Laws of Boolean Algebra
The fundamental laws of Boolean Algebra can be expressed as follows:
The Commutative Law states that swapping the order of the operands in a Boolean expression does not affect the outcome. For instance:
For the OR operator: A + B = B + A:
For the OR operation: A + B is equivalent to B + A.
The Associative Law of multiplication states that the AND operation can be performed on two or more variables without affecting the result. For example:
A * (B * C) is equivalent to (A * B) * C.
The Distributive Law states that multiplying two variables and then adding a third variable yields the same result as multiplying the third variable with each of the original variables separately and then adding them. For example:
A + BC is equivalent to (A + B) (A + C).
Annulment law:
A.0 = 0
A+1=1
Identify law:
A.1 = A
A+0 + A
Idempotent law:
A + A = A
A.A = A
Complement law:
A + A' =1
A.A' = 0
Double negation law:
((A)')' = A
Absorption law:
A. (A+B) = A
A+ AB =A
De Morgan's Law, also known as De Morgan's theorem, is based on the principle of duality. Duality suggests that by swapping operators and variables in a function—such as replacing 0 with 1 and 1 with 0, as well as changing the AND operator to OR and the OR operator to AND—the function's behavior remains consistent.
De Morgan proposed two theorems that are useful for solving algebraic problems in digital electronics. These theorems are expressed as follows:
- "The negation of a conjunction is the disjunction of the negations" means that the complement of the product of two variables is equal to the sum of the complements of the individual variables. For example, (A * B)' = A' + B'.
- "The negation of a disjunction is the conjunction of the negations" means that the complement of the sum of two variables is equal to the product of the complements of each variable. For example, (A + B)' = A' * B'.
