Loogikaalgebra põhiseosed (0)

1.Kontrollida neeldumisseaduse x1  x1x2 = x1  x2 kehtimist võrduse mõlema poole avaldiste tõeväärtustabelite võrdlemise teel.
x1
x2
x1x2
x1  x1x2
x1  x2
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2.Lihtsustada avaldist loogikaalgebra põhiseoste abil:
x1 x2  x1 x3  x2 = x1 x2 x2  x1 x3 = x2  x1 x3
(x2 x1 ) x2  x2 = (x2  x1 ) x2  x2 = x2 x2  x1 x2  x2 = x1 x2  x2 = x1  x2
x1 x2 (x3 x1 )= x1 x2 (x3  x1 )= x1 x2 x3  x1 x2 x1 = x1 x2 x3
x1 ( x1  x2 )  x2 = x1 (x1 x2  x1 x2 )  x2 = x1 (x1 x2  x1 x2 )  x2 = x1 x1 x2  x1 x1 x2  x2 = 0  x1 x2  x2 =x2
x1 x2  x1 x2  x2 = x2  x2 = 1
x2 (x1  x2 )( x2  x3 )= (x1 x2  x2 x2 )( x2  x3 )= (x1 x2  x2 )( x2  x3 )= x2 ( x2  x3 )= x2 x2  x2 x3 = 0  x2 x3 = x2 x3
(x1 x2 )(x1 x3 x2 )= (x1 x2  x1 x2 )( x1 x3  x2 )= (x1 x2  x1 x2 )( x1  x3  x2 )= x1 x2 x1  x1 x2 x3  x1 x2 x2 
 x1 x2 x1  x1 x2 x3  x1 x2 x2 = x1 x2  x1 x2 x3 0 0  x1 x2 x3  x1 x2 = x1 x2  x1 x2
3.Lihtsustada etteantud loogikaavaldis DNK-ks põhiseoste ja tehete asendusseoaste abil:
(x1x2x3x4  x3x4  x1x3 )(x1 x4 )(x1 x4 )= (x1x2x3x4  x3x4  x1x3 )(x1 x4 )(x1 x4 )=
=(x1x2x3x4  x3x4  x1x3 )(0  x4x1  x1x4 0)= x3x4x1 x1x3x4 = x1x3x4
x1 x4  x3 (x1  x2)(x1x2  x4 )= x1 x4 x3 (x2 x1 )(x1 x2 x4 )= x1 x4 x3  x2x1 (x1 x2 x4 )=
= x1 x4 x1 x2 x3 x1 x2 x2 x1 x2 x1 x1 x2  x1 x4 x4  x3 x4  x2 x4  x1 x4 =x1 x2 x4 x1 x2 x3 x3 x4 x2 x4 x1 x4
x1  x1 x2 ( x1 x3 ) x1  x1 x2 x3 x4  x1 = x1 ( x1 x3 ) x1  x1 x2 x3 x4  x1 = (x1 x1x1 x1 x3  x1 x2 x3 x4 )x1 =
=(x1 x1 x1 x3  x1 x2 x3 x4 )x1 =(x1  x1 x2 x3 x4 )x1 = x1 x1  x1 x1 x2 x3 x4 =00=0
(x1x3 )x1 (x1 x2 )(x1 x3 )(x3 x2 )=( x1 x3  x1 x3)x1 (x1 x2 )(x1 x3 )(x3 x2)=
=( x1 x3  x1 x3x1 x2 )(x1 x3  x3 x2)= (x1 x3  x1 x3x1 x2 )(x1 x3  x2)= x1 x3x1  x1 x3 x3  x1 x3 x2  x1 x3 x1 
x1 x3 x3  x1 x3 x2  x1 x2 x1  x1 x2 x3  x1 x2 x2 = x1 x3  x1 x3 x2  x1 x3 x1 x2 x3  x1 x2 x3  x1 x2 =
= x1 x3 x1 x3  x1 x2 x3  x1 x2 = x1 x3 x1 x3  x1 x2
4.Esitada DNK-s funktsioon:
(x1 x2 x3) =x1 x2 x3
x1 x2 x3= (x1 x2 x1 x2)x3 =(x1 x2 x1 x2)x3 (x1 x2 x1 x2)x3 =(x1 x2 x1 x2)x3 (x1 x2 x3 x1 x2 x3)=
=(x1  x2)(x1  x2)x3x1 x2 x3 x1 x2 x3= (x1x1 x1x2  x1x2 x2x2)x3x1 x2 x3 x1 x2 x3=
= x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3
5.Tõestada võrdus avaldise teisenduse teel:
(x1 x2 )( x1 x2 )=0
(x1x2  x1x2)(x1x2  x1x2)= x1x2 x1x2  x1x2 x1x2  x1x2 x1x2  x1x2 x1x2 =0000=0
6.Leia, millised järgnevad võrdused kehtivad:
* x1 x2 = x1 x2 kehtib
x1 x2 = x1x2  x1x2= x1x2 x1x2=(x1 x2)(x1 x2)= x1 x1 x1 x2  x1 x2  x2 x2 = x1 x2  x1 x2
x1 x2= x1 x2  x1 x2= x1 x2  x1 x2
* x1 x2 = x1 x2 kehtib
x1 x2 = x1 x2  x1 x2= x1 x2  x1 x2
* x1 x2 = x1 x2 ei kehti
x1 x2 = x1 x2  x1 x2= x1 x2  x1 x2
* x1 x2 = x1 x2 kehtib
x1 x2 = x1 x2  x1 x2
x1 x2 = x1 x2  x1 x2
* x1 x2x1 x2 = x1  x2 kehtib
x1 x2x1 x2 =( x1 x2 x1  x1 x2 x1) x2 = x1 x2 x1 x2=( x1  x2) x1x2 = x1 x2x2 = x1 x2 x2  x1 x2 x2=
= x1 x2 ( x1  x2) x2= x1 x2  x1 x2  x2= x1 x2  x2= x1  x2
* x1 x1 x2= x1  x2 ei kehti
x1 x1 x2= x1 x1 x2 x1 x1 x2= x1 x2 x1(x1x2)= x1 x20 x1 x2= x1 x2 x1 x2= x1(x2x2)= x1*1= x1
x1  x2 = x1x2
*x1 x2 = x1 x2 ei kehti
x1 x2 = x1 x2  x1 x2
x1 x2 = x1 x2  x1 x2