Summary Logic Gates Identities with Truth Table as proof

LOGIC IDENTITIES
Distributive Law
 +  =  + 
 +   + 
A B C X1 A B C X2
0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0
0 1 1 0 0 1 1 0
1 0 0 0 1 0 0 0
1 0 1 1 1 0 1 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1


 +  =  +  + 
 +   +  + 
A B C X1 A B C X2
0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0
0 1 1 1 0 1 1 1
1 0 0 1 1 0 0 1
1 0 1 1 1 0 1 1
1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1


Associative Law
 =  = 
  
A B C X1 A B C X2 A B C X3
0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0 0 1 0
0 1 0 0 0 1 0 0 0 1 0 0
0 1 1 0 0 1 1 0 0 1 1 0
1 0 0 0 1 0 0 0 1 0 0 0
1 0 1 0 1 0 1 0 1 0 1 0
1 1 0 0 1 1 0 0 1 1 0 0
1 1 1 1 1 1 1 1 1 1 1 1

,  +  +  =  +  +  =  +  + 
++  +  +   +  + 
A B C X1 A B C X2 A B C X3
0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 1 0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1 0 1 0 1
0 1 1 1 0 1 1 1 0 1 1 1
1 0 0 1 1 0 0 1 1 0 0 1
1 0 1 1 1 0 1 1 1 0 1 1
1 1 0 1 1 1 0 1 1 1 0 1
1 1 1 1 1 1 1 1 1 1 1 1


Commutative Law
 = 
 
A B X1 B A X2
0 0 0 0 0 0
0 1 0 0 0 0
1 0 0 0 1 0
1 1 1 0 1 1


+= +
+ +
A B X1 B A X2
0 0 0 0 0 0
0 1 1 0 0 1
1 0 1 0 1 1
1 1 1 0 1 1


ANDed with 1
) = 
)
A 1 X1
0 1 0
1 1 1
• Output will follow the value of input A.