . ~ [(~ p ~ q) (r s)] p q p*p q*q (p*p)*(q*q)(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 .

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(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

77(17.2) qp

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

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

(17.3) (17.4) (17.5) (17.6)

q ~ p

F

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

(~ p q) (~ q p)

: 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 F T T T T 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 q) (p r) F T T F T F F T T T F q (17.1) ~ (p ~ q) ~ (~ p ~ q) p q

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

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


( )

O-NET / A-NET(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 F T F F T F F F r . (~ p r) (~ q r) (~ p ~ q) r (p q) r

78(23.2)1. 2. 3. 4. p ~ q rs r q ~ s p

2. r s

4. ~ s

r 3. r q

q 1. p ~ q ~p

(24.1)

p r 2. s p

1. p (q r) q (p r) 3. q

(24.2)

s r

1. ~ p q 2. q ~ r ~p ~r r p p 1. p ~ q 2. q ~p

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

3.

. . (21) (22.1) 1. p q2. q s p s 3. ~ s ~p

(25) (26)

3.

r

1. p q

(22.2) p (r s)~prs

1. p q 2. ~ p r 3. s ~ r 4. ~ q ?

~p 2. ~ p r

4. ~ q

(23.1) 1. p q2. (p q) r 3. ~ (s r) 4. p

r 3. s ~ r ~ s

1. p q

4. p q 2. (p q) r r 3. ~ s ~ r r~s ~ s

s

~ s . ~ s p . ( ) . s p . ~ r ~ s ~ r ( r ) . p ~ q (27) . x x2 > 0 x = 0 . x = 2 8 > 4, 2 < 4 . x = 1 . x ( )


O-NET / A-NET(28) . x x 2 ... ( x = 3 ) . x ... x 0.5 ... ( x = 2 F F T ) . x ... x ... ( T T) . x ... x 22 ... 22 7 x = 1 x x (29) . x ... x > x2 U = (0, 1) ... . x ... x ... 3 x 1 ... 2, 3, 5 , 8 ... 3 1 . . (30) x (x3 + 5x 1 < 4) x = 1 7 < 4 x(|x2 1| < 0 x > 2) F T T T T 2 (31) x(x 1 ) x = 2 22 1 = 3 x(x + 1 > 0) ( + 1 0 ) 2 2 x( < 0) x = 1 0) x = 0 x(x2 1) T T T (32.3) x(x + 1 > 0 x2 1) x = 0 T (32.4) x(x2 > 0) x = 0 x(x = 0) x = 1 F F Fx 17

79(32.5)x(x2 > 0

x = 0) ( x = 1, 0, 1 ) T (33.1) x y 2 x + 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) . xy(xy = 1) x = 2 y I xy(xy = y) x = 1 xy = y y F T F . xy(xy = 1) xy(xy = y) ( .) F T F . xy(xy = 1) x = 0 y R xy(xy = y) () F T F . xy(xy = 1) x R+ + y R xy(xy = y) () ... . T T T ( )


O-NET / A-NET(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) 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)] (38.8) xyz (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 a =

80(41.4) (41.5) (41.6) (41.7) (41.8)9 9876 + 4 = 88888 ,

9 98765 + 3 = 888888 11 14 = 154 , 11 15 = 165 1089 4 = 4356 , 2 (3) + 2 (9) + 2 (27) + 2 (81 = 3 (81 1 , ) ) 6 7 = 2 (1 + 2 + 3 + 4 + 5 + 6) ,

1089 5 = 54452 (3) + 2 (9) + 2 (27) + 2 (81) + 2 (243) = 3 (243 1)

7 8 = 2 (1 + 2 + 3 + 4 + 5 + 6 + 7)

(42.1,42.2)

2 (42.3) .

(42.4)

()5 6

3 1 3 1

(40.6)

()

(42.5)

(40.7) a = 25 () (40.8) a = 3 3 3 3 3 ( 3 5 ) (40.9) 7 5, 6, _, 8, 9 (42.6) 12, 72 , _, 4032, 362886 9

72 7 = 504 504 8 = 4032 5047 (41.1) 37 12 = 444 , 37 15 = 555 (42.7) (41.2) 9 9999 = 89991 ,9 99999 = 899991

(41.3)

1234 9 = 11111 5 ,

12345 9 = 111111 6

2


( )

O-NET / A-NET(42.8)

81(42.13)

(42.9)

(42.14)

(42.10) (, ) ! (42.11)

2 (42.15)

(42.12)

(42.16) ( 42.10 , ) (42.17)


( )

O-NET / A-NET

82

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 1 and 0 1 or 0 1 nand 0 1 nor 0 1 xor 0 1 0 1 1 0 1

inv

0 False 1 True AND OR ( 1+1 1, 2 ..)

A 1 1 0 0

B 1 0 1 0

A and B (AB) 1 0 0 0

A 1 1 0 0

B 1 0 1 0

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

A 1 0

not A (A ) 0 1

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

, :]

A xor B

A B


( )

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83

G (e,o)

4 eiiea

(Analytic Geometry) , , , y (Plane) Q2 Q1 (Coordinate) (, +) (+, +) (Cartesian Coordinate) 2 x (Origin; O) Q3 O Q4 x y (, ) (+, ) xy 4 (Quadrant; Q) 1, 2, 3, 4 (Ordered Pair) +x +y (2, 4)


( )

O-NET / A-NET

84

4.1 : P, Q P (x, y) Q (2, 4) Q (2, 4) [1] P Q Q (x2,y2)PQ = (x2 x1) 2+ (y2 y1) 2PQ

(Pythagorean Theorem) ( 10) ( 11)

P (x1,y1)

[2] Q (x2,y2)R( x1+ x2 y1+ y2 , ) 2 2

m:n Q (x2,y2) m n P (x1,y1)mx + nx2 my1+ ny2 R( 1 , ) m+n m+n

P (x1,y1)

[3] ( Centroid) 2 : 1 R (x3,y3) P (x1,y1) Cx+x +x y+y +y C ( 1 2 3 , 1 2 3) 3 3

Q (x2,y2)

[4] ( ) ( det 9) T (x5,y5)x1 x2 x 1 = 3 x4 2 x5 x1 y1 y2 y3 y4 y5 y1

P (x1,y1) S (x4,y4) Q (x2,y2) R (x3,y3)

S o! S 1. ae e 3 e eo 3 a 3 a 2. eueo o ai ... eei oae ioo

1 = (x1y2+ x2y3+ x3y4+ x4y5+ x5y1 x2y1 x3y2 x4y3 x5y4 x1y5) 2


( )

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85

4.1(1) P1 (1, 7)

P2 (5, 2)

PP2 1A (2, 7) , B (6, 3) ,

(2) P, Q AB , CD 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 ABC

y D (2,4) C P O B (2,0) Q x

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

(6) A (2, 8) , B (6, 12) , AB BC AP : PB = PQ

P Q 1 : 3 , BQ : BC = 3 : 4

C (2, 4)

(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 (9) P1 (1, 2)

P2 (3, 5)

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

(10) A (2, 1) , B (4, 3) , C (2, 5) (11) (m, n) (4, 1) m n (12) ABC B (6, 7) , C (4, 3) A (4, 5) , (4, 7) ,

P (4/3, 1)

(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

. PQR 15 (16) E (2, 7) A (1, 4) , B (3, 2) , C (1, 3) , D (4, 5) ,

4.2 : P Q P Q PQ L L1 , L2 [1] (Slope; m) (Parallel; ) (Perpendicular; ) -1 Q (x2,y2) y y m = tan = 2 1x2 x1

P (x1,y1)

m > 0 () m < 0 () m = 0 x y m y y1 = m x x1

[2] [2.1] (x1, y1) m m P (x1,y1)

y y1 = m (x x1)

[2.2] (x1, y1) , (x2 , y2) (2.1) Q (x2,y2) y y1 = y2 y1 (x x1) x x 1 2 P (x1,y1) [2.3] (Intercept) Intercept Form y x + y = 1a b

b O a x

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


O-NET / A-NET

87

1. (Common Form) A x + B y + C = 0 2. 3 y = mx + c c y Slope-Intercept Form y y1 = m (x x1) Slope-Point Form S uio! S x y Intercept-Intercept Form + = 1a b

3. m = A B

A C y = x B B c =

C B

eoaao A/B a i x y eoia ao ao Ax+By+C =0 oeo aa..

P (1, 3) Q (5, 9) . P Q m = 9 3 = 3/2 5 1 . L PQ L PQ m = 2 ( 3PQ1

PQ 1

1

L1

)

1

PQ

L (y 6) = 2 (x 3) ... 3 3y 18 = 2x + 6 ... 2x + 3y 24 = 0 L y (0, 1/3) L (1, 2) L . L L m = 1/3 = 2/3 1/25 6 5 5 6

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

x

1/2

L5

L L m = 3/2 : x 1/2 x (1/2, 0) . L L ... x y L 1/2 + 1/3 = 1 2x 3y = 16 5

L6

5

6

5

3 L (y 2) = 2 (x + 1) 3x + 2y = 1 (1/13, 5/13)6


( )

O-NET / A-NET [3] Ax+By+C1=0 d Ax+By+C2=0 [4] P (x1,y1) d Ax+By+C=0d =

88

S oaa! S aoaeaoeo Ax+By+C=0 eo... a A, B ooeoa o eoaoaa

C2 C1 A2+ B2

d =

A x1 + B y1 + C A2+ B2

L : 2x + 3y 24 = 0 . S (2, 5) 1

L1

dSL1 =

2(2) + 3(5) 24 2 +32 2

=

13 = 13

131

. L 2 13 L () 2x + 3y + C = 0 C 1

2 13 =

24 C22 + 32

...

26 = 24 C

2x + 3y + 2 = 0 2x + 3y 50 = 0 . L : 2x + y 6 = 0 L (x , y ) 2 1

C = 2, 50

2 13

1

1

2 13 =

2x1 + 3y1 24 22 + 321 1

... 1 1 1 1 1

(x , y ) 2x + y 6 = 0 ... 2x + y 6 = 0 x = 8, 5 x = 8 y = 22 x = 5 y = 4 ... (8, 22) (5, 4) . L . . L 2 13 1 1

2

1

L 3x + y = 2 3 L 3x + 3y = 18 . L A/B ... 3 /1 = 3 . L x L 3/ 3 = 3 ( y : x ) 3 3 4 34

4


( )

O-NET / A-NET

89

x 60 ( : ) . L L L L ( . .) 3 4

3

4

6 3 2 3 3 +12 2

d =

C2 C1 A2+ B2

... L4

A, B

3x + y = 6 3L3

dL3L4 = = 4 3 = 2 3 2

3

...

L4

2 3 3 . L , x , y 0 (0, 0) x y ... x y x ( y = 0 ) 2 y ( x = 0 ) 2 3 ... (1/2) (2) (2 3) = 2 3 3

[5] m1

m2

tan =

m1 m2 1 + m1m2

A1x + B1y +2 C1 = A2x + B2y + C2 2 2 2A1 + B1 A2+ B2

( ) [6] (Projection) P L Q P (x1,y1) Q L: Ax+By+C=0

Ans1 Ans2

PP2 L Q1Q2 1 P2 (x2,y2) L: Ax+By+C=0 P1 (x1,y1) Q2

Q1( )


O-NET / A-NET

90

( P L ) ( P ) P (x1, y1) y = x ( 45 x) Q (x1+ y1 , x1+ y1)2 2

4.2(17) (18) A (1, 2) , B (2, k) , C (3, 4)

kP (2, 6)

(1, y)

PR

R (4, 2)

y

(19) AB x y x 4 y 3 AB 3 A B (20) A (4, 5) , B (1, 2) , C (2, 8) , D (2, 4)

AB

CD

(21) D ABCD (22) (k, 7) , (3, 2)

A (4, 1) , B (5, 4) , C (1, 2)

(3, 2) , (1, 4)

k

(23) A (1, 5) D (1, m) m (24) L

B (3, 6)

C (m, 4)

C (5, 6)

L

(3, 1)

(25) ABC A (1, 7) , B (8, 6) , C (7, 1) (26) (7) (27) (3, 0)

(0, 2)

(28) L (2, 5) x y (29)

(1, 3)

(6, 8)

x

3x + 4y = 12 D (2, 5)

(30) ABCD A (1, 2) , B (2, 1) , C (3, 6) , P P (31) 2x + 3y + 10 = 0

x+y=1

2x + y = 5


( )

O-NET / A-NET

91

3x 4y + 5 = 0

(32) x y (33) L 2x + 3y + 5 = 0 L x (1, 5)

(34) M 3x 3y + 5 = 7 N L M y N

2x 5y + 7 = 4

(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 (2/3, 1) L1 L2 L3 (37) A (3, k) 1 4 L A x L (38) L A (1, 2) B (2, 1) PQR , 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) X-intercept Y-intercept 9 x (42) [Ent24] L 0.5 C (3, 0) y A AB L B y B x C BC (43) ABC B A AB 3/2 (44) 2x 3y = 6

(3, 5) ,

C

(4, 4) ,

4x 6y = 25

3x 4y 5 = 0

(45) C

Ax + 2y + C = 0


( )

O-NET / A-NET (46) L1

92

L2 4 L 12x 5y 15 = 0 L1 L1 , L2 x L1 L2 (47) A (2, 1) , A BCB (5, 4) , C (2, 3)

P (3, 2)

(48) L 5x 12y + 3 = k L k (49) 2x 4y = 15

4

3x + 4y = 10

3 (2, 3) , (1, 4)

(50) (51) L1 ( 3, 2) , L1 L 2(0, 1 )

5x y = 0

L2

2x 3y + 1 = 0

L1

(52) L1 (2, 3) , (1, 0) L2 O C L1 L 2 30 CO (53) 4x 3y 6 = 0 3x + 4y + 1 = 0

y

(54) A AB

(2, 1)

x B

(5, 6)

(55) A (1, 0) , B (5, 8) , P AB Q B x = 1 PQ PQ (56) (57) (2, 1) (0, 7)

x y = 04x 5y = 6

4.3 : (Conic Section) () 1. () 2. () 3. () 4. 2 ()


( )

O-NET / A-NET

93

(Circle)

(Ellipse)

(Parabola)

(Hyperbola)

x y , , y y [1] 2 y=x y = (x-3)2 (Translate Shift) f (x, y) = 0 x x f (x h, y k) = 0 O (3,0) h, k h y y k 2 y+1 = x y+1 = (x-3)2 ( (h, k) ) x x (0,-1) (3,-1) [2] () (Scale) y = f (x) my = f (nx) m, n 1 ... n m ( m, n 1 ) h, k y y = x2 x y y = (2x)2 y y/4 = x2 x y 3y = x2 x 1 3

O

x

1 2 4


( )

O-NET / A-NET

94

1. * 2. y 2y = (x-3)2-2 ( 2(y+1)=(x-3)2 ) x (3,-1) 1 2

[3] () (2) (Flip) h, k ( x , y ) y y -(y+1) = (x-3)2 y = x2 x O x (3,-1)

4.4 : (Center; C) (Radius; r) () C (0, 0) r x 2 + y 2 = r 2 C (h, k) (x h)2 + (y k)2 = r 2 r C (h,k)(x h)2 + (y k)2 = r 2

C (h, k) r x 2+ y 2+ Dx + Ey + F = 0


( )

O-NET / A-NET

952

(1, 2) (2, 1) Ax + By + Dx + Ey + F = 0 2

(1, 2) (2, 1) 1 + 3 = 10 (x h) + (y k) = r (x 1) + (y +2) = 10 x 2x + 1+ y + 4y + 4 = 10 x + y 2x+ 4y 5 = 02 2 22 2 2 2 2 2 2

2

2

2

x2+ y2+2x 4y 10 = 02

x y (x + 2x) + (y 4y) = 10 () (x + 2x + 1) + (y 4y + 4) = 10 + 1+ 4 (x + 1) + (y 2) = 15 (1, 2) 15 2 2 2 2

1. x 2+ y 2+ Dx + Ey + F = 0 (h, k) = (D/2, E/2) 2.1 3 D, E, F h, k, r 3 3 x 2+ y 2+ Dx + Ey + F = 0 2.2 r 2 h, k h, k r (x h)2 + (y k)2 = r 2 ( ) () P (x1, y1) Q Q d P (x1,y1) Cd = x2+ y2+ Dx1+ Ey1+ F 1 1

d

d =

(x1h)2+ (y1k)2 r2

4.4(58) (58.1) (x 4)(y + 3) = 1 xy = 1 (58.2) y = x + 1 2 y = x (58.3) x 2+ y 2+ 2x 4y + 5 = 9 x 2+ y 2= k


( )

O-NET / A-NET

96

(59) (59.1) (3, 4) (1, 1) (59.2) (1, 1) (2, 2) (59.3) y = 2x (1, 1) (59.4) (6, 3) , (2, 3) (2, 7) (59.5) (1, 5) x 2+ y 2 2x + 2y 8 = 0 x 2+ y 2+ 3x 3y 8 = 0

(60)

(0, 1 )

3x 2+ 3y 2+ 11x + 15y = 9

(61) (61.1) x 2+ y 2= 8 (2, 2) (61.2) x 2+ y 2= 17 4 [Hint: ] (61.3) x 2+ y 2= 16 (1, 8) [Hint: ] (62) [Hint: ] (62.1) 2 (x 2)2 + (y + 1)2 = 1 (x 6)2 + (y 2)2 = 4 1 (62.2) 1 , y = x + 2 , x 2+ y 2 4x + 2y + 1 = 0

(62.3) 3x 2y 6 = 0 , 2x + 3y + 9 = 0 (63) k (64) [Ent32] x 2+ y 2 6x + 8y + k = 0 k > 0

2x 3y + 21 = 0 ,

y = kx

x 2+ y 2 14x + 49 = k 2

(65) C x 2+ 4x + 2 = (y 2+ 8y + 9) OC OC (66) [Ent38] 4/3 x 2+ y 2 4x + 2y = 4 A B D (1, 2) ABD (67) P (x, y) (x 1 2= (1 y)(1+ y) A (1, 0) )

4.5 : (Focus; F) (Directrix; ) (Axis)


( )

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97

(Vertex) V (0, 0) c x = 4 c y ( y, c , c ) y 2 = 4 c x ( x, c , c ) V (h, k) (x h)2 = 4 c (y k) (y k)2 = 4 c (x h) 2

() 2c c c V (h,k)

(x h)2 = 4 c (y k)

F (h,k+c)

V (h, k) c 4c x 2+ Dx + Ey + F = 0

Directrix : y=k-c Axis : x=h

() 2c (y k)2 = 4 c (x h)

c Axis : y=k Directrix : x=h-c V (h,k)

c

F (h+c,k)

V (h, k) c 4c y 2+ Dx + Ey + F = 0

(Latus Rectum) 1. 2. 3 ( D, E, F h, k, c)

(1, 2) (2, 1) Ax + By + Dx + Ey + F = 0 2 2

(x h) = 4 c (y k) (h, k) = (1, 2) (x 1) = 4 c (y +2) c (2, 1) x, y 2

2


( )

O-NET / A-NET(2 1)2 = 4 c (1+2) 4 c = 1/32

98... (x 1) 2 2

= (1/3)(y +2)

3 (x 2x + 1) = y +2 3x 6x y + 1 = 0 (h, k) (2, 1) r

2

x 2 2x 2y 3 = 0

y () x y (x 2x) = 2y + 3 ... (x 2x + 1) = 2y + 3 + 1 (x 1) = 2y + 4 (x 1) = 2 (y + 2) (x 1) = 4 (0.5)(y + 2)2 2 2 2 2

(1, 2) (1, 2 + 0.5) = (1, 1.5) y = 2 0.5 = 2.5 ( 2y + 5 = 0 ) ( )

4.5(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 (4, 3)

2x 2+ 3y = 0

(70) (70.1) x 2 12y = 0 (70.2) y 2 10y + 12x + 61 = 0 (70.3) (4, 2) x 1 = 0 (70.4) x (0, 1/3) (0, 7/6) (71) P (x, y) (71.1) y = 4 (2, 8) (71.2) x = 4 (3, 1) 5 (72) 4y = (x 1 2 )

13 x ( )


O-NET / A-NET (73) (74) (1, 6)2x y = 8

99y 2= 8x

y = 5

y 2 4x 4y = 8

(75) x 2 6x = 6 2y y 2

x = y

(76) x

x 2+ y 2+ 6x = 0

(77) [Ent39] y 2 4y + 8x = 20 P , P, (78) 6 8

4.6 : ( F1, F2 ) (2a) x + y = 1 a b2 2

C (0, 0)

2a 2b y + x = 1 a b2 2

( x)

( y)

()(x h)2 a2

a V2 F2 c

B1 (h,k+b) b C F1 V1 (h,k) (h+c,k) (h+a,k) B2

+

(y k)2 b2

= 1

C (h, k) 2a c = a2 b2 Ax 2+ By 2+ Dx + Ey + F = 0

2b


( )

O-NET / A-NET

100V1 (h,k+a)

math e-book release2.2.04

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