Boolean Algebra is a branch of mathematics that deals with binary values, 0 and 1, and is widely used in digital electronics and computer systems to simplify logical expressions.
Note: In Boolean Algebra, the symbol + represents the OR operation (Boolean addition), and the symbol · represents the AND operation (Boolean multiplication).
The properties (laws) are the basic rules used to simplify Boolean expressions.
Annulment law
A variable ANDed with 0 gives 0, while a variable ORed with 1 gives 1, i.e.,
A.0 = 0
A + 1 = 1
Identity law
In this law variable remains unchanged it is ORed with '0' or ANDed with '1', i.e.,
A.1 = A
A + 0 = A
Idempotent law
A variable remains unchanged when it is ORed or ANDed with itself, i.e.,
A + A = A
A.A = A
Complement law
In this Law if a complement is added to a variable it gives one, if a variable is multiplied with its complement it results in '0', i.e.,
A + A' = 1
A.A' = 0
Double Negation Law
A variable with two negations, its symbol gets cancelled out and original variable is obtained, i.e.,
((A)')'=A
Commutative law
A variable order does not matter in this law, i.e.,
A + B = B + A
A.B = B.A
Associative law
The order of operation does not matter if the priority of variables are the same like '*' and '/', i.e.,
A+(B+C) = (A+B)+C
A.(B.C) = (A.B).C
Distributive law
This law governs the opening up of brackets, i.e.,
A.(B+C) = (A.B)+(A.C)
(A+B)(A+C) = A + BC
Absorption law
The absorption law consists of two dual statements:
X.(X+Y) = X
X+XY = X
De Morgan law
In De Morgan law, the operation of an AND or OR logic circuit is unchanged if all inputs are inverted, the operator is changed from AND to OR, and the output is inverted, i.e.,
(A.B)' = A' + B'
(A+B)' = A'.B'
Consensus theorem
AB + A'C + BC = AB + A'C
➣Practice: Solved Examples
