Boolean Algebra

Boolean algebra is the category of algebra in which the variable’s values are the truth values, true and false, ordinarily denoted 1 and 0 respectively. It is used to analyze and simplify digital circuits. It is also called as Binary Algebra or logical Algebra. It has been fundamental in the development of digital electronics and is provided for in all modern programming languages. It is also used in set theory and statistics.

The important operations performed in boolean algebra are – conjunction (∧), disjunction (∨) and negation (¬). Hence, this algebra is far way different from elementary algebra where the values of variables are numerical and arithmetic operations like addition, subtraction is been performed on them.

Boolean Algebra Operations

The basic operations of Boolean algebra are as follows:

• Conjunction or AND operation
• Disjunction or OR operation
• Negation or Not operation

Below is the table defining the symbols for all three basic operations.

 Operator Symbol Precedence NOT ‘ (or) ¬ Highest AND . (or) ∧ Middle OR + (or) ∨ Lowest

Suppose A and B are two boolean variables, then we can define the three operations as;

• A conjunction B or A AND B, satisfies A ∧ B = True, if A = B = True or else A ∧ B = False.
• A disjunction B or A OR B, satisfies A ∨ B = False, if A = B = False, else A ∨ B = True.
• Negation A or ¬A satisfies ¬A = False, if A = True and ¬A = True if A = False

Boolean Algebra Truth Table

Now, if we express the above operations in a truth table, we get;

 A B A ∧ B A ∨ B True True True True True False False True False True False True False False False False
 A ¬A True False False True

Boolean Algebra Rules

Following are the important rules used in Boolean algebra.

• Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.
• The complement of a variable is represented by an overbar. Thus, complement of variable B is represented as $\bar{B}$. Thus if B = 0 then $\bar{B}$=1 and B = 1 then $\bar{B}$ = 0.
• OR-ing of the variables is represented by a plus (+) sign between them. For example OR-ing of A, B, C is represented as A + B + C.
• Logical AND-ing of the two or more variable is represented by writing a dot between them such as A.B.C. Sometimes the dot may be omitted like ABC.
 Related Links Truth Table Tautology Conjunction Mathematical Logic

Laws of Boolean Algebra

There are six types of Boolean algebra laws. They are:

• Commutative law
• Associative law
• Distributive law
• AND law
• OR law
• Inversion law

Those six laws are explained in detail here.

Commutative Law

Any binary operation which satisfies the following expression is referred to as a commutative operation. Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit.

• A. B = B. A
• A + B = B + A

Associative Law

It states that the order in which the logic operations are performed is irrelevant as their effect is the same.

• ( A. B ). C = A . ( B . C )
• ( A + B ) + C = A + ( B + C)

Distributive Law

Distributive law states the following conditions:

• A. ( B + C) = (A. B) + (A. C)
• A + (B. C) = (A + B) . ( A + C)

AND Law

These laws use the AND operation. Therefore they are called AND laws.

• A .0 = 0
• A . 1 = A
• A. A = A
• $A. \bar{A}= 0$

OR Law

These laws use the OR operation. Therefore they are called OR laws.

• A  + 0 = A
• A + 1 = 1
• A + A = A
• $A + \bar{A}= 1$

Inversion Law

This law uses the NOT operation. The inversion law states that double inversion of variable results in the original variable itself.

• $A+\bar{\bar{A}}=1$

Example of Boolean Algebra Simplication

Question: Simplify the following expression: $c+\bar{BC}$

Solution:

Given: $C+\bar{BC}$

According to Demorgan’s law, we can write the above expressions as

$C+(\bar{B}+ \bar{C})$

From Commutative law:

$(C+\bar{C})+ \bar{B}$

From Complement law

$1+ \bar{B}$ = 1

Therefore, $C+\bar{BC} = 1$

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