Boolean Logic

Definition: What is Boolean Logic?

Like many concepts in computer science, Boolean logic is based on simple yet powerful principles. Named after George Boole, an English mathematician from the 19th century, Boolean Logic, at its core, is about classifying things as true or false. The most basic unit of Boolean Logic is a Boolean value, which can be either true (1) or false (0). 

The Importance of Boolean Logic in Computer Science

Basic Boolean Logic Principles and Rules

• \( A + A = A \) represents the idempotent law.
• \( A + \bar{A} = 1 \) stands for the law of nullity.
• \( A.1 = A \) and \( A.0 = 0 \) depict an invariant law.

Real Life Examples of Boolean Logic

Delving into Boolean Logic Gates

The Different Types of Boolean Logic Gates

• AND Gate - An output is true only when both its inputs are true.
• OR Gate - The output is true if any of its inputs is true.
• NOT Gate (or Inverter) - The output is the inverse of the input.
• NAND Gate - The combination of an AND gate followed by a NOT gate. The output is false only if all its inputs are true.
• NOR Gate - The combination of an OR gate followed by a NOT gate. The output is true only if all its inputs are false.
• XOR Gate (Exclusive OR) - The output is true only if the inputs are different.
• XNOR Gate (Exclusive NOR) - The output is true only if the inputs are similar.

Understanding How Boolean Logic Gates Function

Examples of Boolean Logic Gate Operations

Getting to Know Boolean Logic Operators

The Role of Boolean Logic Operators in Computer Science

Exploring Different Boolean Logic Operators

• AND: Commonly represented as (\( \cdot \)), the AND operator outputs true if all its operands are true. In other words, \( A \cdot B \) equals true only if both A and B are true.
• OR: Usually represented as (+), the OR operator outputs true if at least one of its operands is true. This means that \( A + B \) equals true as long as either A or B is true.
• NOT: Often symbolised as (\( \overline{} \)), the NOT operator inverts the value of its operand. Therefore, if \( A \) is true, then \( \overline{A} \) is false, and vice versa.

Practical Examples of Boolean Logic Operators

num = 10
if num > 0 AND num < 20:
    print("Num is between 0 and 20")
else:
    print("Num is not between 0 and 20")

Boolean Logic Simplification Explanation

The Importance of Boolean Logic Simplification

Steps for Boolean Logic Simplification

• Identity Laws: These state that \( A + 0 = A \) and \( A . 1 = A \).
• Domination Laws: These state that \( A + 1 = 1 \) and \( A . 0 = 0 \).
• Idempotent Laws: These suggest that \( A + A = A \) and \( A . A = A \).
• Double Negation Law: This states that \( \overline{\overline{A}} = A \).
• Complement Laws: These express that \( A + \overline{A} = 1 \) and \( A . \overline{A} = 0 \).
• Commutative Laws: These signify that \( A + B = B + A \) and \( A . B = B . A \).
• Associative Laws: These explain that \( A + (B + C) = (A + B) + C \) and \( A . (B . C) = (A . B) . C \).
• Distributive Laws: These inform us that \( A . (B + C) = (A . B) + (A . C) \) and \( A + (B . C) = (A + B) . (A + C) \).

Practical Example of Boolean Logic Simplification

A . \overline{A + B} (Original Expression)
= A . (\overline{A} . \overline{B})  (De Morgan's Law)
= A.\overline{A} . \overline{B} (Distributive Law)
= 0 . \overline{B} (Complement Law)
= 0 (Domination Law)

Boolean Logic Rules Expounded

Understanding Boolean Logic Rules

Application of Boolean Logic Rules in Computer Science

Illustrative Examples of Boolean Logic Rule Application

  boolean A = true;
  boolean B = false;
  if(A || B) {
    System.out.println("At least one is true");
  } else {
    System.out.println("None of them is true");
  }

  bool x = true;
  bool y = true;
  if(x && y) {
    std::cout << "Both are true";
  } else {
    std::cout << "At least one is false";
  }