ตรรกศาสตร์

1.
2. x + 5 = 5 + x
3.
4.
5.

57
2+3= 4

3 6,7,8,2,1, 5

23 = 8
x + y = 16

1

ชอ่ื .....................................สกลุ ...........................
ชัน้ ....................................เลขที.่ .........................

(8 + 22)3 (10)2

2 x2 − x = 6

7 1,3,5,7,9

 +2

2

pq pq pq

TT T
TF F
FT F
FF F

p
q

pq

p
q

pq

p
q

pq

p
q

pq

3

pq pq pq

TT T
TF T
FT T
FF F

p
q

pq

p
q

pq

p
q

pq

p
q

pq

-

-

4

p q pq pq

pq → p→q
pq
T
TT F
TF T
FT T
FF

2 +3 = 3+ 2 3(2 + 3) = 3(3 + 2)

p 2+3=3+2

q 3(2 + 3) = 3(3 + 2)

p→q 2 +3 = 3+ 2 3(2 + 3) = 3(3 + 2)

3 23 23
p 3
q 23
3
p→q

5

23 3 3
p 23
3
q 23
p→q

p
q

p→q



pq pq pq

TT T
TF F
FT F
FF T

p
q

pq

22 = (− 2)2 2 = −2

p 22 = (− 2)2

6

q 2 = −2

pq 22 = (− 2)2 2 = −2

23 11
23

p 23

pq q 11
23 23

11
23

p
q

pq

p

p ~p

TF
FT

p 5+2=7
~p 5+27

p
~p

7

ช่อื .....................................สกุล...........................
ชั้น....................................เลขท.่ี .........................

20 = 1 22  2k + 1 k = 1,2,3,...

37 73



8

√9 ≠ 3
9

ลำดบั ขั้นการหาค่าความจริงของประพจนท์ ี่มตี วั เชื่อมตง้ั แต่สองตัวข้ึนไป
1. หาคา่ ความจรงิ ของประพจนย์ ่อยในวงเลบ็ ก่อน (ไมม่ ีวงเลบ็ ข้ามไปทขี่ ้อ 2)
2. หาคา่ ความจรงิ ของตัวเชอ่ื ม “~”
3. หาค่าความจริงของตวั เชื่อม “ v ”,“ Λ ”
4. หาคา่ ความจรงิ ของตวั เชอื่ ม “→”

5. หาค่าความจรงิ ของตวั เชอ่ื ม “  ”

8 + 4 =10 34 9

7 + 3 =10 p 8 + 4 =10
q 34 9
r 7 + 3 =10

pq→r

8 + 4 =10 34 9 7 + 3 =10

a,b c (a  b)  c

a
b
c

(a  b)  c

10

(~ p  q)  (p  q) p→q

p→q
p
q

(~ p  q)  (p  q)

(~ p  q)  (p  q)

pq

(~ p →~ q) → (p  q)

pq
p
q

(~ p →~ q) → (p  q)

(~ p →~ q) → (p  q)
~ (p → q) s  t

(p  q) (s  t) → t
~ (p → q)
(p → q)

11

p
q
st

s
t

(p  q) (s  t) → t

(p  q) (s  t) → t

( p → q)  (p  r) → (s → r)

p, q, r s

( p → q)  (p  r) → (s → r)

p
q
r
s

12

(~ q ~ s)  ( p → q)
p, q s

(~ q ~ s)  ( p → q)

p
q
s

p  (~ q → r)→ (~ s → r)

p, q, r s

p  (~ q → r) → (~ s → r)

p
q
r
s

13

ช่อื .....................................สกลุ ...........................
ช้นั ....................................เลขที่..........................

p, q, r, s t

~ (p ~ q)  t

(s ~ p)  (q →~ r)
(p  q)  (t  s)  (q → r) → s

14

pq

~ (p → q) → (p ~ q)

pq qr

(p  q)  (r ~ p)

( p → q)  (q  p)

pq

15

pq p q
pq
pq p→q

pq ~p

p
p

p

T
F

pq pq

pq

TT
TF
FT
FF

16

pq r p,q r

T pq r
T
TTT
F TTF
TFT
T TFF
F FTT
FTF
F FFT
FFF

n
n 2n

(p  q) →~ p pq p

q (p  q) →~ p

p q p  q ~ p (p  q) →~ p

17

( p → q) ~ (q → p) pq

( p → q) ~ (q → p) p

q

p q p → q q → p ~ (q → p) ( p → q) ~ (q → p)

TT T T F F

TF F T F F

FT T F T T

FF T T F F

( p  q)  (q → r) p,q r

p,q r ( p  q)  (q → r)

p q r p  q q → r ( p  q)  (q → r)

TTT T T T

TTF T F F

TFT F T F

TFF F T F

FTT F T F

FTF F F T

FFT F T F

FFF F T F

18

ช่อื .....................................สกุล...........................
ชน้ั ....................................เลขที.่ .........................

pq r s

p → (~ p → q)

(q ~ p)  r

(p → q)  (p  s) → (r → s)

19

≡ pq p ~ q ~ (~ p  q)
~ (~ p  q)
pq p ~ q ~ (~ p  q)

~ p ~ q p ~ q ~ p  q

p ~ q ~ (~ p  q)

p ~ q ~ (~ p  q) p ~ q ~ (~ p  q)

~ (p → q) ~ p →~ q

~ ( p → q) ~ p →~ q

~ ( p → q) ~ p →~ q

p q ~ p ~ q p → q ~ ( p → q) ~ p →~ q

TTF F T F T

TFF T F T T

FTT F T F F

FFT T T F T

~ (p → q) ~ p →~ q

~ (p → q) ~ p →~ q

20

pq ~ (~ p  q) p ~ q

~ (~ p  q) ~ (~ p) ~ q ~ (~ p  q)  p ~ q

~ (~ p  q)  p ~ q ~ (p ~ q) ~ pq
p ~ q
pq ~ (p ~ q) ~ p  q
~ p ~ (~ q)
~ (p ~ q)
~ p  q
~ (p ~ q) ~ pq

รปู แบบประพจนท์ ีส่ มมลู กันทีค่ วรทราบ

1. ~ (~ p)  p
2. p  q  q  p
3. p  q  q  p
4. p  q  q  p
5. p  (q  r)  ( p  q)  r

 pqr

6. p  (q  r)  ( p  q) r

 pqr

7. p  (q  r)  ( p  q) ( p  r)
8. p  (q  r)  ( p  q)  ( p  r)
9. p → q ~ q →~ p

~ p  q

10. p  q  ~p  ~q
 ( p → q)  (q → p)

11. ~ ( p  q) ~ p ~ q
12. ~ ( p  q) ~ p ~ q
13. ~ (p → q)  p ~ q
14. ~ (p  q)  p ~ q

 ~p  q

15. p  p  p
16. p  p  p

21

ช่อื .....................................สกลุ ...........................
ชัน้ ....................................เลขที.่ .........................

p →~ (q → p) ~ p  (q ~ p)
~ p  (~ q  p) ~ p  (q → r)

pq→r

~ p  (q → r)

( p → r)  (q → r) ~(p  q) r
~ (p  q)→ r (~q → ~p)  (~q  p)

pq

( p → q)  (q  ~p)

p → (q  r) (~ r  p) → q

(~p → ~r )  (~p  r) r  p
p →~ (q  r) ~ p  (q ~ r)
(p  q) → r ( p → r)  (q → r)
~ (p ~ q) ~ p  q

22

pq (p → q) p ~ q

(p → q) p ~ q

p q ~ q (p → q) p ~ q

(p → q) p ~ q

(p → q) p ~ q pq ~ p ~ q
q
p

pq ~ p ~ q

p q ~ p ~q pq ~ p ~ q

pq ~ p ~ q

~ pq p ~ q

23

ชอื่ .....................................สกุล...........................
ชั้น....................................เลขที.่ .........................

pq ~ p ~ q

p→q ~ p →~ q

pq ( p  ~q) (q  ~p)

24

p q

( p → q)  p → q

( p → q)  p → q

p q p → q ( p → q)  p ( p → q)  p → q

TTT T T

TFF F T

FTT F T

FFT F T

( p → q)  p → q
( p → q)  p → q

pq

(~p  ~q)  ~( p → q)

(~p  ~q)  ~( p → q)

p q ~ p ~ q ~ p ~ q ( p → q) ~ ( p → q) (~ p ~ q) ~ ( p → q)

TT F F F T F F

TF F T F F T T

FT T F F T F F

FF T T T T F T

(~p  ~q)  ~( p → q)
(~p  ~q)  ~( p → q)

25

pq

(~ p → q) → ( p ~ q)

(~ p → q) → ( p ~ q)

p q ~ p ~ q ~ p → q p ~ q (~ p → q) → ( p ~ q)

TT F F T T T
TF F T T T T
FT T F T F F
FF T T F T T

(~ p → q) → ( p ~ q)
(~ p → q) → ( p ~ q)

p q

(p  q)→ (p → q) (p  q)→ (p → q)
(p  q)→ (p → q)

ขดั แยง้

q

(p  q)→ (p → q)

26

pq

(~ p → q) → ( p ~ q)
(~ p → q) → ( p ~ q)

(~ p → q) → ( p ~ q)

(~ p → q) → ( p ~ q)

(~ p → q) → ( p ~ q)

p q

p  (~p  q) → q p  (~p  q) → q
p  (~p  q) → q

ขัดแยง้ กัน

p

p  (~p  q) → q

27

ช่อื .....................................สกลุ ...........................
ชนั้ ....................................เลขท.่ี .........................

(~ p  q) →~ ( p  q)

(p → q)  (p → r)  p → (q  r)

( p  q)  ~p → q
(~p → ~q) → ( p → q)

28

P1, P2 ,...,Pn

P1, P2 ,...,Pn
C

P1, P2 ,...,Pn C
P1, P2 ,...,Pn
Λ
Λ
→C

(P1  P2  P3  ...  Pn ) → C

pq
p→q
p
q

Λ→

( p → q)  p → q

29

( p → q)  p → q
( p → q)  p → q

ขดั แยง้

( p → q)  p → q

p, q r
p→q
~q→r
~r
~p
Λ→

( p → q)  (~ q → r)  (~ r) →~ p
( p → q)  (~ q → r)  (~ r) →~ p
( p → q)  (~ q → r)  (~ r) →~ p

( p → q)  (~ q → r)  (~ r) →~ p

30

p
q
r

p→q
~q→r
~r
~p

( p → q)  (~q → r)  (~r ) → ~p

( p → q)  (~q → r)  (~r ) → ~p
( p → q)  (~ q → r)  (~ r) →~ p

( p → q)  (~q → r)  (~r ) → ~p

x

P(x) "","","→","" "~"
P(x)
= 0,1,2,3,4,5 P(x) 2x −1 = 5

x=3 x=5
2x −1 = 5
P(x)

x = 3 P(3) 2(3) −1 = 5

31

x = 5 P(5) 2(5) −1 = 5

P(x) x = 3 x = 5

= − 2,−1,0,1,2 P(x) x2  2x

P(− 2) P(2) x2  2x
P(x)
(− 2)2  2(− 2)
P(− 2)

P(2) 22  2(2)

P(− 2) P(2)

32

ช่อื .....................................สกลุ ...........................
ชั้น....................................เลขท.ี่ .........................

• x
•
•
• x + 7 = 6 − 2x
•
•x

= 1,2,5,6,7 P(x) 1 x − x2  6
2

P(2) → P(5)

P(x) x2 − 2x + 2 = 5

P(3)  P(2)

33

 x x
 x x

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

x

xx + x = 2x

 x (x +1)2 = x2 +1

xx + 0 = 2x

xx  I → x  R

xx  N 

xy x+ y =0

xy xy = yx

34

x y y
xy 5
y
x y x=y
y yx
x x+ yQ

x xy  0

x0 y0

xyx + y = 0
xyxy = yx
xyxy = y
xyx + y = 5

xy   x = y
Q y 
x 


x  Iyx + y  Q

x  Ry  Nxy  0 → x  0  y  0

xx  I  x  5 5
xx  I → x  5 5

 yx x2 + y2 = 8 x x2 + y2 = 8
 yx x2 + y2 = 8 x x2 + y2 = 8

xy  Qx + y  R y y2 = x

 x  Ny y2 = x

x

x

y

y

xy

x

35

ชือ่ .....................................สกุล...........................
ชั้น....................................เลขที่..........................

x x +11
x x2 = 2

x y y x2y = x2
xy x+ y 5

x x− y =0

 x x  2 → x2  4

y2 y + 1 = 0

 x x  Q → x2 = 2

yxx − y = 0

 x  Ry  N x2 + y2 = 9

36

xP(x) x P(x)

xP(x) x P(x)

xP(x) x P(x)

xP(x) x P(x)

= −1,0,1

x(x  0) → (x2  0)
 xx  0 → x x2  0

x(x  0) → (x2  0) x2  0

P(x) x  0 Q(x)

x −1

P(− 1) −1 0

Q(− 1) (−1)2  0

P(−1) → Q(−1)

x0

P(0) 0  0
Q(0) 02  0
P(0) → Q(0)

x1

P(1) 1  0
Q(1) 12  0
P(1) → Q(1)

x(x  0) → (x2  0)

 xx  0 → x x2  0

P(x) x  0

P(− 1) −1 0

P(0) 0  0

37

P(1) 1  0

xP(x)

Q(x) x2  0

Q(− 1) (−1)2  0

Q(0) 02  0

Q(1) 12  0

xQ(x)

 xx  0 → x x2  0

= −1,0,1

x(x  0)  (x −1 = 0) x −1= 0
xx  0 xx −1 = 0

x(x  0)  (x −1 = 0)
P(x) x  0 Q(x)

x −1

P(− 1) −1 0

Q(− 1) −1−1= 0

P(−1)  Q(−1)

x0

P(0) 0  0
Q(0) 0 −1 = 0
P(0)  Q(0)

x1

P(1) 1  0
Q(1) 1−1 = 0
P(1)  Q(1)
x(x  0)  (x −1 = 0)

xx  0 xx −1 = 0

P(x) x  0

P(− 1) −1 0

P(0) 0  0

P(1) 1  0

xP(x)

Q(x) x2  0

38

Q(− 1) −1−1= 0

Q(0) 0 −1 = 0

Q(1) 1−1 = 0

xQ(x)

xx  0 xx −1 = 0

 x x Q  x  2

 x x2  9 → x  3

xP(x)

x xQ x  2

 x x Q  x  2

xQ x

x 2 x 2

x= 3

3

3 2

xQ x  2

x xQ x  2

 x x Q  x  2

xP(x)

x x2  9 → x  3

x2  9 x3

 x x2  9 → x  3

x = −4

(− 4)2  9

−43

x2  9 → x  3

x xQ x  2

 x x2  9 → x  3

39

ช่อื .....................................สกลุ ...........................
ชน้ั ....................................เลขท่.ี .........................

xx + 1 = 4 = 1,2,3,4

xx + x = 2x = − 2,−1,0,1,2

xx + 1  x

x x

x(x −1)(x +1) = x2 −1 = − 2,1,3,7

 x 2x2 + 3x +1 = 0 = − 2,1,3,7

40

x x = x =R

 xx  0  x x2  5 =R

x x  x x = 0,1,2

=R

P(x) x
Q(x) x

xQ(x) → P(x)

xP(x)  Q(x)

41

P(x),Q(x) R(x)

pqq p P(x) Q(x)  Q(x) P(x)
pqq p P(x) Q(x)  Q(x) P(x)
P(x) → Q(x) ~ Q(x) →~ P(x)
p → q ~ q →~ p
~ p  q ~ P(x) Q(x)

pqq p P(x)  Q(x)  Q(x)  P(x)
 (P(x) → Q(x))  (Q(x) → P(x))
 ( p → q) (q → p)
~ (p  q) ~ p ~ q ~ (P(x)  Q(x)) ~ P(x) ~ Q(x)
~ (p  q) ~ p ~ q ~ (P(x)  Q(x)) ~ P(x) ~ Q(x)
~ (p → q)  p ~ q ~ (P(x)  Q(x)) ~ P(x) ~ Q(x)
p  (q  r)  ( p  q) ( p  r) P(x)  (Q(x) R(x))  (P(x)  Q(x)) (P(x)  R(x))

xP(x)  Q(x)  xQ(x)  P(x)
x~ (P(x)  Q(x))  x~ P(x) ~ Q(x)
x~ (P(x) → Q(x))  xP(x) ~ Q(x)

x~ (P(x) Q(x)) x~ P(x) ~ Q(x)

x(P(x) → Q(x)) → R(x) x(P(x) ~ Q(x)) R(x)

x~ (P(x) Q(x))

~ (P(x)  Q(x)) ~ P(x) ~ Q(x)

x~ (P(x) Q(x))  x~ P(x) ~ Q(x)

x(P(x) → Q(x)) → R(x)

(P(x) → Q(x)) → R(x) ~ (P(x) → Q(x)) R(x)
 (P(x) ~ Q(x)) R(x)

x(P(x) → Q(x)) → R(x)  x(P(x) ~ Q(x)) R(x)

42

x~ P(x) xQ(x) xQ(x) x~ P(x)

xP(x) → xQ(x) ~ xQ(x) →~ xP(x)

x~ P(x) xQ(x)

pqq p

x~ P(x) xQ(x)  xQ(x) x~ P(x)

xP(x) → xQ(x)

p → q ~ q →~ p

xP(x) → xQ(x) ~ xQ(x) →~ xP(x)

43

ชอื่ .....................................สกุล...........................
ชนั้ ....................................เลขท.ี่ .........................

~ xP(x) xQ(x) ~ xP(x) ~ xQ(x)

xP(x) xQ(x) xR(x) xP(x) xQ(x) xP(x) xR(x)

x~ (P(x) → Q(x)) → Q(x) x~ P(x) Q(x)

 x x  0 → x2  0  x x  0  x2  0

44

xP(x) xP(x)
x~ P(x)
~ xP(x) xP(x)
xP(x)
~ xP(x) x~ P(x) x~ P(x)

xP(x)

~ xP(x) xP(x)

~ xP(x) x~ P(x)

xP(x)

xP(x)

x~ P(x)

~ xP(x) สมมูลกับ x~ P(x)

x x2  0
x2  0
xx + 5 = 7

x

xx + 5  7

xP(x) xP(x)
~ xP(x) สมมูลกับ x~ P(x) x~ P(x)

45

x x +1 5

 x x = 2 2 +1

 x x2 + x = 5

x x +1 5

 x x  2 2 +1

 x x2 + x  5

x(x + 3 = 4) → (x2 = 9) ~ (p → q)  x x + 3 = 4  x2  9
~ (p  q)
x(x = 5) (x =  ) x(x  5)  (x   )
xP(x) xQ(x) pq
xP(x) xQ(x) ~ p ~ q
~ xP(x) → xQ(x) pq
~ p ~ q
x~ (x + 3 = 4) → (x2 = 9) ~ (p → q)
p→q
x~ (x = 5)  (x =  )
~ xP(x) ~ xQ(x)

x~ P(x) x~ Q(x)
~ xP(x) ~ xQ(x)

x~ P(x) x~ Q(x)
~ xP(x) → xQ(x)

xP(x) → xQ(x)

46

ช่อื .....................................สกุล...........................
ช้นั ....................................เลขท.่ี .........................

xx + 2  0

xx  0 → xx  0

 x x2  0 → x  0

xP(x) ~ Q(x)

47

xyP(x, y) xyP(x, y) xy a

b P(a, b ) xy a

xyP(x, y) P(a, b )
= 0,1,2
b

xyx + y  xy

x = 0, y = 1

x+ y =1 xy = 0

x + y  xy x + y  xy
xy

xyx + y  xy

 xy (x + y)2 = x2 + 2xy + y2

(x + y)2 = x2 + 2xy + y2 (x + y)2

 xy (x + y)2 = x2 + 2xy + y2 xy

xyP(x, y)

xyP(x, y) xy a

b P(a, b )

xyP(x, y) xy a

b P(a, b )

= 2,3 xy  x =  
 y 
 

xy  x =  
 y 
 

x y

P(x, y) x =
y

48

x y x =
y
P(2,2) 2 =
P(2,3) 2  xy x2 + y2 = 9
P(3,2) 2 =
P(3,3) 3
3 =
x 2
3 =
3

y

xy  x =  
 y 

 xy x2 + y2 = 9
= 0,2,3

x = 0, y = 3 x2 + y2 = 9
02 + 32 = 9
0+9 = 9 x
x
xy

 xy x2 + y2 = 9

xyP(x, y) xyP(x, y) a
xyP(x, y) yP(a, y) a
yP(a, y)

  
 




49

xyP(x, y) xyP(x, y) xa
xyP(x, y) xa
yP(a, y)
xyx + y = y
yP(a, y)
= 0,1,2,3

x

xy P(x, y)

x=0

y=0 0+0=0

x=0 y =1 0 +1 = 1
y=2 0+2 = 2

y=3 0+3=3

x=0 x+y= y y

xyx + y = y  yx y − x2  5

= I−

y x
y  5+ x2
P(x, y)

y − x2  5

y  5+ x2

yI− y

5+ x2

y x

 yx y − x2  5

50