Basic Theorems of Boolean Algebra

Basic Theorem / Postulates of Boolean Algebra and its Truth Tables

1.       Properties of 0 and 1

1.1.    0 + X = X

0	X	R
0	0	0
0	1	1

1.2.    0.X = 0

0	X	R
0	0	0
0	1	0

1.3.    1 + X = 1

1	X	R
1	0	1
1	1	1

1.4.    1.X = X

1	X	R
1	0	0
1	1	1

2.       Idempotent Law

2.1.    X + X = X

X	X	R
0	0	0
1	1	1

2.2.    X.X = X          

     

X	X	R
1	0	0
1	1	1

3.       Involution

3.1.    (X’)’ = X

X	X’	(X’)’
0	1	0
1	0	1

4.       Complement Law

4.1.    X + X’ = 1

X	X’	R
0	1	1
1	0	1

4.2.    X.X’ = 0

X	X’	R
0	1	0
1	0	0

5.       Commutative Law

5.1.    X + Y = Y + X

X	Y	X+Y	Y+X
0	0	0	0
0	1	1	1
1	0	1	1
1	1	1	1

5.2.    X.Y = Y.X

X	Y	X.Y	Y.X
0	0	0	0
0	1	0	0
1	0	0	0
1	1	1	1

6.       Associative Law

6.1.    X + (Y + Z) = (X + Y) + Z

X	Y	Z	X+Y	Y+Z	X+(Y+Z)	(X+Y)+Z
0	0	0	0	0	0	0
0	0	1	0	1	1	1
0	1	0	1	1	1	1
0	1	1	1	1	1	1
1	0	0	1	0	1	1
1	0	1	1	1	1	1
1	1	0	1	1	1	1
1	1	1	1	1	1	1

  

6.2.    X.(Y.Z) = (X.Y).Z

X	Y	Z	X.Y	Y.Z	X.(Y.Z)	(X.Y).Z
0	0	0	0	0	0	0
0	0	1	0	0	0	0
0	1	0	0	0	0	0
0	1	1	0	1	0	0
1	0	0	0	0	0	0
1	0	1	0	0	0	0
1	1	0	1	0	0	0
1	1	1	1	1	1	1

7.       Distributive Law

7.1.    X(Y + Z) = XY + XZ

X	Y	Z	Y+Z	X.Y	X.Z	X.(Y+Z)	X.Y+X.Z
0	0	0	0	0	0	0	0
0	0	1	1	0	0	0	0
0	1	0	1	0	0	0	0
0	1	1	1	0	0	0	0
1	0	0	0	0	0	0	0
1	0	1	1	0	1	1	1
1	1	0	1	1	0	1	1
1	1	1	1	1	1	1	1

  7.2.    X + (Y.Z) = (X + Y ).(X + Z) 

X	Y	Z	Y.Z	X+Y	X+Z	X+(Y.Z)	(X+Y).(X+Z)
0	0	0	0	0	0	0	0
0	0	1	0	0	1	0	0
0	1	0	0	1	0	0	0
0	1	1	1	1	1	1	1
1	0	0	0	1	1	1	1
1	0	1	0	1	1	1	1
1	1	0	0	1	1	1	1
1	1	1	1	1	1	1	1

 8.       Absorption Law 

8.1.    X + XY = X

X	Y	X.Y	X+(X.Y)
0	0	0	0
0	1	0	0
1	0	0	1
1	1	1	1

  

8.2.    X .(X + Y) = X

X	Y	X+Y	X.(X+Y)
0	0	0	0
0	1	1	0
1	0	1	1
1	1	1	1

 9.       3^rd Distributive Law

9.1.    X + X’Y = X + Y

X	Y	X’	X’.Y	X+ X’Y	X+Y
0	0	1	0	0	0
0	1	1	1	1	1
1	0	0	0	1	1
1	1	0	0	1	1