Upload
ntsatsakis
View
25
Download
0
Embed Size (px)
DESCRIPTION
Discrete Mathematics - Propositional Logic I
Citation preview
5/25/2018 02 DM PropositionalLogic 2802 01
1/55
Module #1 - Logic
2/28/2013 1 1
HY118-
Kees van Deemter,
University of Aberdeen
, 28/02/2013
. e-mail: [email protected]
5/25/2018 02 DM PropositionalLogic 2802 01
2/55
Module #1 - Logic
2/28/2013 2 2
118
118
:
http://www.ics.forth.gr/~argyros/cs118.html Username: cs118
Password: _dm13_
5/25/2018 02 DM PropositionalLogic 2802 01
3/55
Module #1 - Logic
2/28/2013 3 3
Liu, . 1.8 ( )
5/25/2018 02 DM PropositionalLogic 2802 01
4/55
Module #1 - Logic
2/28/2013 4 4
.
: .
.
!
5/25/2018 02 DM PropositionalLogic 2802 01
5/55
Module #1 - Logic
2/28/2013 5 5
:
1. 2.
( 1. )
,
5/25/2018 02 DM PropositionalLogic 2802 01
6/55
Module #1 - Logic
2/28/2013 6 6
, .
:
.
.
.
George Boole(1815-1864)
(280 206 ..)
5/25/2018 02 DM PropositionalLogic 2802 01
7/55
Module #1 - Logic
2/28/2013 7 7
(T) (F)
,
,
5/25/2018 02 DM PropositionalLogic 2802 01
8/55
Module #1 - Logic
2/28/2013 8 8
rock
, 1 + 4 = 2.7 2x2 = x2 + x2
:
; () x := x+1 ()
1 + 2 ( )
5/25/2018 02 DM PropositionalLogic 2802 01
9/55
Module #1 - Logic
2/28/2013 9 9
:p, q, r, ( p = )
:
(..,
)
5/25/2018 02 DM PropositionalLogic 2802 01
10/55
Module #1 - Logic
2/28/2013 10 10
.
.
.
5/25/2018 02 DM PropositionalLogic 2802 01
11/55
Module #1 - Logic
2/28/2013 11 11
n
.., +
1 (.., 3) 2 (.., 3+4)
(Boolean operators)
.
5/25/2018 02 DM PropositionalLogic 2802 01
12/55
Module #1 - Logic
2/28/2013 12 12
.
NOT .
() AND .
() OR .
XOR .
... ... IMPLIES .
IFF .
5/25/2018 02 DM PropositionalLogic 2802 01
13/55
Module #1 - Logic
2/28/2013 13 13
(NOT) .
.. p = .
p = .
NOT: p p
T FF T
T : True; F : False
:
5/25/2018 02 DM PropositionalLogic 2802 01
14/55
Module #1 - Logic
2/28/2013 14 14
(AND)..
p= .q= ,
pq=
.
5/25/2018 02 DM PropositionalLogic 2802 01
15/55
Module #1 - Logic
2/28/2013 15 15
p q pq
F F F
F T F
T F FT T T
5/25/2018 02 DM PropositionalLogic 2802 01
16/55
Module #1 - Logic
2/28/2013 16 16
(OR).p=
.
q= .
pq=
/ .
5/25/2018 02 DM PropositionalLogic 2802 01
17/55
Module #1 - Logic
2/28/2013 17 17
pq p ,
q
.
,
p q pq
F F F
F T TT F T
T T T
AND
5/25/2018 02 DM PropositionalLogic 2802 01
18/55
Module #1 - Logic
2/28/2013 18 18
:
5/25/2018 02 DM PropositionalLogic 2802 01
19/55
Module #1 - Logic
2/28/2013 19 19
.
.. p (p) [
;]
5/25/2018 02 DM PropositionalLogic 2802 01
20/55
Module #1 - Logic
2/28/2013 20 20
T.
..: p (p)
p p p(p)
F T T
T F T
5/25/2018 02 DM PropositionalLogic 2802 01
21/55
Module #1 - Logic
2/28/2013 21 21
.
.., p (p) [
;]
5/25/2018 02 DM PropositionalLogic 2802 01
22/55
Module #1 - Logic
2/28/2013 22 22
F..: p (p)
pp p
(p)F T F
T F F
5/25/2018 02 DM PropositionalLogic 2802 01
23/55
Module #1 - Logic
2/28/2013 23 23
, :
...
T, F
5/25/2018 02 DM PropositionalLogic 2802 01
24/55
Module #1 - Logic
2/28/2013 24 24
(., ).
.
:
5/25/2018 02 DM PropositionalLogic 2802 01
25/55
Module #1 - Logic
2/28/2013 25 25
p q , pq:
p q
p q
5/25/2018 02 DM PropositionalLogic 2802 01
26/55
Module #1 - Logic
2/28/2013 26 26
... pq(p q).
p q ppqq pp qq ppqq ((ppqq))
F FF T
T F
T T
FT
TT
T
T
T
T
T
T
F
F
F
F
F
F
FF
T
T
5/25/2018 02 DM PropositionalLogic 2802 01
27/55
Module #1 - Logic
2/28/2013 27 27
p = ,
q = ,
r =
p =
r p = r p q =
.
.
, ,
5/25/2018 02 DM PropositionalLogic 2802 01
28/55
Module #1 - Logic
2/28/2013 28 28
-:
f g s
f (gs) :
,.......
(fg) s : , .......
fgs !
5/25/2018 02 DM PropositionalLogic 2802 01
29/55
Module #1 - Logic
2/28/2013 29 29
,
.
f g (f)g , (f g)
,
5/25/2018 02 DM PropositionalLogic 2802 01
30/55
Module #1 - Logic
p1 p2p3 ;
2/28/2013 30 30
M d l #1 L i
5/25/2018 02 DM PropositionalLogic 2802 01
31/55
Module #1 - Logic
2/28/2013 31 31
(p1 p2 ) p3 p1 (p2 p3 ) , !
(p1 p2 ) p3 p1 (p2 p3 )
M d l #1 L i
5/25/2018 02 DM PropositionalLogic 2802 01
32/55
Module #1 - Logic
2/28/2013 32 32
p1
p2
p3
(p1
p2
) (p1
p2
)p3
(p2
p3
) p1
(p2
p3
)
F F F F F F F
F F T F F F F
F T F F F F F
F T T F F T F
T F F F F F F
T F T F F F F
T T F T F F FT T T T T T T
p1
p2
p3 ;;;
M d l #1 L i
5/25/2018 02 DM PropositionalLogic 2802 01
33/55
Module #1 - Logic
(p1 p2 ) p3 =p1 (p2p3 );
(
! )
(p1 p2 ) p3p1 (p2p3 )
2/28/2013 33 33
Module #1 Logic
5/25/2018 02 DM PropositionalLogic 2802 01
34/55
Module #1 - Logic
2/28/2013 34 34
1. p1 p2 pn, n .
;2x2x2x x2 (n )
,
2n n
Module #1 Logic
5/25/2018 02 DM PropositionalLogic 2802 01
35/55
Module #1 - Logic
2/28/2013 35 35
(XOR, ) ... (IMPLIES, )
(IFF, )
Module #1 Logic
5/25/2018 02 DM PropositionalLogic 2802 01
36/55
Module #1 - Logic
2/28/2013 36 36
(XOR).
p = 10
q =
p q = 10
(... !)
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
37/55
Module #1 Logic
2/28/2013 37 37
pq
p, q , !
, p q .
,
p q pq
F F F
F T TT F T
T T F
OR.
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
38/55
Module #1 Logic
2/28/2013 38 38
OR XOR!
p = q =
r=
rp q ... ... rp q;
...
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
39/55
Module #1 Logic
2/28/2013 39 39
OR
XOR!
...
p q p"" q
F F F
F T TT F T
T T ?
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
40/55
g
2/28/2013 40 40
1. p
q . pq
;
OXI: p=T, q=T
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
41/55
g
2/28/2013 41 41
2. p
q . p q
;
:
pq :
a)p=T, q=F ( p q )b) p=F, q=T ( p q )
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
42/55
g
2/28/2013 42 42
...
p q p q.
.., .p =
q = .p q = ,
.
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
43/55
2/28/2013 43 43
...
p q
p - q p q
p q p q!
p q p q !
..:
(1=0)
!
p q pqF F T
F T T
T F FT T T
False
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
44/55
2/28/2013 44 44
... 118,
. True False;
,
. True orFalse ;
1+1=6, . True orFalse ;
,
Bill Gates. True orFalse ;
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
45/55
2/28/2013 45 45
;
[ ] [ ]
,
. ,
!
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
46/55
2/28/2013 46 46
...
q T.
pq ; !
p q pqF F T
F T T
T F F
T T T
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
47/55
2/28/2013 47 47
...
p F.
pq;
!
p q pqF F T
F T T
T F F
T T T
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
48/55
2/28/2013 48 48
...
(p
q)
(p
q)
p q pq p p q
F F T T TF T T T T
T F F F F
T T T F T
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
49/55
2/28/2013 49 49
p q
p q
p, q
p, q
p, q p, q
q p
q p
q p
p q
q p q p
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
50/55
2/28/2013 50 50
:
:
,
!...
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
51/55
2/28/2013 51 51
p q p q .
p q p q .
p q
,p q (p q)
p q pq
F F T
F T FT F F
T T T
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
52/55
2/28/2013 52 52
...
... p q,
1. p= q= 2+2 =4
2. p= q= 2+2 =5
3. p=
q=
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
53/55
2/28/2013 53 53
P Q.
PQ
. PQ
PQ .
H PQ
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
54/55
2/28/2013 54 54
P Q P Q
. P Q
P Q
A
B , P Q P Q.
... , P Q PQ P Q.
Module #1 - Logic
5/25/2018 02 DM PropositionalLogic 2802 01
55/55
2/28/2013 55 55
P Q:
P
Q Q P
PQ PQ
.