The document contains examples of logical statements and proofs involving propositional logic and quantifiers. Specifically: 1. It provides examples of common logical rules and forms such as modus ponens, modus tollens, hypothetical syllogism, and others. 2. It demonstrates evaluating the truth values of compound propositional statements. 3. It introduces basic quantifier concepts like domain of discourse and shows examples of quantified statements with universal and existential quantifiers like "For all x" and "There exists an x".

2. 5 2 SC 101

6. y+3>8 Y
x2-6x+4 = 0 x
l?JllhlUlfiJ 131
x + 5 = 5 + x Y
5 6 SC 101

7. ” I 0
,+miUlJ 2.4 n1wwl%99" x+ 2 < 5 LhJ~~hli&l iJil u = {0,1,2,3,4,5} L&La"-
nw8aJ~m6
51 P(x) : x+%<5
” E
FlJlJl4 P(O) : 0 +2 < 5 SiJ
P(1) : 1+2<5 -$44
P(2) : 2+2<5 -95J
P(3) : 3+2<5 Ii-;:
P(4) : 4+2<5 rh
P(S) : 5+2<5 Li&
SC 101 5 7

8. P
‘iT
F
P cl T T
T
5 8 SC 101

20. P 4 -9
T T F
~
T F T
F T F
F F T
1
F I
,
T 1
-(p " q) WC-P h--q)
kPl1s1akPl1s1a
P 4 -P -gP 4 -P -q P v9P v9 i(P vq)i(P vq) -P A -9-P A -9 -(p vq) w c-p A -q)-(p vq) w c-p A -q)
T TT T FF FF TT FF FF TT
T FT F FF TT TT FF PP TT
F TF T TT FF TT FF FF TT
F FF F TT TT FF TT TT TT
70 SC 101

21. -
P
-
T
T
'r
T
F
F
F
F
-
-
q
-
T
T
F
F
T
T
F
F
-
-
I
T
F
T
F
T
F
T
F
-
-
IAr
-
T
F
F
F
r
F
F
F
-
F
--
‘v(q nr)
I-
‘T
1
?
T
1
F
F
I‘
- -C
Lp v (q At-)
F
F
F
F
F
T
T
T
,vq
T
T
T
T
T
(
f
,[P V (q Ar)] 3
(t, va) A CD v r)
T
m
T
T
T
F
F
-[I) V (Cl Ar)] ==F’ (P Vq) A (P V r) rihth
SC 101 7 1

22. 2.5 lkwa~4du~RG (Equivalent Proposition)
7 2 SC 101

23. 7
3 A-q P-4 -(P 3 q)
F T F
T F T
F T F
F T b F
S C 101 7 3

24. -
P
-
T
T
T
T
F
F
F
F
-
-
q
-
T
T
F
F
T
T
F
F
-
-
r
-
T
F
T
F
T
F
T
F
-
XAr
T
F
F
F
T
F
F
F
p=S(q Ar)
T
F
F
F
T
T
T
T
I
P39
T
T
F
F
T
T
T
T
5*r
T
F
T
F
T
T
T
T
P=>q) A (p-r
7 4 SC 101

25. -
9
-
T
F
T
F
-
-
P
-
F
F
T
T
-
-
-9
-
F
T
F
T
-
.9 3P -p A c-q =a PI [-p/IL-q =>p)l =3q
F T
F T
T T
F T
SC 101 7 5

26. P
T
T
F
F
-
-
9
-
T
F
T
F
-
-
P
-
-q
-
F
T
F
T
‘34 -(P =+I) -PA-q (p 3q) a-p A-q:
Y
i%hhlkiis"uns (tautology) ikh%lJ~~li
(1) [p A(p 3 q)l 3 q
(2) [(p =aq)h-qla-p
(3) [!p *q) A cq 3 r)l 3 (p -;sr)
(4) [(P vq) A-ql=+p
(5) [[(p-q) A (i--s)1 A (p VI-11 3 (q v s)
(Modus Ponens)
(Modus Tollens)
(Hypothetical Syllogism)
(Dls]uncti.we Sylloqism)
(Constructive Dllemma)
(6) [[(p=sn)A(r3511 A&q v-s)]*(+p"- 1-1 (DestructiveDilemma)
(7) (P *q) a (-q--p) (Contrapositlve)
(8) (p A q) 9 p (Simplification)
7 6
SC 101

27. Cp*q) A (PA -q) 1
[ (p*q) Aiq=>r)l*[-(p=>r)
-
TTT T F
TTF 7 F
T FT F T
T F F F F
F T T T F
FTF T T
F’FTT T F
F F F T F
SC 101
.
7 7

32. 1 . Modus Ponens
2. Modus Tollens
in? Pa9
-s
He-i -p
3. Hypothetical Syllogism
SC 101

33. 4. Disjunctive Syllogism
-2
WIJ P
5. Constructive Dilemma
x-35
6. Destructive Dilemma
-9 v - s
RB -p v - r
' 7. Contrapositive
lyl P-q
RA -cl=+-P
8. Substraction (simplification)
8 3

34. d v e
-d
. -
. . f
W'Ejd
1 . d V e
2.1s v d
3. -d
4. e
RA -p
G$u’
1. ,-(p A - q)
2. -P v - C-q)
3. -P v 9
4 . q=;s-r
2, 3 disjunctive sylloqism
nTcruw%w"
4.5 Modus Ponens
8 4 SC! 101

35. tny
1) c-g
2)P*"
3) (u A g)*s
4) -s
5) -C-c)
SC 101 8 5

36. - P
1.
c 49
c
24
(u A g) *s
-,S
-(u A g)
-u v - g
8. u3-g
9. P3U
.lO p&-g. .
11. grg-P
2.8 -GbmFhi~ (Quantifier)
1-y 5)
1,2 Modus Ponens
LM? 3)
rwg 4)
4,5 M o d u s Tollens
6. m-pihE
7, fi+ja
rlwj 2 )
9, 8 Hypothetical Syllogism
10, contrapositive
11, 3 Modus Ponens
#
8 6 SC 101

37. Vxip(x)l = P(a) A P (b) A P Cc) / .__
3x[p(xll = P(a)" P(b) " P(c) v _._
SC 101 8 7

38. 7; p(x) : x+2<5-
p(l) : 1+2<5 1hsia-
p(2) : 2+2<5 lh?J-
p(3) : 3+2<5 1hiJ-
lW5lYb p(l) A p(2) A p(3) LhJil
” t:
9wu Vx[(x + 2) (51 lhir
Cmii~ 2 . 3 5 n%uw%es” u ={2,3,4} ~&lr~l~lfla?us~~~a~ Vx[x+2<51
sT%l?
IC p(x) : x+255
p(2) : 2+255 Ih?J
p(3) : 3+2(5 1hiJ
p(4) : 4+2(5 lh lib
88 SC 101

39. n"?ad14 2.36 n?wun%w” u = {1,2,3} sa~slm1~lnm~~~aw14 -3x[x+2 (51
%%l=l
“i*si p(x) : x+255
p(l) : 1 +2~5 lSWCi4
p(2) : 2 + 2 55 lhCi4
p(3) : 3+2<5 1SUCiJ
Fish 3x[x + 2 51( lihil
UVGO -3xcx + 2 51( 1Surh
SC 101 8 9

40. 1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
9 0 SC 101

41. S;(x,L.): x = w
SC 101 9 1

42. 2.1 P(5)
2.2 h Q(2) a& ~(2)
2.3 P(2) M;O R(3)
2 . 4 S(2,3) LLR: R(-1)
2.5 o(4) “niOL& R ( 8 )
3. ~4741 Id~n~~auuoJ~s~T~RI~~~~~~~
3.1 x+l<l, U = r-1,0,1,2,3}
3.2 v2+2y+l = 0, II = N (42?9dX&)
3.3 (X-3) (x+1) = 0, u = i-3,-1,0,
3.4 a3-a = 0 , I! = r-(~~U?UL~UiYJ)
3.5 u - 8 < 4, 1J = {0,2,4,6}
9 2 SC 101

43. SC 101 9 3

44. 1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
(P / r) Q (q v s)
-r v -s
-(p A r) 3 c-q fi p)
[P A (q ,v t11 w s
(par) v -(r*s)
(r as) 3 (p v - p)
[[p + (q v’ r)l A - sl 3 t
-P V P
- P A P
(P Aq)*P
9 4 SC 101

45. 3 . 3 (p-q) 3 (-P V 'i)
3 . 4 (q j p:1 A (p A -q)
(p rq) =+I
4.1 F
4 . 2 P *(q*r)
T
4 . 3 (p A q) * (r v s)
F
4 . 4 c-p v (q Ar) =sP
F
4 . 5 (p vq) =+(r. AS)
F
4 . 6 (p 3 r) A (q * 1^)
T
4 . 7 (p v - q) 3 (I- A (p A Cl
F
4 . 8 p =G-(q =+r)
T
5. b~a4lnln-ml~~duaa 13, q, r
))
5.1 p 3 (q v r)
F
5 . 2 p A (q Vr)
F
5 . 3 (p A 9:' * r
F
5.4 [ (p A Cl) d rl 3 ( --I 3 -P)
F
5 . 5 [(p A q) 3 rl V [p 3 (q +r)l
F
SC 101 95

46. 7.1
7.2
7.3
7.4
7.5
7.6
7.7
7-a
7.9
7 . 1 0
- P *P
(p A q) ==a-p
P A - P
[ (p * q) A PI *q
I (P ‘“g)l 3 [(p 3 q) A (4 3 P)l
[(P =.+q) A -sl =a-P
tp A (q v r)l 3 (p A q) v (p fi r)
(p* q) -c-q q-p)
-(p V q) W.-p A -q
UllJd&Fl 2 . 5
1. ;~S~l~rulsil~st:ws~~~n~or~~~bi~OnYU
1.1 Pvq ; 4 V P
1.2 pv(qAr) ; (P vq) A r
1.3 p A (q A r) ; (p A q) A r
1.4 pV(qhr) i (p v q) A (p V r)
1.5 p A (q V r) ; (p A q) v (p A r)
1.6 PAq ; pa-q
1.7 PAP ; P
1.8 PVP ; P
1.9 Peg ; (P =3 q) A (q ==a p)
1.10 p-q ; -p v q
9 6 SC 101

47. 1.11 -(pVg) ; - P Aq
1.I2 -(p A sq) * ; -p A-9
1.13 -(pvq) ; -P A--9
1.14 -(pAq) ; -P v - q
1.15 (- p) P
2. (P v q) ==+ (P A q)
3. P A - P
4. (p =a ‘~1 H (q 3 P)
5. c-p v 9) =+ (P 3 q)
SC 101 9 7

48. 6. q 3 [P A (P 3 s)l
7. [p * (q A r)ler (p 3 q) A (P 3 r)l
a. I (p =>q) A (- q)l jp
9. PV-P
1.1
1.2
1.3
1.4
1.5
9 8
SC 101

49. 2.1
2 . 2
2 . 3
2 . 4
2.5
2 . 6
2 . 7
2 . 8
SC 101 9 9

50. 5. -Vx[x + 2 2 71
4 6. - 3x[x + 3 635
7. 3x [ -(x + 3 (6)l
8. -Vx[x-2211 '
1 0 0 SC 101