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Introduction

Boolean algebra is a branch of algebra that deals with binary variables and logical operations. It is fundamental in digital logic design, computer architecture, and circuit design.

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Basic Boolean Operations

Boolean algebra operates on binary values: 0 (False) and 1 (True). The fundamental operations are:

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AND (⋅ or ∧)

The AND operation returns 1 only if both inputs are 1.

Truth Table for AND
A | B | A ⋅ B
--|--|------
0 | 0 |  0
0 | 1 |  0
1 | 0 |  0
1 | 1 |  1

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OR (+ or ∨)

The OR operation returns 1 if at least one input is 1.

Truth Table for OR
A | B | A + B
--|--|------
0 | 0 |  0
0 | 1 |  1
1 | 0 |  1
1 | 1 |  1

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NOT (¬ or !)

The NOT operation inverts the input.

Truth Table for NOT
A | ¬A
--|---
0 |  1
1 |  0

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Other Important Boolean Operations

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NAND (⊼)

NAND is the negation of AND.

Truth Table for NAND
A | B | A ⊼ B
--|--|------
0 | 0 |  1
0 | 1 |  1
1 | 0 |  1
1 | 1 |  0

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NOR (⊽)

NOR is the negation of OR.

Truth Table for NOR
A | B | A ⊽ B
--|--|------
0 | 0 |  1
0 | 1 |  0
1 | 0 |  0
1 | 1 |  0

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XOR (⊕)

XOR returns 1 if the inputs are different.

Truth Table for XOR
A | B | A ⊕ B
--|--|------
0 | 0 |  0
0 | 1 |  1
1 | 0 |  1
1 | 1 |  0

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XNOR (⊙)

XNOR returns 1 if the inputs are the same.

Truth Table for XNOR
A | B | A ⊙ B
--|--|------
0 | 0 |  1
0 | 1 |  0
1 | 0 |  0
1 | 1 |  1

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Boolean Algebra Laws

The following laws simplify Boolean expressions:

  • Identity Law: A + 0 = A, A ⋅ 1 = A
  • Null Law: A + 1 = 1, A ⋅ 0 = 0
  • Idempotent Law: A + A = A, A ⋅ A = A
  • Complement Law: A + ¬A = 1, A ⋅ ¬A = 0
  • Commutative Law: A + B = B + A, A ⋅ B = B ⋅ A
  • Associative Law: (A + B) + C = A + (B + C), (A ⋅ B) ⋅ C = A ⋅ (B ⋅ C)
  • Distributive Law: A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C)
  • De Morgan’s Theorems:
    • ¬(A ⋅ B) = ¬A + ¬B
    • ¬(A + B) = ¬A ⋅ ¬B

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Real-World Applications

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Digital Circuits

Boolean algebra is used in designing digital circuits such as:

  • Logic gates (AND, OR, NOT, etc.)
  • Flip-flops and memory storage
  • Arithmetic logic units (ALUs)

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Programming & Conditional Statements

Boolean expressions control loops and conditions in programming:

JavaScript Example
if (isUserLoggedIn && hasPermission) {
  console.log("Access granted");
} else {
  console.log("Access denied");
}

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Database Queries

SQL uses Boolean logic to filter data:

SQL Example
SELECT * FROM users WHERE age > 18 AND city = 'New York';

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Problem-Solving with Boolean Algebra

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Example 1: Simplify the Expression

Simplify A ⋅ (A + B).

Solution: Using the Absorption Law:

A ⋅ (A + B) = A

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Example 2: Circuit Implementation

Design a circuit that outputs 1 only when exactly one of two switches is ON.

Solution: Use the XOR gate (A ⊕ B).

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Conclusion

Boolean algebra is a foundational concept in computer science and engineering. Mastering it allows for efficient circuit design, logical problem-solving, and optimized programming logic.