Discrete Mathematics

Core Concepts

Core Concepts

1. Logic and Proofs

• Propositional Logic: The study of logical statements (propositions) that can be either true or false. It involves logical operators like AND (∧), OR (∨), NOT (¬), and IMPLIES (→).
• Truth Tables: A way to systematically determine the truth value of a complex logical expression.
• Predicate Logic: An extension of propositional logic that allows for variables and quantifiers (∀ for all, ∃ there exists).
• Proof Techniques: Methods to formally prove mathematical statements, such as Direct Proof, Proof by Contradiction, and Proof by Induction.

2. Set Theory

• Sets: A collection of distinct objects, considered as an object in its own right. E.g., A = {1, 2, 3}.
• Operations: Union (∪), Intersection (∩), Difference (-), Complement.
• Venn Diagrams: Visual representations of sets and their relationships.
• Power Set: The set of all subsets of a set.

3. Graph Theory

• Graphs: A structure consisting of a set of vertices (or nodes) and a set of edges that connect pairs of vertices.
• Types of Graphs: Directed, Undirected, Weighted, Bipartite.
• Key Concepts: Paths, Cycles, Connectivity, Trees (a special type of graph with no cycles).
• Applications: Modeling networks, social connections, web links, and navigation systems.

4. Combinatorics (Counting)

• Fundamental Principles: The Product Rule and Sum Rule for counting outcomes.
• Permutations: Arrangements of objects where order matters.
• Combinations: Selections of objects where order does not matter.
• The Pigeonhole Principle: If you have more pigeons than pigeonholes, at least one hole must contain more than one pigeon. A simple but powerful concept.

5. Relations and Functions

• Relations: A set of ordered pairs that defines a relationship between elements of two sets.
• Properties of Relations: Reflexive, Symmetric, Transitive.
• Equivalence Relations: Relations that are reflexive, symmetric, and transitive, used to partition sets into equivalence classes.

Examples & Use Cases

Examples & Use Cases

Logic in Code Conditional statements in programming are a direct application of propositional logic.

// if (isUserLoggedIn && hasAdminRights) { ... }
// This corresponds to the logical expression: p ∧ q
let isUserLoggedIn = true;
let hasAdminRights = false;

if (isUserLoggedIn && hasAdminRights) {
  console.log("Admin access granted.");
} else {
  console.log("Access denied.");
}

Graph Theory in GPS Navigation Cities are vertices, and roads are weighted edges (with weights representing distance or travel time). A GPS finds the shortest path between two cities using algorithms like Dijkstra's, which is a core graph theory algorithm.

Practice Questions

Practice Questions

1. MCQ: Which of the following is NOT a property of an equivalence relation? A) Reflexive B) Symmetric C) Associative D) Transitive

2. Short Answer: A committee of 3 people is to be chosen from a group of 10. Is this a permutation or a combination problem? Why? Calculate the number of possible committees.

3. Logic Puzzle: You have two propositions: P = "It is raining," and Q = "I will take an umbrella." Write the logical expression for "If it is raining, then I will take an umbrella."

Quick Summary

Quick Summary

• Discrete Math is the study of countable structures, fundamental to computer science.
• Logic and Proofs provide the rules for reasoning and formal verification.
• Set Theory is the language used to group and manage collections of objects.
• Graph Theory models networks and relationships, crucial for algorithms and data representation.
• Combinatorics provides the tools for counting and probability, essential for analyzing algorithm performance.