Boolean Algebra

Boolean algebra is the method of expressing logic in a mathematical context. It is a form of symbolic logic in which variables have only two values i.e. either true (1) or false (0). The relationship between the two values is expressed by the boolean operators: AND, OR, and NOT. It is used to solve and minimize the logic equation.

Some Theorem in Boolean Algebra

If A, B, and C are three variables capable of having values 0 and 1 then, the following are some important theorems:

1.Commutation Theorems

A+B=B+A

AB=BA

2.Association Theorems

A+(B+C)=(A+B)+C

A(BC)=(AB)C

3.Distribution Theorems

A+(BC)=(A+B)(A+C)

A(B+C)=AB+AC

4.Absorption Theorems

A+AB=A

A(A+B)=A

5.De morgan’s Theorem

(A+B)’=A’B’

(AB)’=A’+B’

Basic Principle of Boolean Algebra

Three basic logic operations are named AND, OR, and NOT are discussed below:

Logic addition (OR operation)	Logic Multiplication   (AND operation)	Inversion   (NOT operation)
0+0=0 0+1=1 1+0=1 1+1=1	0.0=0 0.1=0 1.0=0 1.1=1	(Not zero)=1
		(Not one)=0

 

De-Morgan’s theorem

De Morgan gave two important theorems that are very helpful for simplifying the logic expressions.

First theorem: 

It states that the complement of a sum equals the product of complements.

Mathematically, (A+B)’=A’.B’

Second theorem:

It states that the complement of the product equals the sum of the complements.

Mathematically, (AB)’=A’+B’

The above theorem can be verified by using truth table. For the truth table, we use a binary number system i.e. both A and B can have value either 0 or 1.

A	B	A’	B’	A+B	(A+B)’	A.B	(A.B)’	A’+B’	A’.B’
0 0 1 1	0 1 0 1	1 1 0 0	1   0 1 0	0   1 1 1	1   0 0 0	0   0 0 1	1   1 1 0	1   1 1 0	1 0 0 0

From the truth table,

(A+B)’=A’.B’ and (AB)’=A’+B’

Hence, De-Morgan’s theorem is verified.