1. The document defines propositional logic concepts such as propositions, truth values, connectives like conjunction (∧), disjunction (∨), implication (→), biconditional (↔), and negation (~). 2. Examples of well-formed formulas are provided using variables like p, q, and connectives. Truth tables are used to evaluate formulas. 3. Equivalences between logical formulas are defined, such as De Morgan's laws, double negation, absorption, implication, and biconditional identities.

1. F 31201 ก 4 WiLa 1
1
ก F F
1.1 F (Propositions or Statements)
F F ก F
F F F F
F กF F (truth value) F
F ก F T F
F F
ก F ก F F F ก ก F a, b, c, , z
F F
1. ก F ( )
2. -5 ( )
3. 0 F ( )
F F F F
1 F กF F F ก F F
F F F F
2 ก F F F F ก F F
F ก F FF
F F 2
ก กF F
x+2=5
F x F F F
F 1 F F F F
1. กF F F ......................................
2. 5+ 6 = 12 ..................................
3. ก 49 7 -7 ......................................
4. x+y+5 =0 ...................................
5. F F .....................................
6. a+0 = a .................................

2. F 31201 ก 4 WiLa 2
ก 1
F F F F
1. 0 F
2. F F
3. A ∩ B = B ∩
4. F
5. 1+5 = 8
6. x + 7 = 8
7. ก
8. ก F F
9. F F Fก 4
10. ก F ก ก ก F
11. F
12. ก F F ก
13. {1,2,3} = {2,3,4}
14. π ก
15. {0} F
16. x x>5
17. F ก F
18. ก ก 4
19. x y F x+y=y+x
20. F

3. F 31201 ก 4 WiLa 3
1.2 ก F
F F กก F F
F ... F ... ก F ก
F F F กF (connectives)
F 0 F F
2 4 F
F 3 F 32
1. ก F F
F 5+ 2 = 2 + 5
3 x 1 = 1x 3
F F F F F
5+ 2 = 2 + 5 3 x 1 = 1x 3
p q F p∧q F (truth table) p∧q F
q p∧q
P
T T T
T F F
F T F
F F F
2. ก F F
F 1+8=8+1
5( 3 + 7 ) = ( 5 x 3 ) + ( 5 x 7 )
F F F F F
1+8=8+1 5( 3 + 7 ) = ( 5 x 3 ) + ( 5 x 7 )
p q F p∨q F (truth table) p∨q F
P q p∨q
T T T
T F T
F T T
F F F

4. F 31201 ก 4 WiLa 4
3. ก F F F ... F ...
F 5<7
5+(-3)<7+(-3)
F F F ... F ... F F F
F 5<7 F 5+(-3)<7+(-3)
F p F q F p→q F (truth table) p→q F
P q p→q
T T T
T F F
F T T
F F T
4. ก F F ก F
F 5( 7 + 3 ) = 5 x 10
7 + 3 = 10
F F ก F F F F
5( 7 + 3 ) = 5 x 10 ก F 7 + 3 = 10
pก F q F p↔q F (truth table) p↔q F
P q p↔q
T T T
T F F
F T F
F F T
1. F ก F F กF F F (atomic statement)
2. ก F ก กก FF F F F ก
5. F
F 2+3=5 2+3≠5
F 2<3 2<3
(F F 2 F F กF 3 F 2 Fก 3 กก F 3 Fก F 2≥3 )

5. F 31201 ก 4 WiLa 5
Fp F ~p F ~p F
P ~p
T F
F T
ก 2
1. F F F ก F
1) 4 5 F .....................................................................................
2) 3 Fก 4 3 F ก F 4 .....................................................................................
3) 4 F F 43 F .....................................................................................
4) F F F ก ........................................................................
5) 3 × 5 = 15 ก F 15 ÷ 3 = 5 .....................................................................................
2. ก F p F 3 q F 3
r F 2 F s F 2
ก FF
1) p ∧ ∼q
2) r⇒s
3) ∼r ⇔ s
4) (p ∧ q) ⇒ r
5) q ⇔ (r ∨ ∼ p) ..
3. ก F FF
1) 4 5 F
2) 2 Fก 3 2 กก F 3 .....
3) F 7 F F 72
4) 2<5 ก F 3 > 5 .......
5) F {1 , 2} = {2 , 1} F {1 , 2} ⊂ {2 , 1} ......

6. F 31201 ก 4 WiLa 6
1.3 ก F F
F F F ก F F F F F ก F
F F F
F 1 F F F
Fp F
Fq
ก F F p∧q
กp q F p∧q
F F
F 2 ก F a, b c F F
F (a ∧ b) ∨ c
กa b F a∧b
ก a∧b c F (a ∧ b) ∨ c
( a ∧ b) ∨ c
T T F
T
T
ก F F F F F
F 3 F ~ ( a → ~ b) a,b F F
กb F ~b
กa ~b F a →~ b
~ ( a → ~ b) F
~ (a → ~ b )
T T
F
F
T

7. F 31201 ก 4 WiLa 7
F 4 ก Fp q r s
F [ ( p ∧ q) ∨ r ] → ( p ∧ s)
[ ( p ∧ q ) ∨ r ] → ( p ∨ s )
T F
F F T T
F T
T
F [ ( p ∧ q) ∨ r ] → ( p ∧ s) F
ก 3
1. ก F P F F Q F F
R F F S F F
F FF
1. [ P ∧ (~ Q)] ↔ ( P ∨ S) 2. ( P → Q) → (S ∨ R)
3. [ P ∨ (~ R)] → S 4. [( P ∨ Q) ∧ (~ R)] → Q
5. ( P ∧ Q) ∨ (~ R) 6. ( P ↔ R) → (Q ∨ S)
7. Q ↔ [( P ∧ S) ∨ R ] 8. ~ ( P ∧ Q) ↔ [(~ P) ∨ (~ Q)]
2. ก F P,Q,R S F P Q F R ∧ (~ S)
F F F F
1. ( P ∨ Q) → ( P∧ ~ R)
2. [( P ∧ S) ∨ (~ R)] → ( R ∧ Q)
3. ( P ↔ R ) → (Q → S)
4. [( R ∧ Q) ∨ (S → P)] → [( P ∧ (S ∨ Q)]
3. F P,Q,R,S F F [( P → Q)].V ( R∨ ~ S )
F P,Q,R,S
4. F P,Q,R,S F P∨Q F (S ∨ R ) ∨ Q F
F P,Q,R,S

8. F 31201 ก 4 WiLa 8
1.4 ก F F
ก F F F ก F F F
F F F F F ก
F F F
F 1 F F F ( p → q ) ∧ ∼q
p q ∼q p→q ( p → q ) ∧ ∼q
T T F T F
T F T F F
F T F T F
F F T T T
F 2 F F F ( p ∧ ∼q ) ∨ ∼r
p q r ∼q ∼r p ∧ ∼q ( p ∧ ∼q ) ∨ ∼r
T T T F F F F
T T F F T F T
T F T T F T T
T F F T T T T
F T T F F F F
F T F F T F T
F F T T F F F
F F F T T F T
F F F F
F 3 F F F F ( p ∧ ∼q ) ∨ ∼r
(p → q) ∧ ∼q
T T T F F
T F F F T
F T T F F
F T F T T

9. F 31201 ก 4 WiLa 9
F 4 F F F F ( p ∧ ∼q ) ∨ ∼r
(p ∧ ∼q ) ∨ ∼r
T F F F F
T F F T T
T T T T F
T T T T T
F F F F F
F F F T T
F F T F F
F F T T T
1.5 F ก
ก F F F F ก ก Fก
F F ก F ก F กF F F ก
F p→q ก ~ p∨q ก ก F F
F 1 ก Fp q F F p→q ~p∨q
p q p→q ~p ~ p∨q
T T T F T
T F F F F
F T T T T
F F T T T
Q F p→q ~p∨q F ก กก F Fp,q
∴ F p→q ก F ~p∨q
ก F F Fa ก Fb F F a≡b

10. F 31201 ก 4 WiLa 10
F 2 F p ∧∼q ก ∼( q→p ) ก F
p q ∼p ∼p ∧q q→p ∼( q→p )
→
T T F F T F ก
T F F F T F p ∧∼q ก ∼( q→p ) F
F T T T F T ก กก
F F T F T F ∴ p ∧∼q ≡ ∼( q→p )
F ก Fก ก F p, q r F
1. p ∧ p ≡ p
2. p ∧ q ≡ q ∧ p
3. p ∨ q ≡ q ∨ p
4. p ∧ q ≡ q ∧ p
5. p → q ≡ ∼( p ∨ q )
≡ ∼ q → ∼p
6. p ↔ q ≡ ( p → q ) ∧ (q → p )
≡ ∼ p ↔ ∼q
7. ∼(∼ p ) ≡ p
8. ∼ (p ∧ q) ≡ ∼p ∨ ∼ q
9. ∼(p ∨ q) ≡ ∼p ∧ ∼ q
10. ∼ ( p → q ) ≡ p ∧ ∼q
11. ∼ ( p ↔ q ) ≡ ∼p ↔ q
≡ p ↔ ∼q
12. p∧ (q ∨ r ) ≡ ( p∧q ) ∨ ( p∧r )
p∨ (q∧r ) ≡ ( p∨q ) ∧ ( p∨r )
13. p→ ( q ∧ r ) ≡ ( p→q ) ∧ ( p→r )
p→ ( q ∨ r ) ≡ ( p→q ) ∨ ( p→r )
14. p→ ( q → r ) ≡ ( p∧q ) → r
15. ( p→q ) ∧ ( q→r ) ≡ p→ r

11. F 31201 ก 4 WiLa 11
ก 4
1. F F F F
1.1 (p → q) → (∼ p ∧∼ q )
1.2 ( p ∧∼ q ) ↔ ( q ∨ p )
1.3 (p∧q) → ( p∨r)
2. F F F ก F
2.1 ∼ p ∧ q ก ∼ (q → p)
2.2 p → q ก ∼ p →∼ q
2.3 (p∧q) → r ก p→ ( q → r)
1.6 F (TAUTOLOGY)
ก F F กก F F
ก F F ก F F F F F
F 1 ก Fp q F
F ( p → q) ∧ p → q
p q p→ q ( p → q) ∧ p ( p → q) ∧ p → q
T T T T T
T F F F T
F T T F T
F F T F T
Q ( p → q) ∧ p → q F กก F Fp q
∴ ( p → q) ∧ p → q F
ก F F ก F F F F
ก F F F F ก F
ก F ก F F FF ก F Fก F
F F F FF F F Fก F F
F F F

12. F 31201 ก 4 WiLa 12
F 2 F [( p → q) ∧ p] →~ q F F
F [( p → q) ∧ p] →~ q F
[ ( p → q ) ∧ p ] → ~ q
F
T F
T T
T T
ก F F [( p → q) ∧ p] →~ q F F F
F F Fก
ก p q F [( p → q) ∧ p] →~ q
F [( p → q) ∧ p] →~ q F F
F 3 F ( p ∧ q) → (q ∨ p ) F F
F ( p ∧ q) → (q ∨ p ) F
( p ∧ q ) → ( q ∨ p )
F
T F
T T F F
Fก
ก F F p q F
ก ก Fก FF [( p → q ) ∧ p ] →~ q
F [( p → q) ∧ p] →~ q F

13. F 31201 ก 4 WiLa 13
(2) F F A∨B
ก A∨B F ก A F B F ก F
A∨B F F F F F
(3) F F A↔B
ก F A↔B F F F F F ก ก Fก F
A ก Bก F F A B ก กก
ก F A↔B T กก
ก F A↔B F
1.7 ก F
ก F ก F ก F( )
F F ก F F F F F ก F
F F ก F F F F F
F F Fก F F
ก F3
1. ก Fก ก F
ก ก F
1. ก ก (modus ponens) 2. ก ก F (modus tollens)
p →q p →q
p ~q
∴q ∴~ p
3. ก ก (law of syllogism) 4. ก ก ก (disjunctive syllogism)
p →q p ∨q
q →r ~ p
∴p → r ∴q
5. ก ก F (conjunctive inference) 6. ก ก ก (inference by cases)
p p →r
q q →r
∴p ∧q ∴p ∨q → r
7. ก ก F F (law of simplification) 8. ก F F
p ∧q p →q
∴p ∴~ q →~ p

14. F 31201 ก 4 WiLa 14
2. ก Fก F
ก F ก F F F p1 , p2 ,..., pn F
F C F Fก F ก F F F ก F
F กF F p1 , p2 ,..., pn F F กF F Cก F
F ก F F F ∧ F F ก F
→ F ก
( p1 , p2 ,..., pn ) → C
F ( p1 , p2 ,..., pn ) → C F
ก F F ก F (valid) FF
( p1 , p2 ,..., pn ) → C F Fก F FF ก F F (invalid)
ก F ก ก F
F 1 Fก F F F
: 1. p→q
2. ~ q
: ~ p
ก F F [( p → q )∧ ~ q ] →~ p F
p q ~p ~q p→q ( p → q )∧ ~ q [( p → q )∧ ~ q ] →~ p
T T F F T F T
T F F T F F T
F T T F T F T
F F T T T T T
ก F F กก Fก F F ก F
F 2 : 1. F กF ก
2. F F กF
:
ก F F
ก F ก F FF ก F p กF
q

15. F 31201 ก 4 WiLa 15
: 1. p → q
2. ~ p
: q
ก F [( p →q)∧ ~ p] → ( q)
F( )
T
T= ~
T = ( p → q)
p F= q
F= p F= q F= p
F F F ก p q FF
F
ก 5
ก F F F
1. 1. p → q
2. q → r
3. ∼r
∼p∨r
2. 1. F 7 F F 7 F 2
2. 7 F F
F 2
3. 1. F กก ก F
2. F F กก ก F F
3. F
F

16. F 31201 ก 4 WiLa 16
1.8
ก F F F F
F F ก ก F F F
F x F กก F 3
a+2= 1
NOTE F P(x) Q(x) ก
F 1 F F
1. ก
2. กก
3. x - 5 = 10
4. F x+2=3 F x-2 = 0
5. π ก
6. 3x = 15 x=3 F F
7. x + 10
8. a + a = 2a
F
1 ก F F F
2 กก F F F
3 x - 5 = 10 x F x ∈R
F F
4 F x+2=3 F x-2=0 x F x ∈R
F F
5 π ก F F
6 3x = 15 x F x ∈R
F F
7 x + 10 F x F F x ∈R
F F F F F
8 a + a = 2a F F F

17. F 31201 ก 4 WiLa 17
1.9 F
F 2
1. F ... ก F ก ก
ก F x ก
x
x F
F ... ก F ก F ∀
F F P(x) x+2 = 3
∀ x P (x) ∀ x[x+2 =3 ] x ก x+2 = 3
2. F ... F
ก F x
x
x F F
F ... F ก F ∃
F F P(x) x-3 = 5
∃ xP(x) ∃ x[x - 3 = 5] x x-3 = 5
F ก F ∀x x ก
F ก F ∃x x
F ก FU ก F
F ก FR
F ก FQ ก
F ก FI Z
F ก FN
F F F
F P(x) x
(1) F ∀x[ p(x)] F ก F x p(x) F ก F U
F F F
∀x[ p(x)] ก F ก x∈U , P(x)
(2) F ∀x[ p(x)] F ก F ก U F F
x p(x) F F F

18. F 31201 ก 4 WiLa 18
∀x[ p(x)] ก F x∈U P(x)
(3) F ∃x [p (x ) ] F ก F ก U F F
x p(x) F F F
∃x[ p( x)] ก F x∈U P(x)
(4) F ∃x [p (x ) ] F ก F x p(x) F ก F U
F F F
∃x[ p( x)] ก F ก x∈U , P(x)
F 1 F F F
1. ∀ x [ x + 8 ≥ 8 ] U = { 0, 2, 4 }
F x∈U ก F x+8≥ 8
x=0 ; 0+8≥8 (T)
x=2 ; 2+8≥8 (T)
x=4 ; 4+8≥8 (T)
F ก F x∈U F x+8≥ 8
∀ x [ x + 8 ≥ 8 ] ; U = { 0, 2, 4 } F
2. ∀x [ x + 8 > 8 ] U = { 0, 2, 4 }
F x∈U ก F x+8>8
x=0 ; 0+8>8 (F)
x=2 ; 2+8>8 (T)
x=4 ; 4+8>8 (T)
F F x∈U F x+8> 8
∀ x [ x + 8 > 8 ] ; U = { 0, 2, 4 } F
3. ∃x[ x 2 = 2 x] U = { -1, 0, 1 }
F x∈U ก F x2 = 2x
x = -1 ; (-1)2 = 2(-1) (F)
x= 0 ; (0)2 = 2(0) (T)
x= 1 ; (1)2 = 2(1) (F)
F F x∈U F x2 = 2x
∃x[ x 2 = 2 x] ; U = { -1, 0, 1 } F

19. F 31201 ก 4 WiLa 19
4. ∃x[ x + 1 = 1] U = R
F F x∈U F x=o F x+1 = 1
∃x[ x + 1 = 1] U= R F
5. ∀ x [ x+1 = x ] U = R
F F x∈U F x=o F x+1 = x
∀ x [x+1 = x ] U = R F
ก 9
1. F FF
1. ∀ x[ x+x = 2x ] ; u = {-2,-1,0,1,2}
2. ∃ x[ (x-1) (x+1) = x2 - 1 ] ; U = { -2,1,3,7}
3. ∃ x[2x2+3x+1 = 0] ; U = {-2,1,3,7}
4. ∀ x[x2+2x+1 = 0] ; u = {-2,1,3,7}
5. ∀ x[ x+3 < 5 ] ; U = R-
2. F U = {-2,-1, 0, 1, 2} P(x) x ≥0 ;
Q(x) x/4
R(x) x2-4 =0
F FF
1. ∀x[ P( x)∨ ~ R( x)]
2. → P(x)]
∃ x[ R(x)
3. ∀ x[ Q(x) ↔ P(x)]
3. F U=R P(x) x ก ;
Q(x) x ก
F FF
1. ∀ x[P(x)] ∨ ∀ x [Q(x0]
2. ∀ x[P(x) → Q(x) ]
3. ∀ x(P(x) ∨ Q(x)]
4. ∀ x(P(x)]∧ ∀ x[Q(x)]
5. ∀ x[P(x) ∧Q(x)]
______________ ^__^ ______________

20. F 31201 ก 4 WiLa 20
F F F
ก ก FU
1. F ∀x∀y [ p( x, y )] F ก F F x p(x,y) F
กa U F F F ∀y  p ( a, y ) 
 
2. F ∀x∀y [ p( x, y )] F ก F กb U F F
x p(x,y) F F F ∀y [ p(b, y)]
3. F ∀y∀x [ p( x, y )] F ก F F y p(x,y) F
กa U F F F ∀x  p ( x, a ) 
 
4. F ∀y∀x [ p( x, y )] F ก F กb U F F
y p(x,y) F F F ∀x [ p( x, b)]
5. F ∃x∃y [ p( x, y )] F ก F กa U F F
a x p(x,y) F F F ∃y [ p(a, y )]
6. F ∃x∃y [ p( x, y )] F ก F F x p(x,y) F กa
U F F F ∃y  p ( a, y )
 
7. F ∃y∃x [ p( x, y )] F ก F กa U F F
a y p(x,y) F F F ∃x [ p( x, a)]
8. F ∃y∃x [ p( x, y )] F ก F F y p(x,y) F กa
U F F F ∃x  p ( x, a ) 
 
9. F ∀x∃y [ p( x, y )] F ก F F x p(x,y) F กa
U F F F ∃y  p ( a, y )
 
10. F ∀x∃y [ p( x, y )] F ก F กb U F F
x p(x,y) F F F ∃y [ p(b, y )]
11. F ∀y∃x [ p( x, y )] F ก F F y p(x,y) F กa
U F F F ∃x  p ( x, a ) 
 
12. F ∀y∃x [ p( x, y )] F ก F กb U F F
y p(x,y) F F F ∃x [ p( x, b)]
13. F ∃x∀y [ p( x, y )] F ก F กa U F F
x p(x,y) F F F ∀y [ p(a, y)]
14. F ∃x∀y [ p( x, y )] F ก F F x p(x,y) F กb
U F F F ∀y  p ( b, y ) 
 
15. F ∃y∀x [ p( x, y )] F ก F กa U F F
y p(x,y) F F F ∀x [ p( x, a)]

21. F 31201 ก 4 WiLa 21
16. F ∃y∀x [ p( x, y )] F ก F F y p(x,y) F กb
U F F F ∀x  p ( x, b ) 
 
1.10 ก ก F F
ก ก ก ก F ก F
ก
ก F F ก FF F
ก F F F ก
ก F p(x),q(x) p(x,y)
1. ∀x[ p(x) ∧ q (x)] ≡ ∀x[ p(x)] ∧ ∀x[ q (x)]
2. ∃x[ p( x) ∨ q ( x)] ≡ ∃x[p( x)] ∨ ∃x[q ( x)]
3. ~ ∀x[ p(x)] ≡ ∃x[ ~ p(x)]
4. ~ ∃x[p( x)] ≡ ∀x[ ~ p( x)]
5. ~ ∀x∀y [ p( x, y)] ≡ ∃x∃y [ ~ p( x, y)]
6. ~ ∀x∃y [ p( x, y)] ≡ ∃x∀y [ ~ p( x, y)]
7. ~ ∃x∀y [ p( x, y)] ≡ ∀x∃y [ ~ p( x, y)]