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.