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Module #1 - Logic

2/28/2013 1 1

HY118-

Kees van Deemter,

University of Aberdeen

, 28/02/2013

. e-mail: argyros@csd.uoc.gr

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118

118

:

http://www.ics.forth.gr/~argyros/cs118.html Username: cs118

Password: _dm13_

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Liu, . 1.8 ( )

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.

: .

.

!

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:

1. 2.

( 1. )

,

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, .

:

.

.

.

George Boole(1815-1864)

(280 206 ..)

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(T) (F)

,

,

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rock

, 1 + 4 = 2.7 2x2 = x2 + x2

:

; () x := x+1 ()

1 + 2 ( )

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:p, q, r, ( p = )

:

(..,

)

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.

.

.

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n

.., +

1 (.., 3) 2 (.., 3+4)

(Boolean operators)

.

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.

NOT .

() AND .

() OR .

XOR .

... ... IMPLIES .

IFF .

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(NOT) .

.. p = .

p = .

NOT: p p

T FF T

T : True; F : False

:

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(AND)..

p= .q= ,

pq=

.

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p q pq

F F F

F T F

T F FT T T

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(OR).p=

.

q= .

pq=

/ .

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pq p ,

q

.

,

p q pq

F F F

F T TT F T

T T T

AND

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:

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.

.. p (p) [

;]

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T.

..: p (p)

p p p(p)

F T T

T F T

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.

.., p (p) [

;]

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F..: p (p)

pp p

(p)F T F

T F F

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, :

...

T, F

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(., ).

.

:

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p q , pq:

p q

p q

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... 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

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p = ,

q = ,

r =

p =

r p = r p q =

.

.

, ,

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-:

f g s

f (gs) :

,.......

(fg) s : , .......

fgs !

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,

.

f g (f)g , (f g)

,

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p1 p2p3 ;

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M d l #1 L i

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(p1 p2 ) p3 p1 (p2 p3 ) , !

(p1 p2 ) p3 p1 (p2 p3 )

M d l #1 L i

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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

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(p1 p2 ) p3 =p1 (p2p3 );

(

! )

(p1 p2 ) p3p1 (p2p3 )

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1. p1 p2 pn, n .

;2x2x2x x2 (n )

,

2n n

Module #1 Logic

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(XOR, ) ... (IMPLIES, )

(IFF, )

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(XOR).

p = 10

q =

p q = 10

(... !)

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pq

p, q , !

, p q .

,

p q pq

F F F

F T TT F T

T T F

OR.

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OR XOR!

p = q =

r=

rp q ... ... rp q;

...

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OR

XOR!

...

p q p"" q

F F F

F T TT F T

T T ?

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g

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1. p

q . pq

;

OXI: p=T, q=T

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g

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2. p

q . p q

;

:

pq :

a)p=T, q=F ( p q )b) p=F, q=T ( p q )

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g

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...

p q p q.

.., .p =

q = .p q = ,

.

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...

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

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... 118,

. True False;

,

. True orFalse ;

1+1=6, . True orFalse ;

,

Bill Gates. True orFalse ;

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;

[ ] [ ]

,

. ,

!

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...

q T.

pq ; !

p q pqF F T

F T T

T F F

T T T

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...

p F.

pq;

!

p q pqF F T

F T T

T F F

T T T

Module #1 - Logic

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...

(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

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p q

p q

p, q

p, q

p, q p, q

q p

q p

q p

p q

q p q p

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:

:

,

!...

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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

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...

... p q,

1. p= q= 2+2 =4

2. p= q= 2+2 =5

3. p=

q=

Module #1 - Logic

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P Q.

PQ

. PQ

PQ .

H PQ

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P Q P Q

. P Q

P Q

A

B , P Q P Q.

... , P Q PQ P Q.

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P Q:

P

Q Q P

PQ PQ

.