Machine Translated by Google

Machine Translated by Google

•
•
•
•
•
•
Machine Translated by Google

•
•
•1+1=2
•2+2=3
Machine Translated by Google

•
•
•
•4>2
n
•2 ÿ 100
•
•
•
Machine Translated by Google

• p, q, . . .
•
Machine Translated by Google

¬p

p ¬p
Machine Translated by Google

pÿq pÿq
pq pq

pqp ÿ q pqp ÿ q
Machine Translated by Google

pÿq pÿq
pq p q

pqp ÿ q pqp ÿ q
Machine Translated by Google

pÿq

• p q
• p q
• p q
• qp
• p q
• q ¬p
•
• 2+3=5
Machine Translated by Google

pÿq

• qÿp
• ¬q ÿ ¬p
•
pÿq
Machine Translated by Google

•
Machine Translated by Google

pÿq
p q

pqp ÿ q

• p q
• p q
• pq
Machine Translated by Google

•
• ¬
• ÿ, ÿ, ÿ
•ÿ

•ÿ
Machine Translated by Google

• p
• q
•r
• p ÿ (q ÿ ¬r)
Machine Translated by Google
Machine Translated by Google

(p ÿ ¬q) ÿ (p ÿ q)

pq ¬qp ÿ ¬qp ÿ q (p ÿ ¬q) ÿ (p ÿ q)

¬p ÿ (¬q ÿ r)

pq r ¬p ¬q ¬q ÿ r ¬p ÿ (¬q ÿ r)

(p ÿ q) ÿ ¬q
(p ÿ r) ÿ (r ÿ ¬p) (p
ÿ q) ÿ (q ÿ p) (p ÿ
¬q) ÿ (¬p ÿ q) (p ÿ
¬q) ÿ (q ÿ ¬p) ¬(¬p ÿ
¬q) (p ÿ q)
ÿ (p ÿ q) (p ÿ q) ÿ
(r ÿ q)
Machine Translated by Google

•
•

•
•
•
Machine Translated by Google

•
•

AB AB
A B B

•
•
Machine Translated by Google

•
•

• p ÿ ¬p
• p ÿ ¬p
Machine Translated by Google

(p ÿ q) ÿ (p ÿ
q) (p ÿ q) ÿ (p
ÿ q) p ÿ (¬q
ÿ p) p ÿ (p
ÿ q) p ÿ (p
ÿ p) (p ÿ q) ÿ [(p ÿ r) ÿ (q ÿ r)]
Machine Translated by Google

2x + 5x + 6 = 0

mx2 + 2x ÿ 1 = 0

2x +1>0

20
3 2 30

10
2 ÿ1

x x
Machine Translated by Google

p
pÿq qp ÿ q

¬(p ÿ q) ¬p ÿ ¬q
Machine Translated by Google

p ÿ T ÿ pp ÿ F ÿ p

pÿTÿTpÿFÿ
F
p ÿ p ÿ pp ÿ p ÿ p

¬(¬p) ÿ p
Machine Translated by Google

pÿ ÿ qÿ
qp ÿ q ÿ pq ÿ p

(p ÿ q) ÿ r ÿ p ÿ (q ÿ r) (p ÿ q) ÿ r ÿ p ÿ (q
ÿ r) p ÿ (q ÿ r) ÿ (p ÿ q) ÿ (p ÿ r) p ÿ (q ÿ r)

ÿ (p ÿ q) ÿ (p ÿ r) ¬(p ÿ q) ÿ ¬(p ÿ q) ÿ p ÿ
(p ÿ q) ÿ p ÿ (p ÿ q) ÿ

¬p ÿ ¬q
¬p ÿ ¬q

p
p
Machine Translated by Google

p ÿ ¬pp ÿ T
ÿ ¬pp ÿ ÿ F
q ¬p ÿ q ÿ

(p ÿ q) ÿ (p ÿ r) ÿ p ÿ (q ÿ r)
(p ÿ r) ÿ (q ÿ r) ÿ (p ÿ q) ÿ r
(p ÿ q) ÿ (p ÿ r) ÿ p ÿ (q ÿ r)
(p ÿ r) ÿ (q ÿ r) ÿ (p ÿ q) ÿ r
pÿqÿ ÿ (p ÿ q) ÿ (q ÿ p)
pÿq (¬p ÿ q) ÿ (p ÿ ¬q)
Machine Translated by Google

¬(p ÿ (¬p ÿ q)) ¬p ÿ ¬q

¬(p ÿ (¬p ÿ q)) ÿ ¬p ÿ ¬(¬p ÿ q)

ÿ ¬p ÿ [¬(¬p) ÿ ¬q]

ÿ ¬p ÿ (p ÿ ¬q)

ÿ (¬p ÿ p) ÿ (¬p ÿ ¬q)

ÿ F ÿ (¬p ÿ ¬q) ¬p ÿ p ÿ F

ÿ ¬p ÿ ¬q F

¬(p ÿ (¬p ÿ q)) ¬p ÿ ¬q

p ÿ (¬q ÿ r)
pÿq ¬p ÿ q

¬(p ÿ (¬q ÿ r)) ÿ

p ÿ (q ÿ r) ÿ (¬p ÿ ¬q ÿ r) (p ÿ
q) ÿ rp ÿ q ÿ
(¬p ÿ ¬q ÿ r) [[[(p ÿ q) ÿ
r] ÿ [(p ÿ r) ÿ ¬r]] ÿ ¬q] ÿ s
Machine Translated by Google

¬(p ÿ q) ¬p ÿ q
(p ÿ q) ÿ (p ÿ r) p ÿ (q ÿ r)
(p ÿ r) ÿ (q ÿ r) (p ÿ q) ÿ r
(p ÿ q) ÿ (p ÿ r) p ÿ (q ÿ r)
¬p ÿ (q ÿ r) q ÿ (p ÿ r)
pÿq (p ÿ q) ÿ (q ÿ p)
Machine Translated by Google

p ÿ (p ÿ q) p ÿ q
p ÿ q ¬p ÿ (p ÿ q)
p ÿ q ¬p ÿ ¬q
¬p ¬(p ÿ q) ÿ (¬p ÿ q)
[(p ÿ q) ÿ (q ÿ r) ÿ (r ÿ p)] [(p ÿ q)
ÿ (q ÿ r) ÿ (r ÿ p)]
[(p ÿ q) ÿ (q ÿ r) ÿ (r ÿ p)] [(p ÿ q) ÿ (q ÿ r) ÿ (r ÿ p)]
Machine Translated by Google

ax2 + bx + c = 0 ac < 0
x y n (x + y)

ÿ2 ÿ2ÿ2

3n + 2 n
Machine Translated by Google

p q
• p
• q

(p ÿ q) ÿ r
pÿq
pÿq
pÿq
pÿq
¬p ÿ ¬q

¬p ÿ (p ÿ q)
Machine Translated by Google

PQR
•P
•Q
•R

P ÿ R ÿ ¬Q
Machine Translated by Google

x ÿ {x}
{x} ÿ {x} {x}
ÿ {x} {x} ÿ
{{x}} ÿ ÿ {x}

a
b
c
d
Machine Translated by Google

(p ÿ ¬q) ÿ
qp ÿ (p ÿ
q) ¬p ÿ (p ÿ
q) ¬(p ÿ q) ÿ q
Machine Translated by Google

• p ABC q
• ABC
• r ABC 60o

ABC 60o

(p ÿ q) ÿ r
(p ÿ r) ÿ q
(p ÿ r) ÿ q
q ÿ (p ÿ r)
Machine Translated by Google
Machine Translated by Google

ÿx ÿ N, x2 + 5x + 6 ÿx ÿ R,
x2 + x + 1 ÿ 0
3
ÿn ÿ N, (n ÿ n)
ÿn ÿ Nÿ, n2 ÿ 1 ÿx, ÿy
2
ÿ R, x2 + y > 2xy
ÿr ÿ Q, 3 < r < ÿ
ÿn ÿ N, n2 + 1 ÿx ÿ
2 <9
R, |x| < 3 ÿ x
2 2
ÿa, b ÿ R, (a + b) > 2(a +b 2)

n x
2
f(x) = x ÿ 2x + n
x y xÿy
x y xÿy
x y xÿy
x y xÿy