Discrete Mathematics is the backbone of many fields in computer science, cryptography, and artificial intelligence. It provides the fundamental concepts used in logic, set theory, graph theory, and probability. In this article, I will explore key topics in Discrete Mathematics, summarizing my learning journey and sharing useful insights.
Set theory is the study of collections of objects, called sets. A set can be finite (e.g., {1,2,3}) or infinite (e.g., the set of natural numbers).
Set Operations
Union (A ∪ B): Combines all elements of both sets.
Intersection (A ∩ B): Includes only elements common to both sets.
Difference (A - B): Elements in A that are not in B.
Complement (A'): Elements not in A.
The subset (⊆) and superset (⊇) relationships describe how sets relate to one another.
Logic and Propositional Statements
Logic forms the foundation of mathematical reasoning. A proposition is a statement that is either true or false.
Logical Operators
Negation (¬P): Reverses truth value.
AND (P ∧ Q): True if both are true.
OR (P ∨ Q): True if at least one is true.
Implication (P → Q): "If P, then Q."
3. Counting and Probability
Counting techniques help solve combinatorial problems.
Factorial (n!)
Represents the product of all integers from 1 to n. Example:
5!=5×4×3×2×1=120
Permutations (P(n, r))
Represents the number of ways to arrange r elements from a set of n when order matters:
Combinations (C(n, r))
Represents the number of ways to choose r elements when order does not matter:
Graph Theory
Graph theory studies how objects (nodes) are connected through relationships (edges).
Basic Graph Components
Nodes (Vertices): The points in a graph.
Edges: The connections between nodes.
Directed vs. Undirected Graphs: Whether edges have direction or not.
Graph theory is widely used in networking, AI, and social media analysis.
Boolean Algebra
Boolean Algebra simplifies logical expressions using binary values (0 and 1).
Basic Operations
NOT (¬A): Inverts a value.
AND (A ⋅ B): True if both inputs are 1.
OR (A + B): True if at least one input is 1.
XOR (A ⊕ B): True if only one input is 1.
Proof Techniques
Mathematical proofs are essential for verifying the correctness of logical statements.
Types of Proofs
Direct Proof: Uses logical steps to show truth.
Indirect Proof (Contradiction): Assumes the opposite and finds a contradiction.
Proof by Induction: Used for sequences and recursive formulas.
Induction Process
Base Case: Show it's true for the first value (n = 1).
Inductive Hypothesis: Assume it's true for n = k.
Inductive Step: Prove it's true for n = k+1.