In order to deal mathematically with the digital circuits, We need to use the Boolean Algebra. It is a tool for the analysis and design of digital systems. Boolean Algebra differs in a major way from ordinary algebra because Boolean constant and variables are allowed to have only two possible values 0 or 1.
Laws of Boolean Algebra:- The three basic lows of Boolean Algebra are the same as in ordinary Algebra;
- Commutative Laws
- Associative Laws
- Distributive Laws
1.Commutative Laws This Law state that
A + B = B + A
Law2: For multiplication:- This law state that
A . B = B . A
Law1: For Addition:- This Law state that
A + ( B + C ) = ( A + B ) + C
Law2: For Multiplication:- This law state that
A ( B C ) = ( A B )C
Distributive Law:- this law state that
A (B + C) = A.B + A.C
Properties of Boolean Algebra:-
1. A + 0 = A2. A + 1 = 13. A . 0 = 04. A . 1 = A
5. A + A = A6. A + Ã = 17. A . A = A
8. A . Ã = 09.
10. A + AB = A
A(1+B) = A
A(1) = A
A = A11. A + ÃB = A + B
According to rule 10 put ( A+AB) in place of A
So. A+AB + ÃB = A+B
A+B ( A+Ã) = A+B
A +B(1) = A+B
A+B = A+B12. (A+B)(A+C) = A+BC
AA + AC + AB + BC = A+BC
A+AC+AB+BC = A+BC
A(1+C)+AB+BC = A+BC
A+AB+BC = A+BC
A (1+B) +BC = A+BC
A +BC = A+BCDuality Theorem:- According to the duality Theorem a Boolean relation can be written to form another Boolean relation by
- Changing each OR sign to an AND sign.
- Changing each AND sign to an OR sign
- Complementing any 0 or 1 appearing in the expression.
For example: Distributive Law state that.
A(B+C) = AB+BC
A+(B.C) = (A+B).(A+C)