Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

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Discrete Mathematics


Discrete mathematics is a branch of mathematics that deals with separate and isolated values. It is fundamentally different from continuous mathematics, which involves elements that can easily change. Discrete mathematics is used heavily in computer science, as digital systems use discrete data. It is also the mathematical foundation for algorithms and data structures.

Characteristics of discrete mathematics

Discrete mathematics includes some specific features that make it different from continuous mathematics:

  • It deals with countable, discrete values rather than continuous categories.
  • This often involves mathematical structures such as graphs, integers, and logical statements.
  • It has a wide range of practical applications in computer science; for example, in cryptography, network design, and optimization problems.

Fundamental concepts in discrete mathematics

1. Sets and set theory

Sets are collections of objects. Set theory is the study of sets. A set is represented by curly braces {} and the objects in it are called elements.

Example: Let A = {1, 2, 3, 4} Here, 1, 2, 3 and 4 are elements of the set A.

Operations on sets include union, intersection, and difference:

  • Union: All elements that are in any set.
    A ∪ B = {elements in A} or {elements in B}.
  • Intersection: Only elements that are in both sets.
    A ∩ B
  • Difference: Elements in one set but not in the other.
    A - B
A B

2. Arguments and propositions

Logic deals with the study of reasoning and arguments. Propositions are statements that are either true or false. Logical operators such as AND, OR and NOT are used to form logical statements.

Example: “It is raining” is a proposition. Its truth value can be true or false.

  • NOT - negates the statement.
    If P is true, then NOT P is false.
  • AND - Joins two statements and is true if both are true.
    P AND Q - True if both P and Q are true.
  • OR - joins two statements and is true if at least one is true.
    P OR Q - True if P is true or Q is true.
Truth False

Logical connectives can also be represented using truth tables. For two propositions P and Q, the truth table for P AND Q is:

PQP and Q
,
TTT
TFF
FTF
FFF

3. Functions and relationships

A function is a type of relation where each element in the domain is related to exactly one element in the co-domain. A function is often expressed as f(x) = y.

Example: f(x) = x + 2, if x = 3, then f(x) = 5

A relation is a connection between the elements of two or more sets. They are often represented as ordered pairs.

Example: Consider the relation R = {(1, 2), (3, 4)}, here (1, 2) and (3, 4) are ordered pairs.

4. Graph theory

Graph theory involves the study of graphs, which are mathematical structures used to model paired relationships between objects. A graph is made up of vertices (nodes) and edges (connections).

Graphs can be classified into different types:

  • Undirected Graph: A graph whose edges have no direction.
  • Directed graph (digraph): A graph in which each edge has a direction.
  • Weighted graph: A graph in which edges have weights (values).

5. Combinatorics

Combinatorics is the branch of mathematics that deals with combinations of objects belonging to a finite set according to certain restrictions, such as graph theory.

  • Permutations: Different ways to arrange objects.
    nPr where n is the total number of items and r is the selection.
  • Combinations: Methods of selecting items without considering the order.
    nCr where n is the total number of objects and r is the selection.

Example: If there are a total of 5 books, in how many ways can you arrange 3 books on a shelf? This is a permutation problem: 5P3.

6. Number theory

Number theory studies the properties and relationships of numbers, especially integers. It includes concepts such as divisibility, prime numbers, and modular arithmetic.

Example: Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
The first few prime numbers are: 2, 3, 5, 7, 11, 13, etc.

Modular arithmetic involves arithmetic with remainders. This is similar to how a clock "turns around" every 12 hours.

Example: 7 mod 5 = 2 because when you divide 7 by 5 the remainder is 2.

7. Boolean algebra

Boolean algebra is a subfield of algebra in which the values of variables are truth values (true and false), usually denoted by 1 and 0, respectively.

ABA and BA or BNo A
00001
01011
10010
11110

Applications of discrete mathematics

Computer science

Discrete mathematics is fundamental in computer science. Algorithms, data structures, and complexity theory are all based on discrete mathematics concepts. For example, sorting algorithms are based on discrete mathematics ideas about ordering and permutation.

Coding principles

Coding theory, which is used in data compression, error detection and error correction, relies on the principles of discrete mathematics. It deals with the properties and design of codes and their suitability for specific applications.

Cryptography

Cryptography, which is the science of coding and understanding information, is based on number theory and algorithmic principles.

Crypto public key private key

Network

Graph theory and traversal algorithms from discrete mathematics are important for the analysis and design of networks such as social networks, computer networks, and utility networks.

Conclusion

Discrete mathematics provides important tools for understanding systems in which different phenomena are interconnected. It is integral to fields ranging from computer science to engineering and economics. Its applicability goes beyond just academic interest, being used in real-world problem-solving in data analysis and technology advancement.


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