ΠΛΗ31 ΜΑΘΗΜΑ 2.2

31 2:

2.2: .. ..

.

.

1. 1.

2. -

3.

2 , 31, 2.2: wff

2.

1.

2.

3. Clausal Form

3.

1. ...

2. ...

.

.

wff

(-)

3 , 31, 2.2: wff

(-)

.

1. :

4 , 31, 2.2: wff

1

2 OR

3 De Morgan OR AND

~~)(~~~)(~

)(~~ ~

4

5

6

7

8

~~)(~

)()()()()()(

)()()()(

~~ B

~~

~~

xx

xx

xxx

xxx

}{}{

.

1. 1.-

5 , 31, 2.2: wff

.

.

1. 2.

,

.

:

6 , 31, 2.2: wff

)]](~),([()([~ yyxyxx pipi :

:

)]](~),([()([~ yyxyxx pipi

)]](~),([()([~ yyzyzz pipi

)]](~),([()([~ wwzwzz pipi

.

2.

3 :

clausal form

7 , 31, 2.2: wff

wff

8 , 31, 2.2: wff

: (), :

i :

Bj (literal)

n ...21

m ...21

.

2. 1.

:

.. :

...

)),(~)(()),()(( bobpamparentbobmanpamtomparenttomman

))()(~)(())(~)()(( dQdTdTdTcTaQ

.

2. 2.

9 , 31, 2.2: wff

: (), :

i :

B (literal)

n ...21

m ...21 Bj (literal)

:

.. :

...

)),(~)(()),()(( bobpamparentbobmanpamtomparenttomman

))(~)(~)(())()(~)(())(~)()(( dRcQaRdQdTdTdTcTaQ

.

2. 3. Clausal Form

10 , 31, 2.2: wff

nm ...... 2121

m...21

n ...21

212121212121 ~~)(~ =

.

1. wff

wff 7 :1. 2. 3. 4.

11 , 31, 2.2: wff

4. 5. 6. 7. AND

.

1. wff 1.

12 , 31, 2.2: wff

1 :

1 OR: ~

:: :

: OR:

)])],(~)([~)],(),([~)](~),([()([

yxyyxyyxyyyxyxx

pipipipi

)])],(~)([)],(),([~

)](~),([()([~

yxyyxyyxy

yyxyxx

pipi

pipi

.

1. wff 2.

13 , 31, 2.2: wff

2 : 2 .

De Morgan:

()

~~

~~

xx

xx

~~)(~~~)(~

()

:

De Morgan:

)])],(~)([)],(),([~

)](~),([()([~

yxyyxyyxy

yyxyxx

pipi

pipi

)])],(~)([)],(),([~

)](~),([()([~

yxyyxyyxy

yyxyxx

pipi

pipi

)])],(~)([)],(~),([~

)](~),([()([~

yxyyxyyxy

yyxyxx

pipi

pipi

.

1. wff 3. ()

14 , 31, 2.2: wff

3 : ()

,

..: )()]([ AQxQx

,

..:

)()]([ AQxQx

),()],([ AyyQxyQyx ),()],([ BAQyxQyx

)]),(([)],([ xxfQxxyQyx )]),,(([)],([ xzxfQzxxyQyzx

)],,(),,(([)],([ wyxgyxfQwyxkzQkwzyx

)]),(([)],([ xxparentxxyparentyx

.

1. wff 3. ()

15 , 31, 2.2: wff

()

:

)])],(~)([)],(~),([~

)](~),([()([~

yxyyxyyxy

yyxyxx

pipi

pipi

:

)])],(~)([)],(~),([~

))]((~))(,(([)([~

yxyyxyyxy

xxxxx

pipi

pipi

.

1. wff 4.

16 , 31, 2.2: wff

4 : 4 ,

! !

()

:

)])],(~)([)],(~),([~

))]((~))(,(([)([~

yxyyxyyxy

xxxxx

pipi

pipi

)])],(~)([)],(~),([~

))]((~))(,(([)([~

zxzz

xyyxyxxxxx

pipi

pipi

.

1. wff 5.

17 , 31, 2.2: wff

5 : 5

, ..

. .

()

)])],(~)([)],(~),([~

))]((~))(,(([)([~

zxzz

xyyxyxxxxx

pipi

pipi

)])],(~)([)],(~),([~

))]((~))(,(([)([~

zxz

xyyxxxxxzyx

pipi

pipi

.

1. wff 6.

18 , 31, 2.2: wff

6 :

:

:

()

)()()( )()()()(

()

:

:

)])],(~)([)],(~),([~

))]((~))(,(([)([~

zxz

xyyxxxxxzyx

pipi

pipi

)]]],(~)([)([~)]],(~),([~)([~

))]((~))(,(([)([[~

zxzx

xyyxxxxxxzyx

pipi

pipi

)]],(~)()([~)],(~),(~)([~

))]((~)([~))](,()([[~

zxzx

xyyxxxx

xxxzyx

pipi

pipi

.

1. wff 7. AND

19 , 31, 2.2: wff

7 : AND 7 AND

2 ( ( ).()

AND. :

)]],(~)()([~)],(~),(~)([~

))]((~)([~))](,()([[~

zxzx

xyyxxxx

xxxzyx

pipi

pipi

),(~)()(~),(~),(~)(~))((~)(~

))(,()(~

zvzv

uyyuuww

xxx

pipi

pipi

20 , 31, 2.2: wff

.

2. 1. ..

:

21 , 31, 2.2: wff

.

2. 1. ..

22 , 31, 2.2: wff

.

2. 2. ..

. ,

... :

. 1

23 , 31, 2.2: wff

:

(Q T) R, (Q P) R, S (T P), Q, S, T

.

. 2

24 , 31, 2.2: wff

... :( ) ( ) ( ) ( )( )( )( )wuRwvRwvtRvutRut ,,,,

25 , 31, 2.2: wff

:1: 2: 3: 4: 5: 6: () . : /1,__/2, /1 /2

. 3

: /1,__/2, /1 /2

26 , 31, 2.2: wff

()