Boolean Logic Notes – Class 11 CS (083) | Computer System Organization
These Boolean Logic Notes are the third part of Unit 1 – Computer System Organization of the CBSE Class 11 Computer Science (Code 083) curriculum.
These Boolean Logic Notes include well-labeled diagrams, truth tables, examples, and illustrations that help you understand concepts quickly and retain them for a longer time. All concepts are explained in a simple, pointwise, and exam-oriented manner according to the latest CBSE Class 11 Computer Science (083) syllabus.
Before starting this chapter, make sure you have completed the first two parts of Unit 1:
• Chapter 1 – Computer System Organization
• Chapter 2 – Encoding Schemes
If you haven’t studied them yet, I strongly recommend doing so first. The links to both chapters are provided below.
Chapter-1 Computer System Notes
Chapter-2 Encoding Scheme and Number System Notes
Boolean Logic
Boolean Algebra
- Boolean Algebra is a branch of mathematics used to represent and solve logical expressions. It deals with only two values:
- 1 (True / Yes / ON)
- 0 (False / No / OFF)
- It is widely used in digital electronics, computer programming, and logic circuit design.
Components of Boolean Algebra
Boolean Algebra consists of:
Elements
- Elements are the variables or constants used in Boolean Algebra.
They can have only two possible values:- 0 (False / OFF)
- 1 (True / ON)
- Example
- A = 0
- B = 1(Here, A and B are Boolean variables.)
Operators
- Operators are symbols used to perform logical operations on Boolean variables.
- There are three fundamental Boolean operators:
| Operator | Symbol | Function |
| AND | . | Logical Multiplication |
| OR | + | Logical Addition |
| NOT | ‘ or ¯ | Complementation |
- Example
- A . B → AND operation
- A + B → OR operation
- A’ → NOT operation
Axioms and Theorems
- Axioms and Theorems are the rules and laws used to simplify Boolean expressions.
- Examples of Boolean Laws
- A + 0 = A
- A . 1 = A
- A + A’ = 1
- A . A’ = 0
- These laws help in simplifying logical expressions and designing digital circuits.
Boolean Variable
- A Boolean variable is a symbol used to represent a logical quantity.
- It can take only two values from the set {0,1}.
- Boolean constants are single binary digits (bits), i.e., 0 and 1.
Boolean Operators
There are three fundamental Boolean operators:
AND Operator
- AND is a binary operator used for logical multiplication.
- It is represented by the symbol.
Truth Table of AND Operation
| X | Y | X . Y |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR Operator
- OR is a binary operator used for logical addition.
- It is represented by the symbol +
Truth Table of OR Operation
| X | Y | X + Y |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOT Operator
- NOT is a unary operator used to complement a variable.
- It is represented by ‘ or ¯
- Complement means the inverse of a variable or constant.
Truth Table of NOT Operation
| X | X’ |
| 0 | 1 |
| 1 | 0 |
In Boolean Algebra, since variables can have only values 0 or 1, their complements will always be the opposite value.
Axioms and Theorems of Boolean Algebra
Axioms and Theorems are the fundamental rules used in Boolean Algebra to simplify Boolean expressions and design digital circuits.
Axiom
An axiom is a statement that is accepted as true without proof.
Theorem
A theorem is a statement that can be proved using axioms and other previously proved theorems.
The rules used in Boolean Algebra are known as Boolean Laws.
Some Boolean laws are axioms, while others are theorems derived from those axioms.
Examples of Boolean Laws
| Boolean Expression | Law | Description |
| A + 0 = A A . 1 = A | Identity Law | Adding 0 or multiplying by 1 leaves the value unchanged. |
| A + B = B + A A . B = B . A | Commutative Law | The order of variables can be changed. |
| A . (B + C) = A . B + A . C A + (B . C) = (A + B) . (A + C) | Distributive Law | Multiplication distributes over addition and vice versa. |
| A + A’ = 1 A . A’ = 0 | Complement Law | A variable combined with its complement gives identity values. |
| A + A = A A . A = A | Idempotent Law | Repeating the same variable does not change the result. |
| A + 1 = 1 A . 0 = 0 | Null Element (Annulment) Law | Any variable ORed with 1 becomes 1, and ANDed with 0 becomes 0. |
| (A’)’ = A | Involution Law | Double complement of a variable gives the original value. |
| A + (B + C) = (A + B) + C A . (B . C) = (A . B) . C | Associative Law | Grouping of variables does not affect the result. |
De Morgan’s Laws
De Morgan’s Laws are important Boolean laws used to simplify logical expressions. They state that:
- The complement of an OR operation becomes the AND of the complements.
- The complement of an AND operation becomes the OR of the complements.
Truth Table
- A truth table is a table showing all possible combinations of input values and their corresponding outputs.
- The number of columns depends on the number of Boolean variables.
- For n variables, the total number of possible combinations is: 2n
Truth Table for AND, OR, NAND and NOR
| A | B | A AND B | A OR B | NOT(A AND B) | NOT(A OR B) |
| T | T | T | T | F | F |
| T | F | F | T | T | F |
| F | T | F | T | T | F |
| F | F | F | F | T | T |
Truth Table for A + B + C and A.B.C
For three variables A, B, and C: possible combinations obtained are.
| A | B | C | A.B.C | A + B + C |
| T | T | T | T | T |
| T | T | F | F | T |
| T | F | F | F | T |
| T | F | T | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
| F | T | F | F | T |
| F | F | F | F | F |
Truth Table for AB + C
| A | B | C | AB | AB + C |
| T | T | T | T | T |
| T | T | F | T | T |
| T | F | F | F | T |
| T | F | T | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
| F | T | F | F | T |
| F | F | F | F | F |
Logic Circuits
- A logic circuit is a digital electronic circuit made using logic gates to perform logical operations on data.
- It is also called a logic network or structure of logic gates.
- Electronic devices such as computers, calculators, robots, and video games use logic circuits in their functioning.
Logic Gates
- Logic gates are basic electronic circuits that perform Boolean operations such as AND, OR, and NOT.
- A logic gate produces output either as logic 0 or logic 1 depending on the input signals.
- Most logic gates have:
- Two inputs
- One output
Types of Logic Gates
There are seven basic logic gates:
- AND Gate
- OR Gate
- NOT Gate
- NAND Gate
- NOR Gate
- XOR Gate
- XNOR Gate
AND Gate
- An AND gate performs logical multiplication.
- It may have two or more inputs and one output.
Truth Table
| X | Y | X . Y |
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
OR Gate
- An OR gate performs logical addition.
- It may have two or more inputs and one output.
Truth Table
| X | Y | X + Y |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
NOT Gate
- A NOT gate is also known as an inverter.
- It reverses the logic state of the input.
- It has only one input and one output.
Truth Table
| X | X’ |
| 0 | 1 |
| 1 | 0 |
NAND Gate
- NAND gate is formed by combining NOT and AND operations.
- It can have two or more inputs and one output.
Truth Table
| X | Y | (X . Y)’ |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
NOR Gate
- NOR gate is formed by combining NOT and OR operations.
- It can have two or more inputs and one output.
Truth Table
| X | Y | (X + Y)’ |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
XOR Gate
- XOR stands for Exclusive OR.
- The output is true when only one of the inputs is true.
- The output becomes false when both inputs are same.
Truth Table
| X | Y | X ⊕ Y |
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
