Boolean Logic

Overview

Boolean logic is a form of algebra centered around three main logical operators: AND, OR, and NOT. It is the fundamental building block of all digital computers, as it allows the CPU to make decisions based on binary values (true or false, 1 or 0).

Core Concepts

Boolean Values: Data that can only have one of two possible values: True or False.
Basic Logical Operators:
- AND ($\land$): Result is true only if all operands are true.
- OR ($\lor$): Result is true if at least one operand is true.
- **NOT ($

eg$)**: Inverts the value (True $

ightarrow$ False, False $

ightarrow$ True).

Derived Operators:
- XOR (Exclusive OR): Result is true if exactly one operand is true (but not both).
- NAND (Not AND): The inverse of AND; true unless all inputs are true.
- NOR (Not OR): The inverse of OR; true only if all inputs are false.
Truth Tables: A mathematical table used to determine the output of a logical expression for all possible combinations of inputs.

Logic Laws

Identity Law: $A \land ext{True} = A$; $A \lor ext{False} = A$.
Null Law: $A \land ext{False} = ext{False}$; $A \lor ext{True} = ext{True}$.
Idempotent Law: $A \land A = A$; $A \lor A = A$.
De Morgan’s Laws: Critical for simplifying complex logic:
- $

eg(A \land B) =

eg A \lor

eg B$

eg(A \lor B) =

eg A \land

eg B$

Code Examples

// Basic Logic in TypeScript
const isAdult = true;
const hasTicket = false;

// AND: Must have both
const canEnter = isAdult && hasTicket; // false

// OR: Must have at least one
const canEnterWithGuestPass = isAdult || hasTicket; // true

// NOT: Invert the state
const isBanned = false;
const canAccess = !isBanned; // true

// XOR (Implemented using != since there is no native XOR for booleans in TS)
const hasApple = true;
const hasOrange = false;
const hasExactlyOneFruit = hasApple !== hasOrange; // true

Use Cases

Conditional Statements: The core of if, while, and for loop conditions.
Circuit Design: Logic gates (AND, OR, NOT gates) in CPU hardware.
Search Queries: Using “AND”, “OR”, and “NOT” to filter results in databases or search engines.
Permission Systems: Checking if a user has (isAdmin OR isEditor) AND isActive.

Gotchas

Truthiness vs. Booleans: In languages like JavaScript, non-boolean values can be treated as true or false (e.g., 0 is falsy, 1 is truthy). This is different from strict Boolean logic.
Short-Circuiting: In A && B, if A is false, B is never evaluated. This can lead to bugs if B was a function that needed to execute for its side effects.