Mathematics for Machine Learning and Data Science

Mathematics for Machine Learning and Data Science is the foundational toolkit of mathematical concepts essential for understanding, developing, and analyzing data-driven algorithms. This specialized area primarily draws from linear algebra for representing data and operations in vector spaces, multivariable calculus for optimizing model parameters through techniques like gradient descent, and probability and statistics for modeling uncertainty and evaluating performance. Together, these core pillars provide the theoretical underpinnings required to build and interpret machine learning models and make sense of complex datasets.

1. Foundations of Mathematical Notation and Concepts
   1. Set Theory Basics
      1. Sets and Elements
         1. Definition of a Set
         2. Notation for Sets
         3. Membership and Non-membership
         4. Empty Set and Universal Set
      2. Subsets, Supersets, and the Power Set
         1. Subset and Proper Subset
         2. Superset and Proper Superset
         3. Power Set Definition
         4. Cardinality of Power Set
      3. Set Operations
         1. Union
         2. Intersection
         3. Complement
         4. Difference
         5. Symmetric Difference
         6. Cartesian Product
         7. Venn Diagrams
      4. Properties of Set Operations
         1. Commutative Laws
         2. Associative Laws
         3. Distributive Laws
         4. De Morgan's Laws
   2. Essential Mathematical Notation
      1. Summation Notation
         1. Basic Sigma Notation
         2. Indexing and Limits
         3. Properties of Summation
         4. Double Summation
      2. Product Notation
         1. Basic Pi Notation
         2. Indexing and Limits
         3. Properties of Product Notation
      3. Basic Logical Operators
         1. Conjunction (AND)
         2. Disjunction (OR)
         3. Negation (NOT)
         4. Implication
         5. Biconditional (Equivalence)
      4. Quantifiers
         1. Universal Quantifier
         2. Existential Quantifier
         3. Nested Quantifiers
      5. Floor and Ceiling Functions
      6. Absolute Value and Sign Function
   3. Functions and Relations
      1. Definition of Functions
      2. Domain and Range
      3. One-to-One and Onto Functions
      4. Inverse Functions
      5. Composition of Functions
      6. Relations and Equivalence Relations
   4. Review of Single-Variable Calculus
      1. Limits and Continuity
         1. Definition of a Limit
         2. One-Sided Limits
         3. Limits at Infinity
         4. Continuity at a Point
         5. Properties of Continuous Functions
         6. Intermediate Value Theorem
      2. Derivatives and Rules of Differentiation
         1. Definition of the Derivative
         2. Power Rule
         3. Product Rule
         4. Quotient Rule
         5. Chain Rule
         6. Implicit Differentiation
         7. Higher-Order Derivatives
         8. Applications of Derivatives
      3. Integrals and the Fundamental Theorem of Calculus
         1. Indefinite Integrals
         2. Definite Integrals
         3. Properties of Integrals
         4. Fundamental Theorem of Calculus
         5. Integration by Substitution
         6. Integration by Parts
         7. Applications of Integration
      4. Series and Sequences
         1. Arithmetic and Geometric Sequences
         2. Convergence and Divergence
         3. Taylor Series
         4. Maclaurin Series