Important Rules of Boolean Algebra

Definition of Boolean Algebra

Boolean Algebra are a set of rules that are used to simplify a complex logic expression without changing it’s functionality.
Boolean Algebra was introduced by the English Mathematician George Boole in 1847.

Rules of Boolean Algebra

1. Complement Rule:
   Example –
   0′ = 1,
   1′ = 0,
   (A’)’ = A,
2. AND Rule:
   Example –
   A.A = A,
   A.0 = 0,
   A.1 = A,
   A.A’ = 0,
3. OR Rule:
   Example –
   A + A = A,
   A + 0 = A,
   A + 1 = 1,
   A + A’ = 1,
4. Distributive Law:
   Example –
   A(B + C) = A.B + A.C,
   A + B.C = (A + B)(A + C),
   A + A’.B = A + B, Similarly, A’ + A.B = A’ + B,
5. Commutative Law:
   Example –
   A + B = B + A,
   A.B = B.A,
6. Associative Law:
   Example –
   (A.B)C = A(B.C),
7. De Morgan’s Law:
   Example –
   (A + B)’ = A’.B’,
   (A.B)’ = A’ + B’,
8. Redundancy Theorem or Consensus Theorem:
   Redundancy Theorem is a trick but it will only apply when all the below conditions are satisfied.

• Three variables must be present
• Each variable must be repeated twice
• One variable must be complimented

Then we can keep the complimented variable and remove the extra variable which is the redundant variable.
A.B + A’.C + B.C = A.B + A’.C

Priority of Logic Gates

When we have multiple Logic Gates in a Digital Circuit or Logical Operation it follows the below priority or order of execution:

1. NOT
2. AND
3. OR

Conclusion

Above rules help us greatly minimizing Boolean Algebraic expressions, which in real-life translates to less cost, and more hardware efficiency.