(2)_______________________________ 20062006

1. 2 1. {a, b, c}2. {x|x }

*x A x A2.

1. (), {1, 3, 5}2. {1, 2, 3, ...}

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

A B 4.

A B (A B) A B5. (U) U6.

A(P(A)) = {x|x A}

2006 _______________________________ (3)2006

1. P(A) P(A)2. A P(A), A - B P(A), AI B P(A)3. A B P(A) P(B)4. P(A)U P(B) P(AU B)5. P(A)I P(B) = P(AI B)6. P(A) - P(B) P(A - B)7. P(A - B) (P(A) - P(B))U {}8. n(P(A)) = 2nA

1. UIA = A2. UUA = U3. U - A = A4. IA = 5. IA = A6. A - = A7. - A = 8. A B = AI B9. A B = B A10. (AU B) = AI B11. (AI B) = AU B12. (A - B) = (AI B) = AU B13. AI (BU C) = (AI B)U (AI C)14. AU (BI C) = (AU B)I (AU C)15. A - (BU C) = (A - B)I (A - C)16. A - (BI C) = (A - B)U (A - C)17. A AU B, B AU B, AI B A, AI B B18. A = B A = B, A B B A19. A B B A, A B AI B = A20. A B AU B = B, A B A - B = 21. AI B = A B, AU B = U A B22. AU B = A = B = 23. A B B C A C ( Subset)

(4)_______________________________ 20062006

24. A B C D

DB CADBCA

IIUU

*

DB CADB CA

IIUU

A B C D

25. A B

C B C A 3.CB CA 2.CB CA 1.

--IIUU

26. (AU B) - C = (A - C)U (B - C)27. (AI B) - C = (A - C)I (B - C)28. A A B B 29. A B A B 30. A B AU B 31. A B AU B 32. A B AI B 33. n(AU B) = n(A) + n(B) - n(AI B)34. n(AU BU C) = n(A) + n(B) + n(C) - n(AI B) - n(AI C) - n(BI C) + n(AI BI C)35. n(AI B) = n(A) + n(B) - n(AU B)36. n(A - B) = n(A) - n(AI B)37. n(A) = m n(P(A)) = 2m = A

1. A, B, C, D (AI C) - (BU D) 1) (A B)I (D - C) 2) (A - B)I (C - D)3) (A B)U (D - C) 4) (A - B)U (C - D)

2. S = {1, 2, 3, 4, 5, 6, 7}P(S) = S

X = {A P(S)| 1 A 7 A} Y = {A X| A 6} X Y () 1) 16, 5 2) 16, 6 3) 32, 5 4) 32, 6

3. 100 6

2006 _______________________________ (5)2006

202322 119

U

1) 2) 1993) 10 4) 64

4. A = {0, 1} B = {0, {1}, {0, 1}} 1) A P(B) 2) {1} P(A)I P(B)3) P(AI B) = 2 4) P(AU B) = 8

5. 180 95 92 125 52 43 - 57 180 1) 20 2) 25 3) 30 4) 35

6. R A = {x R|3x2 + x - 2 > 0} B = {x R||3 - 2x| 4}

. B - A =

32 ,2

1- . AU B =

21 , -- U

,321) . . 2) . . 3) . . 4) . .

7. A = {1, a, 2, b, 3, c}, B = {1, 2} S A SI B 1) 48 2) 32 3) 24 4) 16

8. . 20 . . 17 . 15

1) 6 2) 12 3) 26 4) 32

(6)_______________________________ 20062006

9. A = {a, {a}, {b}, {b, c}} 1) (A - {b, c})U {b} = {a, b, {a}, {b}, {b, c}}2) (A {b, c})U {b} = {a, {a}, {b}}3) (A - {a, {b}}) - {a} = {{b, c}}4) (A - {a, {b}}) - {a} = {b, c}

10. A 2x x3 +- 0 B 2x 21 - 1(A - B) 1) (-, -2)U (-1, ) 2) (-, -2)U [-1, )3) (-, -2]U (-1, ) 4) (-, -2]U [-1, )

11. 1 500 3 5 1) 167 2) 200 3) 233 4) 266

12. 80 3 1 30 20 18 3 1) 3 2) 12 3) 14 4) 15

1. 3) 2. 4) 3. 3) 4. 3) 5. 1) 6. 3) 7. 1) 8. 1) 9. 1) 10. 4)11. 3) 12. 2)

2 1.

9

9198 1 + 9 + 8 = 18 198 9 = 2227963 2 + 7 + 9 + 6 + 3 = 27 27963 9 = 31075220 5 + 2 + 2 + 0 = 9 5220 9 = 580

2006 _______________________________ (7)2006

9 9

2.

1. p1

2. p2Mk. pk

q p1, p2, ... , pk

(p1 p2 ... pk) q

1 1. 2.

2

1. A2. B A B

2 1. 2.

3

1. A2. B3. C

B

A

(8)_______________________________ 20062006

1. BA

2. 2 BA

C

BA C

(1) (2)

1. a

1. -15, -5, 5, 15, a2. 1, 4, 9, 16, a3. 1, 3, 6, 10, a4. 2, 6, 12, 20, a5. 1, 5, 14, 30, a

2. 1. 1.

2.

2. 1. 2.

3. 1. 4

2. 4

4. 1. 2 2. 29 2

29

2006 _______________________________ (9)2006

1. 1. 25 2. 25 3. 15 4. 30 5. 552. 1. 2.

3. 4.

... ,15

3 ,32 ()

0.1010010001... Surd 2 , 3 5 , ...1. U U

1. a, b U a * b U2. a, b, c U (a * b) * c = a * (b * c)3. a, b u a * b = b * a4. I U x U I * x = x * I5. a a-1 a * a-1 = a-1 * a = I

2. 1. a < b a + c < b + c2. a < b c > 0 ac < bc3. a < b c < 0 ac > bc4. 0 < a < b 0 < b1 < a15. a < b < 0 b1 < a1 < 0

(10)______________________________ 20062006

6. 0 < a < b 0 < c < d 0 < ac < bd7. a < b < 0 c < d < 0 ac > bd > 08. a > 1 an1 > an2 (n1 > n2)9. 0 < a < 1 an1 < an2 (n1 > n2)10. a < 0 a2N > a (N I+)((-2)2 > -2)11. x2N < a > 0 - 2N a < x < 2N a12. x2N > a > 0 x > 2N a x < - 2N a13. x2N+1 < a R x < 12N a+14. x2N+1 > a R x > 12N a+

3.

x |x| =

=

0 x x,0 x 0, 0 x x,

-1. |x| 02. |x2N| = |x|2N = x2N3. k > 0

|x| = k x = k x = -k|x| < k -k < x < k|x| > k x > k x < -k

4. a2 < b2 |a| < |b|5. |x + y| = |x| + |y| xy > 0 |(-3) + (-2)| = |-3| + |-2| = 56. |x + y| = ||x| - |y|| xy < 0 |(-5) + 2| = ||-5| - |2|| = 37. |xy| = |x||y|8. y

x = |y||x| y 0

1. a2 - b2 = (a - b)(a + b)2. (a + b)2 = a2 + 2ab + b23. (a - b)2 = a2 - 2ab + b24. a3 + b3 = (a + b)(a2 - ab + b2)5. a3 b3 = (a - b)(a2 + ab + b2)6. (a + b)3 = a3 + 3a2b + 3ab2 + b37. (a - b)3 = a3 - 3a2b + 3ab2 - b3

2006 ______________________________ (11)2006

2 ax2 + bx + c = 0 a, b, c a 0 2 2 1. ax2 + bx + c

x2 + 4x - 32 = 0(x + 8)(x - 4) = 0

x = -8, 42. ax2 + bx + c (

) x = 2a

4ac b b 2 --

x2 + 8x - 5 = 0 a = 1, b = 8, c = -5x = 2(1)

5)4(1)( 8 8 2 ---

= 220 64 8 +-

= 284 8 - = 2

212 8 -

= -4 21= -4 + 21 , -4 - 21

2 , () ax2 + bx + c = 0 1. + = - ab

2. = ac , x2 + x - 1 = 0 2 + 2

+ = -1 ...(1) = -1 ...(2)

(1)2 2 + 2 + 2 = 1 ...(3)2(2) 2 = -2 ...(4)(3) - (4) 2 + 2 = 3

(12)______________________________ 20062006

f(x) f(x) f(x) = (x - a1)(x a2) ... (x - ak) a1 < a2 < a3 ... < ak

2ka -x

+ - +-

1ka - ka

x ak f(x) 0(+)(f(ak) = 0) ak-1 x ak f(x) 0(-)(f(ak-1) = 0) ak-2 x ak-1 f(x) 0(+)(f(ak-2) = 0)

1 x2 - x - 2 0

x2 - x - 2 0(x - 2)(x + 1) 0

+ - +-1 2

[-1, 2] 2 2 x 3 2x x

2-

-- 0 2)(x

1)3)(x (x -

- + 0*

+- +

-1 2-

3

[-1, 2)[3, )

1. am an = am+n 2. n

m

aa = am-n

3. (am)n = amn 4. (ab)n = anbn

5.n

ba

= nn

ba 6. a0 = 1 (a 0)

7. a-n = na1 8. ax = ay a 0, 1 x = y

9. ax = bx |a| |b| x = 0

2006 ______________________________ (13)2006

n n 1 a b b n a

bn = a* n * n a n a n a 2 4 2 2 -2

2 4 2 2 4 4 - 4 n a b n n (n > 1)1. ( n a )n = a n a

2. n n a = a a < 0 n

|a| a < 0 n 3. n ab = n a n b

4. n ba = n

n

ba , b 0

a a1/n m n n > 0 nm

am/n = (a1/n)m = ( n a )m

am/n = (am)1/n = n m a

Ex 1 6 = 2.45 233 18

-+

233 18

-+ =

233 2 9

-+

=233 23

-+ ,

232 3

++

= 236 3 6 63

-+++

= 4 6 + 9 = 4 2.45 + 9 = 18.8

a a 0

(14)______________________________ 20062006

Ex 2 y 24y 4y-3 = 8y+1 24y (22)y-3 = (23)y+1

26y-6 = 23y+36y - 6 = 3y + 3

3y = 9y = 3

Ex 3 2x = 3y = 12z z *1) 2y x

xy+ 2) 2y x

xy- 3) x2y

xy- 4) 2x y

xy+

2x = 3y = 12z = (22 3)z = 22z 3z 3y = 2x

(3y)z/y = (2x)z/y

3z = 2xz/y 2x = 22z 2xz/y = 22z+xz/y

2z + yxz = x

z(2y + x) = xyz = 2yx

xy+

1. a, b x2 - x - 3 = 0 a2 + b2 1) 7 2) 6 3) 5 4) 4

2. ax2 - 4x + c = 0 ac 1) 2 2) 3 3) 4 4) 6

3. 1 x 2x x x23+

-- = 0 1) -1 2) 1 3) 2 4) -2

4. 1 |x| 2 3x -- 2 1) (-1, 0]U (1, ) 2) (-2, 0]U (1, ) 3) (1, ) 4) (2, )

5. 1 x x2+ > x

1) (-, -2) 2) (-10, -1) 3) (-2, 1) 4) (1, )

2006 ______________________________ (15)2006

6. A 3x + 5 < x 7B x1 < 21 AI B

1) (-6, 0) 2) (2, ) 3) (-, 0)U (2, ) 4) [-6, 0)U (2, )7. 3 |x + 1| 7

1) (-10, 4) 2) (-9, -2)U (1, 7) 3) (-5, 8) 4) (-10, -3)U (3, 8)8. 2x + 2x+1 + 2x+2 + 2x+3 = 120 2x + x

1) 7 2) 9 3) 11 4) 13

9. x18 - x + 3 = 0 1) 7 2) 9 3) 13 4) 15

10. 56 14 2

52 6

--+

1) 25 3 - 2) 2

5 3 + 3) 252 3 + 4) 2

5 32 +

11. 4110x x 2 +- - 4110x x 2 ++ = 8 3x + 25 1) 5 2) -3 3) 1 4) -1

1. 1) 2. 3) 3. 3) 4. 1) 5. 1) 6. 4) 7. 2) 8. 3) 9. 2) 10. 2)11. 1)

1. (a, b) (a, b) = (c, d) a = c b = d

2. A BA B = {(x, y)|x A y B}

(16)______________________________ 20062006

3. A B(r : A B) A B Dr = {x|(x, y) r} Rr = {y|(x, y) r} r = {(x, y)|y = r(x)} r-1 = {(x, y)|x = r(y)} r-1 = {(y, x)|y = r(x)} Dr = 1rR- Rr = 1rD-

4. 1. f (x, y) f (x, z) f y = z2. A B(f : A B)

f A B Df = A, Rf B3. f 1-1 f(x1) = f(x2) x1 = x24. A B

f A B Df = A Rf = B5. f R R f x1, x2 Df x1 < x2

f(x1) < f(x2)y

x

)f(x2)f(x1

1x 2x

6. f R R f x1, x2 Df x1 < x2 f(x1) < f(x2)

y

x)f(x2

)f(x1

1x 2x

2006 ______________________________ (17)2006

1. y = ax + b a b a 0 a = 0 y = b y = ax + b a y b x - ab

2. y = ax2 + bx + c a, b, c a 0 2

1. a > 0

4a b 4ac ,2a

b 2--

ymin = 4a b 4ac2-

2. a < 0

4a b 4ac ,2a

b 2--

ymax = 4a b 4ac2-

3. y = ax , a > 0 a 1 2

1. a > 1 y = 2x

x y (0, 1)

2. 0 < a < 1 y = x21

x y (0, 1)

4a b 4ac ,2a

b 2--

4a b 4ac ,2a

b 2--

y

x(0, 1)

0

y

x(0, 1)

0

(18)______________________________ 20062006

4. y = |x - a| + c a c (a, c)

* a > 0, c > 0 y (0, a + c)

5.

f(x) =