Boolean Algebra

Boolean algebra refers to the branch of mathematics that basically deals with operations on various logical values with binary variables. Binary numbers represent the Boolean variables in order to represent truths: 0 = false and 1 = true. Remember that elementary algebra deals with all the numerical operations, but the Boolean algebra deals with all the logistical operations.

In this article, we will take a look at Boolean Algebra according to the GATE Syllabus for CSE (Computer Science Engineering). Read ahead to learn more.

Table of Contents

What is Boolean Algebra?
- Rules in Boolean Algebra
- Boolean Laws
Important Boolean Theorems
Practice Problems on Boolean Algebra
FAQs

What is Boolean Algebra?

We use Boolean Algebra to analyze the digital (logic) circuits and simplify them. It only makes use of the binary numbers 0 and 1. And it is also called Binary or Logical Algebra. George Boole invented Boolean algebra in 1854.

Rules in Boolean Algebra

Here are some of the very important rules that we use in Boolean algebra:

The used variable can only have two values. These are Binary 0 for LOW and Binary 1 for HIGH.
An overbar (-) represents the complement of a variable. Thus, the complement of any variable X would be represented as X̄. Thus X̄ = 1 if X = 0 and X̄ = 0 if X = 1.
We represent the ORing of the variables by inserting a (+) plus sign between them. For instance, the ORing of X, Y, Z is represented as X + Y + Z.
We represent the logical ANDing of the multiple variables by using a dot between each of them, like X.Y.Z. Sometimes the dot can even be omitted like XYZ.

Boolean Laws

Six types of Boolean Laws are there in total:

Commutative Law

A binary operation that satisfies the expression given below is known as a commutative operation.

(i) X.Y = Y.X (ii) X + Y = Y + X

The commutative law states that when we change the sequence of a variable, it does not cause any effect on the actual output of the given logic circuit.

Associative Law

The associative law states that the order of performing the logic operations is irrelevant since their effect remains the same.

(i) (X.Y).Z = X.(Y.Z) (ii) (X + Y) + Z = X + (Y + Z)

Distributive Law

The distributive law states the condition given below:

X.(Y + Z) = X.Y + X.Z

AND Law

This law makes use of the AND operation. Thus, it is known as the AND law.

(i) X.0 = 0 (ii) X.1 = X

(iii) X.X = X (iv) X.X̄ = 0

OR Law

This law makes use of the OR operation. It is thus known as the OR law.

(i) X + 0 = X (ii) X + 1 = 1

(iii) X + X = X (iv) X + X̄ = 1

INVERSION Law

The inversion law makes use of the NOT operation. It basically states that the double inversion of any given variable would result in the actual/ original variable itself.

Important Boolean Theorems

Following are a few very important and commonly used Boolean Theorems.

Boolean Theorems/Function	Description
Boolean Functions	Boolean Expressions and Function, NAND Gates realization and K-Map
De Morgan’s Theorems	De Morgan’s Theorems: Theorem 1 and Theorem 2

Do you wish to dive deeper into these concepts? Learn more about these theorems on our detailed article pages. Visit Boolean Functions and De Morgan’s Theorems to learn more about these in detail. Let us now understand the practical implication of Boolean Algebra.

Practice Problems on Boolean Algebra

1. F(A,B,C,D) = A`B`C`D`. The degree of function here is ________.

a) 1

b) 4

c) 2

d) 5

Answer – (b) 4

2. A Boolean expression represents a _____________ value.

a) Negative

b) Positive

c) Boolean

d) Recursive

Answer – (c) Boolean

3. Which of these refer to the canonical forms of a Boolean Expression?

a) SOM and POM

b) MAX and MIN

c) NOR and XNOR

d) OR and XOR

Answer – (a) SOM and POM

4. The ___________ of every variable in a direct or a complemented form refers to a maxterm.

a) moduler

b) product

c) subtraction

d) addition

Answer – (d) addition

FAQs

What are the basic properties of Boolean algebra?

A Boolean Algebra must always follow the three basic properties. They are: commutative, associative, and distributive.

What is Boolean algebra? Give an example.

How many laws are there in Boolean algebra, and what are they?

There are six types of Boolean laws in total, and they are:

Commutative Law
Associative Law
Distributive Law
AND Law
OR Law
INVERSION Law

Keep learning and stay tuned to get the latest updates on the GATE Exam along with Eligibility Criteria, GATE Syllabus for CSE (Computer Science Engineering), GATE CSE Notes, GATE CSE Question Paper, and more.

Also Explore,