Boolean Laws

There are six types of Boolean Laws.

Commutative law

Any binary operation which satisfies the following expression is referred to as commutative operation.

Commutative Law

Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.

Associative law

This law states that the order in which the logic operations are performed is irrelevant as their effect is the same.

Associative Law

Distributive law

Distributive law states the following condition.

Distributive Law

AND law

These laws use the AND operation. Therefore they are called as AND laws.

AND Law

OR law

These laws use the OR operation. Therefore they are called as OR laws.

OR Law

INVERSION law

This law uses the NOT operation. The inversion law states that double inversion of a variable results in the original variable itself.

NOT Law

De Morgan's Theorems

De Morgan has suggested two theorems which are extremely useful in Boolean Algebra. The two theorems are discussed below.

Theorem 1

De Morgan Theorem 1

  • The left hand side (LHS) of this theorem represents a NAND gate with inputs A and B, whereas the right hand side (RHS) of the theorem represents an OR gate with inverted inputs.

  • This OR gate is called as Bubbled OR.

De Morgan Theorem 1 Diagram

Table showing verification of the De Morgan's first theorem −

De Morgan Theorem 1 Verification Table

Theorem 2

De Morgan Theorem 2

  • The LHS of this theorem represents a NOR gate with inputs A and B, whereas the RHS represents an AND gate with inverted inputs.

  • This AND gate is called as Bubbled AND.

De Morgan Theorem 2 Diagram

Table showing verification of the De Morgan's second theorem −

De Morgan Theorem 2 Verification Table


Applications of De Morgan’s Theorems and Boolean Laws

  1. Circuit Simplification: De Morgan’s theorems help simplify complex logic circuits by transforming AND gates into OR gates (or vice versa) and minimizing the use of NOT gates.

  2. Logic Gate Design: These theorems are crucial when designing and optimizing digital circuits using NAND or NOR gates, which are the building blocks of many circuits.

  3. Programming: In software development, Boolean laws are used to optimize conditional statements, improve code efficiency, and reduce complexity.

  4. Error Detection: By applying these laws, it's easier to detect and correct logical errors in Boolean expressions or circuit designs.