Boolean Algebra

Boolean Variables

for base system r there will be r symbols used to represent the system

• for r=2 symbols are: 0, 1
• for r = 10, {0, 1, 2,...9}
• for r = 8, {0, 1, 2,...,7 }

Boolean Operators

• Unitory: NOT
• Binary: AND, OR

OR operator

The result of OR operation is ONE if atleast one variable is ONE

A	B	A+B
0	0	0
0	1	1
1	0	1
1	1	1

important properties

• when A = 0, y = B
• when A = 1, y = 1
• when A = B = x, y = x
• when A != B, y = 1

In OR implementation the enable input=0 and disable input=1

AND operator

The result of AND operation is ZERO if atleast one variable is ZERO

A	B	A+B
0	0	0
0	1	0
1	0	0
1	1	1

important properties

• when A = 0, y = 0
• when A = 1, y = B
• when A = B = x, y = x
• when A != B, y = 0

In OR implementation the enable input=1 and disable input=0

NOT operator

it changes the input ONE to ZERO and ZERO to ONE, denoted by y = A'

A	y = A'
0	1
1	0

Boolean Algebra Properties

OR	AND
1. A+A+A.....A = A	A.A.....A = A
2. A+0 = A	A.1 = A
1. A+1 = 1	A.0 = 0
1. A.A' = 0	A + A' = 1
5. A'' = A
6. A + BC = (A+B)(B+C)	A(B+C) = AB + AC
7. A + A'B = A+B	A(A'B) = AB
8. A'AB = A'+B	A'(A+B) = A'B
9. A + AB = A	A(A+B) = A
10. AB + A'C + BC = AB + A'C ( consensus property)	(A+B)(A' + C)(B+C) = (A+B)(A'+C)
11. (x+y)(x' + z) = xz + x'y	xy + x'z = (x+z)(x'+y)
12. (A+B+C+....)' = A'.B'.C'....	(ABC....)' = A' + B' + C'... (demorgan's property)