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 are been performed on them.

 

Boolean Algebra

 

Basic operations of Boolean Algebra

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

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

Also, read:

Rules in Boolean Algebra

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.

Laws of Boolean Algebra

There are six types of Boolean laws. Th0se 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.

  • \(A+\bar{\bar{A}}=1\)

Boolean Algebra Example

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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