Published by: BhumiRaj Timalsina
Published date: 30 Jun 2021
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.
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’
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 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.