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รายวชิ า ค31102 คณติ ศาสตร์ 2

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ช้ันมัธยมศึกษาปี ท่ี 4

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1. (a, b) (b, a) ab bd
2. (a, b) (c, d) ac bd
3. (a, b) (c, d) ac

2. 1

1 A 1, 7 B 0, 6

1) A B
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B
1) A
A
1, 0 ,(1, 6),(7, 0),(7, 6) B

2) (6, 1)

3. 1
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2 (a, b) (4, 5) ab

(a, b) (4, 5) a4 b5
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6.

3 (x 2, 3) (5, y 1) x y

(x 2, 3) (5, y 1)

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8. 1 1 2
9. 1 1, 0 ,(1, 6),(7, 0),(7, 6) A

AB (a, b)

BB

aA bB A B
B B
A

10. 4

4A 1, 3, 5 B 2, 4, 6, 8
1) B
B
2) B A
3) A A

1)

11

A B= 1,2 ,(1, 4),(1, 6),(1, 8),
3,2 ,(3, 4),(3, 6),(3, 8),
5,2 ,(5, 4),(5, 6),(5, 8)

2) B A= (2,1),( 2, 3)(2, 5),(4,1),(4, 3),(4, 5),
(6,1),(6, 3),(6, 5),(8,1),(8, 3),(8, 5)

3) A A 1,1 ,(1, 3),(1, 5),
3,1 ,(3, 3),(3, 5),
5,1 ,(5, 3),(5, 5)

11. 4
12. A B B A

AB

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

13. 4
14. 5

5A 1, 3, 5 B 2, 4, 6, 8

1) n(A B)
2) n(B A)
3) n(A A)

1) n(A B) n(A) n(B)
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2) n(B A) n(B) n(A)
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3) n(A A) n(A) n(A)
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Ordered Pairs

( , ), , ), ( , )

(, )

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(a, b) a b

1. (a, b) (b, a) ab bd
2. (a, b) (c, d) ac bd
3. (a, b) (c, d) ac

1 A 1, 7 B 0, 6
1)
2) A B
B A

2 (a, b) (4, 5) ab

14

3 (x 2, 3) (5, y 1) xy

(Cartesian product)

A a, b, c B x, y

A B

(a, x),(a, y),(b, x),(b, y),(c, x),(c, y)

A

B AB

A B (a, b)
B AB
aA bB

A

A B (a, b) a A b B

B A (b, a) b B a A

4A 1, 3, 5 B 2, 4, 6, 8

1) A B
2) B A
3) A A

15

AB

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

5A 1, 3, 5 B 2, 4, 6, 8

1) n(A B) 2) n(B A) 3) n(A A)

.................... 16
......... .........
1 A
1) A 1,2, 3 B

B 1, 0,1

2) A 7,1, 5 B 2, 3

2. x y
1) (x, 3) (2, y)

2) (x y,1) (5, x y)

3) (x 1, y 2) (5, 3)

17

3. A B B A AB

1) A a, b B c, d, e

2) A 3, 2, 1 B 2, 3

3) A x I x2 9 0 B x I x2 1 0

4) A 1,2, 3 B x I x2 4

4. A 1,2, 3,...,10 B a, b, c
1) A B

2) B A

3) A A

4) B B

1. A 18
B
1) A 1,2, 3 B 1, 0,1
B
A
B
(1, 1),(1, 0),(1,1),(2, 0),(2,1),(3, 1)(3, 0) (3, 1)

2) A 7,1, 5 B 2, 3

( 7, 2),( 7, 3),(1, 2),(1, 3),(5, 2) A
(5, 3)

2. x y x2 3 y
1) (x, 3) (2, y)

(x, 3) (2, y)

(x, y) (2, 3)

2) (x y,1) (5, x y)

(x y,1) (5, x y) (1)
xy 5

xy 1 (2)

(1) (2); 2y 4

y2

y2 (2) x 3

(x, y) (3,2)

3) (x 1, y 2) (5, 3)

(x 1, y 2) (5, 3) 5
x1 6
x

y2 3

y1

(x, y) (6,1)

19

3. A B B A AB

1) A a, b B c, d, e
(a, c),(a, d),(a, e),(b, c),(b, d),(b, e)
AB

B A (c, a),(c, b),(d, a),(d, b),(e, a),(e, b)

2) A 3, 2, 1 B 2, 3

A B ( 3, 2),( 3, 3),( 2, 2),( 2, 3),( 3, 2),( 3, 3)

B A (2, 3),(2, 2),(2, 1),(3, 3),(3, 2),(3, 1)

3) A x I x2 90 B x R x2 1 0
1
A x2 9 0

B x2 9
x2 3, 3

A 3, 3
x2 1 0

x2 1 x2
B

AB BA

4) A 1, 2, 3 B x I x2 4

B x2 4

x 2, 2

B 2, 2

A B (1, 2),(2, 2),(3, 2),(1, 2),(2, 2),(3, 2)

B A ( 2,1),( 2, 2),( 2, 3),(2,1),(2, 2),(2, 3)

20

4. A 1,2, 3,...,10 B a, b, c
1) A B

n(A B)=n(A) n(B)
=10 3
30

2) B A

n(B A)=n(B) n(A)
=10 3
30

3) A A

n(A A)=n(A) n(A)
=10 10
= 100

4) B B

n(B B)=n(B) n(B)
=3 3
=9

21

12

2 31102 4/4

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(K) ( 60%) ......
2) 35 ..... 35 ..... 100

(P)

1) 35 ..... 35 ..... 100 ......

2)

(A) 35 ..... 35 ..... 100 ......
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(K) ( 60%) ......
2) 38 ..... 38 ..... 100

(P)

1) 38 ..... 38 ..... 100 ......

2)

(A) 38 ..... 38 ..... 100 ......
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(K) ( 60%) ......
2) 26 ..... 26 ..... 100

(P)

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(A) 26 ..... 26 ..... 100 ......
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1.2

4.1

2

AB

r AB r AB

r AA r A

3
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1.

:
1.
2.
3.
4.
5.

:
1.

4

30
5

2

2
2

2

6. 2 22
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1

1 A 1, 3, 5 B 2, 4, 6, 8 AB

A B = 1,2 ,(1, 4),(1, 6),(1, 8), 3,2 ,(3, 4),(3, 6),(3, 8), 5,2 ,(5, 4),(5, 6),(5, 8)

r 1,2 ,(1, 8) 3,2 ,(3, 4),(5, 4),(5, 6) r AB r

AB

1.

AB AB r AB A
r AA
r
r

2. 2

2 A 1,2,3 , B 4,5,6

1) r1 1, 4 ,(2,5),(3,6)
2) r2 1,1 ,(2, 2),(3,3)
3) r3 4,6 ,(5,6)
4) r4 4,1 ,(6,3)
1) r1 1, 4 ,(2,5),(3,6)
r1
AB

32

2) r2 1,1 ,(2, 2),(3,3) A
3) r3 r2 B
4) r4 4,6 ,(5,6) BA
r3
4,1 ,(6,3)
r4

3. 2
4. 3

3 A 2,4 B 1,3,6,8 r

1) A B
2) B A
3) A B
4) A A

r
1) A B 2,1 ,(2,3),(2,6),(2,8),(4,1),(4,3),(4,6),(4,8)

r1 A B

r1 (2,1),(4,1),(4,3) r1 A B

r1 (x, y) A B x y

2) BA 1, 2 ,(1, 4),(3, 2),(3, 4),(6, 2),(6, 4),(8, 2),(8, 4)

r2 BA

3) r3 r2 (1, 2),(1, 4),(3, 4) r2 B A

r2 (x, y) B A x y B

A

r3
r3 (x, y) A B x2 y

4) r4 A A

r4 (4, 2) r4 A A

r3 (x, y) A A x 2y

5. 3 33
6.
r
r = (x, y) y x2 5
mb
r = (m, b) b 5m () m
b 5m b
(m, b)
b 5m r = (m, b) b 5m (1, 5)
(2, 10)
m b (5, 25)
(8, 40)
1 5 (10, 50)
2 10
5 25 )
8 40
10 50

7. 4 x 0}
4 r = {(x, y) y 1 x
(y
5

(x )

x y (x, y)
(0, 0)
() ()
0 0 (1, 1)
5
1 1
5 (2, 25)
2 2 (5, 1)
5
5 (10, 2)
10 1
15 (15, 3)
2

3

34

r = {(x, y) y x x 0}
1x
5

8. 4
9. 5

5

r = ( 2, 2),( 1, 1),(0, 0),(1,1), 2, 2

xy 10. 6
-2 -4 5
-1 -2 r = (x, y) I I y 2x
00 11.
12
24 6

35

12. 6
13. 7

7 r = (x, y) R R y 2x

xy
-2 -4
-1 -2
00
12
24

14. 7
15. 8

36

8 r = (x, y) y x2

16. 8
17. 9

9 r = (x, y) 1 x 4

18. 9

4.
5.

6. 2( )

37
9

2 2- 2

22

2

-

-
-

-

38

AB AB r AB A
r AA
r
r

1 A 1,2,3 , B 4,5,6

1) r1 1, 4 ,(2,5),(3,6)
2) r2 1,1 ,(2, 2),(3,3)
3) r3 4,6 ,(5,6)
4) r4 4,1 ,(6,3)

39

2 A 2,4 B 1,3,6,8 r

1) AB
2)
3) BA
4)
AB
r
AA

r = (x, y) y x2

5

r = (m, b) b 5m () m
b 5m b

b 5m r = (m, b) b 5m 40

mb r

(x, y)

3 r = {(x, y) y 1 x x 0}

5

(x ) (y )

.

.

41

4 r = ( 2, 2),( 1, 1),(0, 0),(1,1), 2, 2

5 r = (x, y) I I y 2x

6 r = (x, y) R R y 2x

..

42

7 r = (x, y) y x2

8 r = (x, y) 1 x 4

.................... 2 43
.........
.........

1. A 1, 3, 6, 9 B 2, 4, 6, 8

1) r1 A B
2) r2 A B
3) r3 A B
4) r4 B A

2.

1) r1 = {(2, 6), (2, 5), (2, 4), (3, 6), (3, 7)}

2) r2 = {(x, y) A B y x }, A={1, 2, 3}, B = {1, 2}

3) r3 = {(x, y) A A x+ y = 4 } A={1, 2, 3, 4, 5}

3.

1) r1 = {(x, y) R R y = x }
2) r2 = {(x, y) R R y = 2x - 1}

3) r3 = {(x, y) R R y = 1 x2 }
2

4) r4 = (x, y) 1 y 4

44

2

1. A 1, 3, 6, 9 B 2, 4, 6, 8

1) r1 AB

r1 r1 y 2, 4, 6, 8
(x, y) A B x B
r1
A 1, 3, 6, 9
2) r2
r1
r2 (3, 2),(6, 2),(6, 4),(9, 2),(9, 4),(9, 6),(9, 8)

r2 AB

3) r3 r2
(x, y) A B x y
r3
A 1, 3, 6, 9 B 2, 4, 6, 8
r3
r2
4) r4 (1, 2),(1, 4),(1, 6),(1, 8),(3, 6),(3, 8),(6, 8)

r4 r3 AB
(x, y) A B x y}
r4
A 1, 3, 6, 9 B 2, 4, 6, 8

r3
(1, 2),(1, 4),(1, 6),(1, 8),(3, 6),(6, 6)

r4 BA
(x, y) B A x
y}
A 1, 3, 6, 9
B 2, 4, 6, 8

r4

(2, 6)(6, 6)

45

2.
1) r1 = {(2, 6), (2, 5), (2, 4), (3, 6), (3, 7)}

r1 = {(2, 6), (2, 5), (2, 4), (3, 6), (3, 7)}

2) r2 = {(x, y) A B y x }, A={1, 2, 3}, B = {1, 2}

r2 = {(1, 1), (1, 2), (2, 2)}

46

3) r3 = {(x, y) A A x+ y = 4 } A={1, 2, 3, 4, 5}

r3 = {(1, 3), (2, 2), (3, 1)}

3.
1) r1 = {(x, y) R R y = x }

xy

--
-

47

2) r2 = {(x, y) R R y = 2x - 1}

xy

--
--

-

3) r3 = {(x, y) R R y = 1 x2 }
2

xy

-
-

/2

Y

X

48

4) r4 = (x, y) 1 y 4

49

22

2 31102 4/4

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9-10 2564 8.30-9.20/10.10-11.00 . 1/3

39 7

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