Boolean Algebra: Introduction, Calculation & Examples

Boolean algebra is a significant part of mathematics that focuses on dealing with operations that involve binary variables in specific. Application of boolean algebra contributes towards analysing and the interpretation of digital gates or circuits. Boolean Algebra is commonly referred to as binary algebra or logical algebra. It plays an important role in the process of developing different digital electronics and different programming languages as well. It is further extensively used in set theory and statistics as well.

Some of the important operations that are performed in boolean algebra are – disjunction (∨), conjunction (∧), and negation (¬). this approach is considered to be significantly different from elementary algebra, where the variable’s values are numerical and arithmetic operations. Besides, subtraction is being performed on them.

Boolean Algebra: Operations

Look at the following points to know the basic operations of boolean algebra:

1. Conjunction or AND operation
2. Disjunction or OR operation
3. Negation or NOT operation

In the following table, we have defined the symbols for all three basic operations of Boolean Algebra:

Assume, P and Q are two boolean variables, then the three operations can be defined as-

• P conjunction Q or P AND Q, satisfies P ∧ Q = True, if P = Q = True or else P ∧ Q = False.
• Negation P or ¬P satisfies ¬P = False, if P = True and ¬P = True if P = False.
• P disjunction Q or P OR O, satisfies P ∨ Q = False, if P = Q = False, else P ∨ Q = True.

Boolean Expression: Overall Details

A logical statement that helps to understand the value of boolean number, either be False or True, is referred as a boolean expression. Sometimes, we used synonyms to express the statement- ‘No’ for ‘False’ and ‘Yes’ for ‘True .’ Moreover, 1 and 0 used for digital circuits for False and True, respectively.

Boolean expressions are the statements that use logical operators, i.e., AND, XOR, OR, and NOT. Thus, if we write ‘X’ AND ‘Y’ = True, then it is a boolean expression.

Boolean Algebra: List of Terminologies

In the following, students can see some important terminologies of Boolean algebra-

1. Boolean Algebra- It is the branch of algebra that deals with binary variables and logical operations.
2. Boolean Function- It is consists of logical operators, binary variables, constants such as 0 and 1, the parenthesis symbols, and equal to the operator.
3. Literal- It is a compliment of a variable and a variable.
4. Complement- It is described as the inverse of the variable represented by a bar over the variable.
5. Boolean Variables- Alphabets that specify the logical quantities like- 0 and 1 are defined as a variable or a symbol.
6. Truth Table- It provides all the possible values of logical variables and the combination of the variables. It is possible to transform the boolean equation into a truth table. The number of rows in the truth table should be equal to 2n, where “n” is the number of variables in the equation.

Boolean Algebra Truth Table

By expressing the above operation in the truth table, we get:

Boolean Algebra: Rules

Check the points mentioned below to know the rules of Boolean Algebra:

• Variable have only two values. Binary 0 for LOW and Binary 1 for HIGH.
• An overbar represents the complement of a variable. Thus, the complement of variable Q is denoted as Q. Thus, if Q = 0, then Q=1 and Q = 1, then Q= 0.
• OR-ing of the variables indicated by a plus (+) sign between them. For example- the OR-ing of P, Q, R is denoted as P + Q + R.
• The Logical AND-ing of the two or more variables described by writing a dot between them, such as P.Q.R. Sometimes, the dot may be omitted like PQR.

Boolean Algebra: Laws

There are six types of Boolean Algebra Laws-

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

Commutative Law

Commutative law asserts that changing the sequence of the variables does not affect the output of a logic circuit.

1. P.Q = Q.P
2. P + Q = Q + P

Associative Law

Associative Law asserts that the order in which the logic operations are implemented is irrelevant as their effect is the same.

1. (P.Q).R = P.(Q.R)
2. (P + Q) + R = P + (Q + R)

Distributive Law

Following conditions asserts the distributive law-

1. P.(Q + R) = (P.Q) + (P.R)
2. P + (Q.R) = (P + Q) . ( P + R)

AND Law

‘AND’ operations are used in these types of law. It has the following conditions-

1. P.0 = 0
2. P.1 = P
3. P.P = P
4. P.P=0

OR Law

‘OR’ operations are used in these types of law. It has the following conditions-

1. P+0 = P
2. P+1 = 1
3. P+P = P
4. P+P= 1

Inversion Law

A ‘NOT’ operation is used in the inversion law. The inversion law asserts that double inversion of variable results in the original variable itself.

1. P + P= 1

FAQs On Boolean Algebra

Frequently asked questions related to boolean algebra is listed as follows:

We conclude that Boolean Logic is a kind of algebra in which the variables have a logical value of ‘TRUE’ or ‘FALSE.’ Also, AND = Can be considered BOTH. It claims that both or all objects (search terms) be present in the results.