Boolean Algebra Theorems and Laws of Boolean AlgebraPublished on 24/2/2012 and last updated on 25/8/2018
There are also few theorems of Boolean algebra, that are needed to be noticed carefully because these make calculation fastest and easier. Boolean logic deals with only two variables, 1 and 0 by which all the mathematical operations are to be performed. Boolean algebra or switching algebra is a system of mathematical logic to perform different mathematical operations in binary system. There only three basis binary operations, AND, OR and NOT by which all simple as well as complex binary mathematical operations are to be done. There are many rules in Boolean algebra by which those mathematical operations are done. In Boolean algebra, the variables are represented by English Capital Letter like A, B, C etc and the value of each variable can be either 1 or 0, nothing else. In Boolean algebra an expression given can also be converted into a logic diagram using different logic gates like AND gate, OR gate and NOT gate, NOR gates, NAND gates, XOR gates, XNOR gates etc.
Some basic logical Boolean operations, AND Operation OR Operation Not Operation
Some basic laws for Boolean Algebra A . 0 = 0 where A can be either 0 or 1. A . 1 = A where A can be either 0 or 1. A . A = A where A can be either 0 or 1. A . Ā = 0 where A can be either 0 or 1. A + 0 = A where A can be either 0 or 1. A + 1 = 1 where A can be either 0 or 1. A + Ā = 1 A + A = A A + B = B + A where A and B can be either 0 or 1. A . B = B . A where A and B can be either 0 or 1. The laws of Boolean algebra are also true for more than two variables like,
Cumulative Law for Boolean AlgebraAccording to Cumulative Law, the order of OR operations and AND operations conducted on the variables makes no differences.
Associative Laws for Boolean AlgebraThis law is for several variables, where the OR operation of the variables result is same though the grouping of the variables. This law is quite same in case of AND operators.
Distributive Laws for Boolean AlgebraThis law is composed of two operators, AND and OR. Let us show one use of this law to prove the expression
Redundant Literal RuleFrom truth table,
|A||B||ĀB||A + ĀB|
Absorption Laws for Boolean AlgebraProof from truth table,
Examples of Boolean AlgebraThese are another method of simplifying complex Boolean expression. In this method we only use three simple steps.
- Complement entire Boolean expression.
- Change all ORs to ANDs and all ANDs to ORs.
- Now, complement each of the variable and get final expression.