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main.toc
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\contentsline {part}{\numberline {I}Introductory Topics}{17}{part.1}%
\contentsline {chapter}{\numberline {1}Mathematical Proof Strategy}{19}{chapter.1}%
\contentsline {section}{\numberline {1.1}Propositions and Axiomatic System}{19}{section.1.1}%
\contentsline {section}{\numberline {1.2}Direct Proof}{23}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Exercises}{24}{subsection.1.2.1}%
\contentsline {section}{\numberline {1.3}Proof by Cases}{25}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Exercises}{26}{subsection.1.3.1}%
\contentsline {section}{\numberline {1.4}Indirect Proof}{28}{section.1.4}%
\contentsline {subsection}{\numberline {1.4.1}Proof by Contradiction}{28}{subsection.1.4.1}%
\contentsline {subsection}{\numberline {1.4.2}Proof by Contrapositive}{32}{subsection.1.4.2}%
\contentsline {subsection}{\numberline {1.4.3}Exercises}{32}{subsection.1.4.3}%
\contentsline {section}{\numberline {1.5}Mathematical Induction}{33}{section.1.5}%
\contentsline {subsection}{\numberline {1.5.1}Framework of MI}{34}{subsection.1.5.1}%
\contentsline {subsection}{\numberline {1.5.2}Strong Mathematical Induction}{35}{subsection.1.5.2}%
\contentsline {subsection}{\numberline {1.5.3}Exercises}{36}{subsection.1.5.3}%
\contentsline {chapter}{\numberline {2}Set, Sequence, Function, and Summation}{43}{chapter.2}%
\contentsline {section}{\numberline {2.1}Set}{43}{section.2.1}%
\contentsline {section}{\numberline {2.2}Properties of Sets with Proofs}{44}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}Exercises}{47}{subsection.2.2.1}%
\contentsline {section}{\numberline {2.3}Function: a perspective from Set Theory}{49}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Function and Operation on Function}{49}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Elementary Functions and More on Cartesian Product}{50}{subsection.2.3.2}%
\contentsline {subsection}{\numberline {2.3.3}Partial and Total Function}{55}{subsection.2.3.3}%
\contentsline {subsection}{\numberline {2.3.4}Injective, Surjective, and Bijective Function}{56}{subsection.2.3.4}%
\contentsline {subsection}{\numberline {2.3.5}Exercises}{57}{subsection.2.3.5}%
\contentsline {section}{\numberline {2.4}Summation}{63}{section.2.4}%
\contentsline {subsection}{\numberline {2.4.1}Sigma Notation}{64}{subsection.2.4.1}%
\contentsline {subsection}{\numberline {2.4.2}Properties and Techniques of Sigma Notation}{64}{subsection.2.4.2}%
\contentsline {subsubsection}{\numberline {2.4.2.1}Manipulation of Sigma Notation}{65}{subsubsection.2.4.2.1}%
\contentsline {subsubsection}{\numberline {2.4.2.2}Multiple Sums}{66}{subsubsection.2.4.2.2}%
\contentsline {subsection}{\numberline {2.4.3}Exercises}{67}{subsection.2.4.3}%
\contentsline {section}{\numberline {2.5}Sequence}{72}{section.2.5}%
\contentsline {subsection}{\numberline {2.5.1}Introduction}{72}{subsection.2.5.1}%
\contentsline {subsection}{\numberline {2.5.2}Special Sequences}{73}{subsection.2.5.2}%
\contentsline {subsubsection}{\numberline {2.5.2.1}Algorithmic Sequence}{73}{subsubsection.2.5.2.1}%
\contentsline {subsubsection}{\numberline {2.5.2.2}Geometric Sequence}{75}{subsubsection.2.5.2.2}%
\contentsline {subsubsection}{\numberline {2.5.2.3}characteristic Sequence}{76}{subsubsection.2.5.2.3}%
\contentsline {subsection}{\numberline {2.5.3}Exercises}{76}{subsection.2.5.3}%
\contentsline {chapter}{\numberline {3}Algorithm and Number System}{81}{chapter.3}%
\contentsline {section}{\numberline {3.1}Numbers}{81}{section.3.1}%
\contentsline {subsection}{\numberline {3.1.1}Typology of Numbers}{81}{subsection.3.1.1}%
\contentsline {subsection}{\numberline {3.1.2}The Real Number System}{83}{subsection.3.1.2}%
\contentsline {subsection}{\numberline {3.1.3}Floor, Ceiling, and Remainder}{85}{subsection.3.1.3}%
\contentsline {subsubsection}{\numberline {3.1.3.1}Properties of Integer Function}{85}{subsubsection.3.1.3.1}%
\contentsline {subsubsection}{\numberline {3.1.3.2}Remainder and Integer Function}{86}{subsubsection.3.1.3.2}%
\contentsline {subsection}{\numberline {3.1.4}exercises}{87}{subsection.3.1.4}%
\contentsline {section}{\numberline {3.2}Algorithm and Algorithm Analysis}{89}{section.3.2}%
\contentsline {subsection}{\numberline {3.2.1}Algorithm}{89}{subsection.3.2.1}%
\contentsline {subsubsection}{\numberline {3.2.1.1}What is an Algorithm?}{89}{subsubsection.3.2.1.1}%
\contentsline {subsubsection}{\numberline {3.2.1.2}Pseudocode}{90}{subsubsection.3.2.1.2}%
\contentsline {subsection}{\numberline {3.2.2}Algorithm Analysis}{91}{subsection.3.2.2}%
\contentsline {subsubsection}{\numberline {3.2.2.1}Time Complexity}{92}{subsubsection.3.2.2.1}%
\contentsline {subsubsection}{\numberline {3.2.2.2}Space Complexity}{93}{subsubsection.3.2.2.2}%
\contentsline {subsection}{\numberline {3.2.3}Exercises}{93}{subsection.3.2.3}%
\contentsline {chapter}{\numberline {4}Inequality}{97}{chapter.4}%
\contentsline {section}{\numberline {4.1}Inequality basics}{97}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}Exercises}{98}{subsection.4.1.1}%
\contentsline {section}{\numberline {4.2}Solving Quadratic Inequality}{99}{section.4.2}%
\contentsline {section}{\numberline {4.3}important Inequalities}{100}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}The Triangle Inequality}{100}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}The Arithmetic-Geometric Mean Inequality}{102}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}Exercises}{106}{subsection.4.3.3}%
\contentsline {subsection}{\numberline {4.3.4}Cauchy-Schwarz Inequality}{111}{subsection.4.3.4}%
\contentsline {subsection}{\numberline {4.3.5}Rearrangement Inequality}{113}{subsection.4.3.5}%
\contentsline {subsection}{\numberline {4.3.6}Exercises}{114}{subsection.4.3.6}%
\contentsline {chapter}{\numberline {5}Complex Number}{125}{chapter.5}%
\contentsline {section}{\numberline {5.1}Algebra of Complex Number}{126}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}Exercises}{127}{subsection.5.1.1}%
\contentsline {section}{\numberline {5.2}Point representation of Complex Number}{129}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}Exercises}{132}{subsection.5.2.1}%
\contentsline {section}{\numberline {5.3}Vector and Polar Form}{135}{section.5.3}%
\contentsline {subsection}{\numberline {5.3.1}Vector Form of Complex Number}{135}{subsection.5.3.1}%
\contentsline {subsection}{\numberline {5.3.2}Polar Form of Complex Number}{137}{subsection.5.3.2}%
\contentsline {subsubsection}{\numberline {5.3.2.1}The Polar Coordinate}{137}{subsubsection.5.3.2.1}%
\contentsline {subsubsection}{\numberline {5.3.2.2}Polar Expression of Complex Number}{137}{subsubsection.5.3.2.2}%
\contentsline {subsection}{\numberline {5.3.3}Exercises}{140}{subsection.5.3.3}%
\contentsline {section}{\numberline {5.4}Exponential Form}{144}{section.5.4}%
\contentsline {subsection}{\numberline {5.4.1}De Moivre's Theorem}{146}{subsection.5.4.1}%
\contentsline {subsection}{\numberline {5.4.2}Exercises}{147}{subsection.5.4.2}%
\contentsline {section}{\numberline {5.5}Finding Complex Roots}{150}{section.5.5}%
\contentsline {subsection}{\numberline {5.5.1}Solving Quadratic Equations Over the Complex Numbers}{150}{subsection.5.5.1}%
\contentsline {subsection}{\numberline {5.5.2}Solving Polynomial Equations on Complex Number}{153}{subsection.5.5.2}%
\contentsline {subsection}{\numberline {5.5.3}Solving Equation with De Moivre's Theorem}{157}{subsection.5.5.3}%
\contentsline {subsection}{\numberline {5.5.4}Exercises}{160}{subsection.5.5.4}%
\contentsline {part}{\numberline {II}Further Discrete Mathematics and Theories}{167}{part.2}%
\contentsline {chapter}{\numberline {6}Boolean Algebra and Further Logic}{169}{chapter.6}%
\contentsline {section}{\numberline {6.1}Boolean Expression and Truth Table}{169}{section.6.1}%
\contentsline {subsection}{\numberline {6.1.1}Property of Algebra Operation}{169}{subsection.6.1.1}%
\contentsline {subsection}{\numberline {6.1.2}Boolean Expression and Truth Table}{170}{subsection.6.1.2}%
\contentsline {subsection}{\numberline {6.1.3}Boolean Identities}{172}{subsection.6.1.3}%
\contentsline {subsection}{\numberline {6.1.4}Exercises}{175}{subsection.6.1.4}%
\contentsline {section}{\numberline {6.2}Boolean Function}{178}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}Representation of Boolean Function}{179}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}Properties of Boolean Function}{181}{subsection.6.2.2}%
\contentsline {subsection}{\numberline {6.2.3}Simplification of Boolean Function}{185}{subsection.6.2.3}%
\contentsline {subsubsection}{\numberline {6.2.3.1}Simplification by Boolean Laws (Algebra)}{185}{subsubsection.6.2.3.1}%
\contentsline {subsubsection}{\numberline {6.2.3.2}Simplification by Karnaugh Maps}{185}{subsubsection.6.2.3.2}%
\contentsline {subsection}{\numberline {6.2.4}Simplification by Quine-McCluskey Method}{188}{subsection.6.2.4}%
\contentsline {subsection}{\numberline {6.2.5}Exercises}{189}{subsection.6.2.5}%
\contentsline {section}{\numberline {6.3}Predicates and Quantifiers}{194}{section.6.3}%
\contentsline {subsection}{\numberline {6.3.1}Predicate}{194}{subsection.6.3.1}%
\contentsline {subsection}{\numberline {6.3.2}Quantifier}{195}{subsection.6.3.2}%
\contentsline {subsection}{\numberline {6.3.3}Nested Quantifier}{197}{subsection.6.3.3}%
\contentsline {section}{\numberline {6.4}Inference and Deduction}{197}{section.6.4}%
\contentsline {chapter}{\numberline {7}Preliminary Number Theory and Cryptography}{199}{chapter.7}%
\contentsline {section}{\numberline {7.1}Divisibility and Modular Arithmetic}{199}{section.7.1}%
\contentsline {subsection}{\numberline {7.1.1}Division and Divisibility}{200}{subsection.7.1.1}%
\contentsline {subsection}{\numberline {7.1.2}Modular Arithmetic}{201}{subsection.7.1.2}%
\contentsline {subsection}{\numberline {7.1.3}Exercises}{204}{subsection.7.1.3}%
\contentsline {section}{\numberline {7.2}Number Representations and Algorithms}{207}{section.7.2}%
\contentsline {subsection}{\numberline {7.2.1}Representations of Numbers and Base Conversion}{207}{subsection.7.2.1}%
\contentsline {subsection}{\numberline {7.2.2}Base Conversion}{208}{subsection.7.2.2}%
\contentsline {subsection}{\numberline {7.2.3}Operation Algorithms of Number}{211}{subsection.7.2.3}%
\contentsline {subsection}{\numberline {7.2.4}Modular Exponentiation Algorithm}{213}{subsection.7.2.4}%
\contentsline {paragraph}{Time Complexity of Algorithm 5}{215}{paragraph*.4}%
\contentsline {subsection}{\numberline {7.2.5}Exercises}{216}{subsection.7.2.5}%
\contentsline {section}{\numberline {7.3}Primes and Greatest Common Divisors}{216}{section.7.3}%
\contentsline {subsection}{\numberline {7.3.1}Primes and Related Algorithms}{216}{subsection.7.3.1}%
\contentsline {subsection}{\numberline {7.3.2}Greatest Common Divisors and Least Common Multiples}{219}{subsection.7.3.2}%
\contentsline {subsection}{\numberline {7.3.3}Exercises}{222}{subsection.7.3.3}%
\contentsline {section}{\numberline {7.4}Solving Congruence}{222}{section.7.4}%
\contentsline {subsection}{\numberline {7.4.1}Linear Congruence}{222}{subsection.7.4.1}%
\contentsline {paragraph}{Solution:}{223}{paragraph*.5}%
\contentsline {subsection}{\numberline {7.4.2}The Chinese Remainder Theorem}{226}{subsection.7.4.2}%
\contentsline {subsection}{\numberline {7.4.3}Fermat's Little Theorem}{230}{subsection.7.4.3}%
\contentsline {chapter}{\numberline {8}Relation}{233}{chapter.8}%
\contentsline {section}{\numberline {8.1}NBG Set Theory and Binary Relation}{233}{section.8.1}%
\contentsline {subsection}{\numberline {8.1.1}Class}{234}{subsection.8.1.1}%
\contentsline {subsection}{\numberline {8.1.2}Binary Relations, Composition and Inverse}{238}{subsection.8.1.2}%
\contentsline {subsection}{\numberline {8.1.3}Mapping, Composition, and Inverse}{241}{subsection.8.1.3}%
\contentsline {subsection}{\numberline {8.1.4}Families of Sets}{243}{subsection.8.1.4}%
\contentsline {subsection}{\numberline {8.1.5}Reflexivity, Symmetry, and Transitivity}{246}{subsection.8.1.5}%
\contentsline {subsection}{\numberline {8.1.6}Exercises}{247}{subsection.8.1.6}%
\contentsline {section}{\numberline {8.2}Representation of Relations}{248}{section.8.2}%
\contentsline {subsection}{\numberline {8.2.1}Representation By Matrix}{248}{subsection.8.2.1}%
\contentsline {subsection}{\numberline {8.2.2}Representation By Digraph}{251}{subsection.8.2.2}%
\contentsline {subsection}{\numberline {8.2.3}Exercises}{253}{subsection.8.2.3}%
\contentsline {section}{\numberline {8.3}Closure of Relations}{255}{section.8.3}%
\contentsline {subsection}{\numberline {8.3.1}Exercises}{255}{subsection.8.3.1}%
\contentsline {section}{\numberline {8.4}Equivalence Relations}{255}{section.8.4}%
\contentsline {subsection}{\numberline {8.4.1}Equivalence}{256}{subsection.8.4.1}%
\contentsline {subsection}{\numberline {8.4.2}Equivalence Classes}{257}{subsection.8.4.2}%
\contentsline {subsection}{\numberline {8.4.3}Exercises}{259}{subsection.8.4.3}%
\contentsline {section}{\numberline {8.5}Order Relations}{262}{section.8.5}%
\contentsline {subsection}{\numberline {8.5.1}Partial, Total, and Well Ordering}{262}{subsection.8.5.1}%
\contentsline {subsection}{\numberline {8.5.2}Lexicographic Order}{264}{subsection.8.5.2}%
\contentsline {subsection}{\numberline {8.5.3}Hasse Diagram}{265}{subsection.8.5.3}%
\contentsline {subsection}{\numberline {8.5.4}Maximal and Minimal Elements}{265}{subsection.8.5.4}%
\contentsline {subsection}{\numberline {8.5.5}Lattices}{265}{subsection.8.5.5}%
\contentsline {subsection}{\numberline {8.5.6}Topological Sorting}{265}{subsection.8.5.6}%
\contentsline {subsection}{\numberline {8.5.7}Exercises}{265}{subsection.8.5.7}%
\contentsline {section}{\numberline {8.6}Special Types of Relations}{267}{section.8.6}%
\contentsline {subsection}{\numberline {8.6.1}Recursive Relations}{267}{subsection.8.6.1}%
\contentsline {subsection}{\numberline {8.6.2}\(n\)-ary Relations}{267}{subsection.8.6.2}%
\contentsline {subsection}{\numberline {8.6.3}Exercises}{267}{subsection.8.6.3}%
\contentsline {chapter}{\numberline {9}Graph Theory}{269}{chapter.9}%
\contentsline {chapter}{\numberline {10}Basics of Abstract Algebra}{271}{chapter.10}%
\contentsline {section}{\numberline {10.1}Fundamentals of Algebraic Structures}{272}{section.10.1}%
\contentsline {subsection}{\numberline {10.1.1}Groups}{272}{subsection.10.1.1}%
\contentsline {subsection}{\numberline {10.1.2}Rings}{272}{subsection.10.1.2}%
\contentsline {subsection}{\numberline {10.1.3}Fields}{272}{subsection.10.1.3}%
\contentsline {subsection}{\numberline {10.1.4}Exercises}{272}{subsection.10.1.4}%
\contentsline {section}{\numberline {10.2}Operations on Algebraic Structures}{272}{section.10.2}%
\contentsline {subsection}{\numberline {10.2.1}Homomorphisms}{272}{subsection.10.2.1}%
\contentsline {subsection}{\numberline {10.2.2}Isomorphisms}{272}{subsection.10.2.2}%
\contentsline {subsection}{\numberline {10.2.3}Exercises}{272}{subsection.10.2.3}%
\contentsline {section}{\numberline {10.3}Applications of Algebraic Structures}{272}{section.10.3}%
\contentsline {subsection}{\numberline {10.3.1}Cryptography}{272}{subsection.10.3.1}%
\contentsline {subsection}{\numberline {10.3.2}Coding Theory}{272}{subsection.10.3.2}%
\contentsline {subsection}{\numberline {10.3.3}Exercises}{272}{subsection.10.3.3}%
\contentsline {chapter}{\numberline {11}Introductory Topology and Category Theory}{273}{chapter.11}%
\contentsline {section}{\numberline {11.1}Basic Topology}{274}{section.11.1}%
\contentsline {subsection}{\numberline {11.1.1}Introduction to Topological Spaces}{276}{subsection.11.1.1}%
\contentsline {subsubsection}{\numberline {11.1.1.1}Open and Closed Sets}{276}{subsubsection.11.1.1.1}%
\contentsline {subsubsection}{\numberline {11.1.1.2}Basis for a Topology}{276}{subsubsection.11.1.1.2}%
\contentsline {subsection}{\numberline {11.1.2}Continuity and Limits}{276}{subsection.11.1.2}%
\contentsline {subsubsection}{\numberline {11.1.2.1}Continuous Functions}{276}{subsubsection.11.1.2.1}%
\contentsline {subsubsection}{\numberline {11.1.2.2}Limit Points and Convergence}{276}{subsubsection.11.1.2.2}%
\contentsline {subsection}{\numberline {11.1.3}Compactness and Connectedness}{276}{subsection.11.1.3}%
\contentsline {subsubsection}{\numberline {11.1.3.1}Compact Spaces}{276}{subsubsection.11.1.3.1}%
\contentsline {subsubsection}{\numberline {11.1.3.2}Connected Spaces}{276}{subsubsection.11.1.3.2}%
\contentsline {subsection}{\numberline {11.1.4}Applications of Topology}{276}{subsection.11.1.4}%
\contentsline {subsubsection}{\numberline {11.1.4.1}Topology in Computer Science}{276}{subsubsection.11.1.4.1}%
\contentsline {subsubsection}{\numberline {11.1.4.2}Topology in Physics}{276}{subsubsection.11.1.4.2}%
\contentsline {section}{\numberline {11.2}Category Theory Fundamentals}{276}{section.11.2}%
\contentsline {subsection}{\numberline {11.2.1}Introduction to Categories}{277}{subsection.11.2.1}%
\contentsline {subsubsection}{\numberline {11.2.1.1}Objects and Morphisms}{277}{subsubsection.11.2.1.1}%
\contentsline {subsubsection}{\numberline {11.2.1.2}Examples of Categories}{277}{subsubsection.11.2.1.2}%
\contentsline {subsection}{\numberline {11.2.2}Functors and Natural Transformations}{277}{subsection.11.2.2}%
\contentsline {subsubsection}{\numberline {11.2.2.1}Definition of Functors}{277}{subsubsection.11.2.2.1}%
\contentsline {subsubsection}{\numberline {11.2.2.2}Natural Transformations between Functors}{277}{subsubsection.11.2.2.2}%
\contentsline {subsection}{\numberline {11.2.3}Limits and Colimits}{277}{subsection.11.2.3}%
\contentsline {subsubsection}{\numberline {11.2.3.1}Universal Properties}{277}{subsubsection.11.2.3.1}%
\contentsline {subsubsection}{\numberline {11.2.3.2}Construction of Limits and Colimits}{277}{subsubsection.11.2.3.2}%
\contentsline {subsection}{\numberline {11.2.4}Applications of Category Theory}{277}{subsection.11.2.4}%
\contentsline {subsubsection}{\numberline {11.2.4.1}Category Theory in Programming Languages}{277}{subsubsection.11.2.4.1}%
\contentsline {subsubsection}{\numberline {11.2.4.2}Category Theory in Logic and Set Theory}{277}{subsubsection.11.2.4.2}%
\contentsline {part}{\numberline {III}Single-variable Calculus}{279}{part.3}%
\contentsline {chapter}{\numberline {12}Function Monotonicity, Parity and Periodicity}{281}{chapter.12}%
\contentsline {section}{\numberline {12.1}Monotonicity of Function}{281}{section.12.1}%
\contentsline {section}{\numberline {12.2}Parity of Function}{281}{section.12.2}%
\contentsline {section}{\numberline {12.3}Periodicity of Function}{281}{section.12.3}%
\contentsline {chapter}{\numberline {13}Abstract and Piece-wise Function}{283}{chapter.13}%
\contentsline {section}{\numberline {13.1}Abstract Function}{283}{section.13.1}%
\contentsline {section}{\numberline {13.2}Piece-wise Function}{283}{section.13.2}%
\contentsline {chapter}{\numberline {14}Limit and Continuity}{285}{chapter.14}%
\contentsline {section}{\numberline {14.1}Limit of Sequence}{285}{section.14.1}%
\contentsline {section}{\numberline {14.2}limit of Function}{285}{section.14.2}%
\contentsline {section}{\numberline {14.3}Continuity}{285}{section.14.3}%
\contentsline {section}{\numberline {14.4}Application of Limit}{285}{section.14.4}%
\contentsline {chapter}{\numberline {15}Differential Calculus}{287}{chapter.15}%
\contentsline {section}{\numberline {15.1}Derivative Basics}{288}{section.15.1}%
\contentsline {subsection}{\numberline {15.1.1}Definition of Derivative}{288}{subsection.15.1.1}%
\contentsline {subsection}{\numberline {15.1.2}Geometric Meaning of Derivative}{288}{subsection.15.1.2}%
\contentsline {subsection}{\numberline {15.1.3}Physical Meaning of Derivative}{288}{subsection.15.1.3}%
\contentsline {section}{\numberline {15.2}Basic Derivative Rules}{288}{section.15.2}%
\contentsline {subsection}{\numberline {15.2.1}Derivatives of Elementary Functions}{288}{subsection.15.2.1}%
\contentsline {subsection}{\numberline {15.2.2}Product Rule, Quotient Rule, Chain Rule}{288}{subsection.15.2.2}%
\contentsline {section}{\numberline {15.3}Higher-order Derivatives}{288}{section.15.3}%
\contentsline {subsection}{\numberline {15.3.1}Second-order Derivatives and Applications}{288}{subsection.15.3.1}%
\contentsline {subsection}{\numberline {15.3.2}Calculation and Significance of Higher-order Derivatives}{288}{subsection.15.3.2}%
\contentsline {section}{\numberline {15.4}Derivatives of Abnormal Function}{288}{section.15.4}%
\contentsline {subsection}{\numberline {15.4.1}Derivatives of Implicit Functions}{288}{subsection.15.4.1}%
\contentsline {subsection}{\numberline {15.4.2}Derivatives of Parametric Equations}{288}{subsection.15.4.2}%
\contentsline {section}{\numberline {15.5}Differentiation}{288}{section.15.5}%
\contentsline {section}{\numberline {15.6}Related Rates}{288}{section.15.6}%
\contentsline {subsection}{\numberline {15.6.1}Relationships between Rates of Change of Different Quantities}{288}{subsection.15.6.1}%
\contentsline {section}{\numberline {15.7}Taylor Series}{288}{section.15.7}%
\contentsline {subsection}{\numberline {15.7.1}Taylor Expansion of Functions}{288}{subsection.15.7.1}%
\contentsline {subsection}{\numberline {15.7.2}Application of Taylor Series}{288}{subsection.15.7.2}%
\contentsline {section}{\numberline {15.8}Applications of Differential Calculus}{288}{section.15.8}%
\contentsline {subsection}{\numberline {15.8.1}Tangents and Normals of Curves}{288}{subsection.15.8.1}%
\contentsline {subsection}{\numberline {15.8.2}Mean Value Theorem in Differential Calculus}{288}{subsection.15.8.2}%
\contentsline {subsubsection}{\numberline {15.8.2.1}Fermat's Theorem}{288}{subsubsection.15.8.2.1}%
\contentsline {subsubsection}{\numberline {15.8.2.2}Rolle's Theorem}{288}{subsubsection.15.8.2.2}%
\contentsline {subsubsection}{\numberline {15.8.2.3}Lagrange's Mean Value Theorem}{288}{subsubsection.15.8.2.3}%
\contentsline {subsubsection}{\numberline {15.8.2.4}Cauchy's Mean Value Theorem}{288}{subsubsection.15.8.2.4}%
\contentsline {subsubsection}{\numberline {15.8.2.5}Indeterminate Form Limit}{288}{subsubsection.15.8.2.5}%
\contentsline {subsection}{\numberline {15.8.3}Extrema Problems and Optimization}{288}{subsection.15.8.3}%
\contentsline {chapter}{\numberline {16}integral calculus}{289}{chapter.16}%
\contentsline {section}{\numberline {16.1}Fundamentals of Integration}{290}{section.16.1}%
\contentsline {subsection}{\numberline {16.1.1}Definition of the Integral}{290}{subsection.16.1.1}%
\contentsline {subsection}{\numberline {16.1.2}Properties of Integrals}{290}{subsection.16.1.2}%
\contentsline {subsection}{\numberline {16.1.3}The Fundamental Theorem of Calculus}{290}{subsection.16.1.3}%
\contentsline {section}{\numberline {16.2}Techniques of Integration}{290}{section.16.2}%
\contentsline {subsection}{\numberline {16.2.1}Basic Integration Formulas}{290}{subsection.16.2.1}%
\contentsline {subsection}{\numberline {16.2.2}Integration by Substitution}{290}{subsection.16.2.2}%
\contentsline {subsection}{\numberline {16.2.3}Integration by Parts}{290}{subsection.16.2.3}%
\contentsline {subsection}{\numberline {16.2.4}Trigonometric Integrals}{290}{subsection.16.2.4}%
\contentsline {subsection}{\numberline {16.2.5}Partial Fractions}{290}{subsection.16.2.5}%
\contentsline {section}{\numberline {16.3}Applications of Integration}{290}{section.16.3}%
\contentsline {subsection}{\numberline {16.3.1}Area Under Curves}{290}{subsection.16.3.1}%
\contentsline {subsection}{\numberline {16.3.2}Volumes of Solids of Revolution}{290}{subsection.16.3.2}%
\contentsline {subsection}{\numberline {16.3.3}Arc Length and Surface Area}{290}{subsection.16.3.3}%
\contentsline {subsection}{\numberline {16.3.4}Center of Mass and Moments}{290}{subsection.16.3.4}%
\contentsline {section}{\numberline {16.4}Improper Integrals}{290}{section.16.4}%
\contentsline {subsection}{\numberline {16.4.1}Convergence and Divergence of Improper Integrals}{290}{subsection.16.4.1}%
\contentsline {subsection}{\numberline {16.4.2}Applications of Improper Integrals}{290}{subsection.16.4.2}%
\contentsline {section}{\numberline {16.5}Numerical Integration Methods}{290}{section.16.5}%
\contentsline {subsection}{\numberline {16.5.1}The Trapezoidal Rule}{290}{subsection.16.5.1}%
\contentsline {subsection}{\numberline {16.5.2}Simpson's Rule}{290}{subsection.16.5.2}%
\contentsline {chapter}{\numberline {17}Differential Equation}{291}{chapter.17}%
\contentsline {chapter}{\numberline {18}Infinite Series}{293}{chapter.18}%
\contentsline {part}{\numberline {IV}Multi-variable and Vector Calculus}{295}{part.4}%
\contentsline {chapter}{\numberline {19}integral calculus}{297}{chapter.19}%
\contentsline {chapter}{\numberline {20}Introduction to Multivariable Functions}{299}{chapter.20}%
\contentsline {subsection}{\numberline {20.0.1}Concepts of Multivariable Functions}{299}{subsection.20.0.1}%
\contentsline {subsection}{\numberline {20.0.2}Graphs and Contour Plots}{299}{subsection.20.0.2}%
\contentsline {chapter}{\numberline {21}Partial Derivatives}{301}{chapter.21}%
\contentsline {subsection}{\numberline {21.0.1}Definition and Interpretation}{301}{subsection.21.0.1}%
\contentsline {subsection}{\numberline {21.0.2}Higher-Order Partial Derivatives}{301}{subsection.21.0.2}%
\contentsline {subsection}{\numberline {21.0.3}Chain Rule in Multiple Variables}{301}{subsection.21.0.3}%
\contentsline {chapter}{\numberline {22}Multiple Integrals}{303}{chapter.22}%
\contentsline {subsection}{\numberline {22.0.1}Double Integrals}{303}{subsection.22.0.1}%
\contentsline {subsubsection}{\numberline {22.0.1.1}Iterated Integrals}{303}{subsubsection.22.0.1.1}%
\contentsline {subsubsection}{\numberline {22.0.1.2}Double Integrals over General Regions}{303}{subsubsection.22.0.1.2}%
\contentsline {subsection}{\numberline {22.0.2}Triple Integrals}{303}{subsection.22.0.2}%
\contentsline {subsubsection}{\numberline {22.0.2.1}Cylindrical and Spherical Coordinates}{303}{subsubsection.22.0.2.1}%
\contentsline {chapter}{\numberline {23}Vector Calculus}{305}{chapter.23}%
\contentsline {section}{\numberline {23.1}Vector Fields}{305}{section.23.1}%
\contentsline {section}{\numberline {23.2}Gradient, Divergence, and Curl}{305}{section.23.2}%
\contentsline {section}{\numberline {23.3}Line and Surface Integrals}{305}{section.23.3}%
\contentsline {part}{\numberline {V}Linear Algebra}{307}{part.5}%
\contentsline {chapter}{\numberline {24}Vectors Space and the Geometry of Space}{309}{chapter.24}%
\contentsline {chapter}{\numberline {25}Matrices and Systems of Equations}{311}{chapter.25}%
\contentsline {chapter}{\numberline {26}Determinant of Matrix}{313}{chapter.26}%
\contentsline {chapter}{\numberline {27}Orthogonality}{315}{chapter.27}%
\contentsline {chapter}{\numberline {28}Linear Transformations}{317}{chapter.28}%
\contentsline {chapter}{\numberline {29}Eigenvalues and Eigenvector}{319}{chapter.29}%
\contentsline {chapter}{\numberline {30}Singular Value Decomposition}{321}{chapter.30}%
\contentsline {chapter}{\numberline {31}Complex Vector and Matrices}{323}{chapter.31}%
\contentsline {chapter}{\numberline {32}Matrix Differential Calculus}{325}{chapter.32}%
\contentsline {part}{\numberline {VI}Probability and Combinatorics}{327}{part.6}%
\contentsline {chapter}{\numberline {33}Introduction to Counting and Probability }{329}{chapter.33}%
\contentsline {section}{\numberline {33.1}Counting Principal}{329}{section.33.1}%
\contentsline {subsection}{\numberline {33.1.1}Principal of Counting}{329}{subsection.33.1.1}%
\contentsline {subsection}{\numberline {33.1.2}Pigeonhole Theorem}{333}{subsection.33.1.2}%
\contentsline {subsection}{\numberline {33.1.3}Exercises}{334}{subsection.33.1.3}%
\contentsline {section}{\numberline {33.2}Combination and Permutation with applications}{337}{section.33.2}%
\contentsline {subsection}{\numberline {33.2.1}Permutation}{337}{subsection.33.2.1}%
\contentsline {subsection}{\numberline {33.2.2}Combination}{339}{subsection.33.2.2}%
\contentsline {subsection}{\numberline {33.2.3}Further Interpretation of Counting with Set Theory}{342}{subsection.33.2.3}%
\contentsline {subsection}{\numberline {33.2.4}Binomial and Multinomial Theorem}{344}{subsection.33.2.4}%
\contentsline {subsection}{\numberline {33.2.5}Catalan Number}{348}{subsection.33.2.5}%
\contentsline {subsection}{\numberline {33.2.6}String Number}{348}{subsection.33.2.6}%
\contentsline {subsection}{\numberline {33.2.7}Exercises}{348}{subsection.33.2.7}%
\contentsline {section}{\numberline {33.3}Axioms of Probability}{351}{section.33.3}%
\contentsline {subsection}{\numberline {33.3.1}Sample Space and Events}{352}{subsection.33.3.1}%
\contentsline {subsection}{\numberline {33.3.2}Probability Axioms}{353}{subsection.33.3.2}%
\contentsline {subsection}{\numberline {33.3.3}Exercises}{357}{subsection.33.3.3}%
\contentsline {section}{\numberline {33.4}Finding Probability with Counting}{369}{section.33.4}%
\contentsline {subsection}{\numberline {33.4.1}Some Basic Problems}{369}{subsection.33.4.1}%
\contentsline {subsection}{\numberline {33.4.2}Further Problems}{371}{subsection.33.4.2}%
\contentsline {subsection}{\numberline {33.4.3}Exercises}{373}{subsection.33.4.3}%
\contentsline {chapter}{\numberline {34}Conditional Probability and Independence of Events}{377}{chapter.34}%
\contentsline {section}{\numberline {34.1}Conditional Probability}{378}{section.34.1}%
\contentsline {subsection}{\numberline {34.1.1}Basic Conditional Probability}{378}{subsection.34.1.1}%
\contentsline {subsection}{\numberline {34.1.2}Exercises}{385}{subsection.34.1.2}%
\contentsline {section}{\numberline {34.2}Bayes's Theorem}{385}{section.34.2}%
\contentsline {subsection}{\numberline {34.2.1}Bayes's Theorem and Bayesian Thinking}{385}{subsection.34.2.1}%
\contentsline {subsection}{\numberline {34.2.2}Exercises}{391}{subsection.34.2.2}%
\contentsline {section}{\numberline {34.3}Independence of Events}{391}{section.34.3}%
\contentsline {subsection}{\numberline {34.3.1}Definition of Independence}{391}{subsection.34.3.1}%
\contentsline {subsection}{\numberline {34.3.2}Multiple Independence}{392}{subsection.34.3.2}%
\contentsline {subsection}{\numberline {34.3.3}Exercises}{395}{subsection.34.3.3}%
\contentsline {section}{\numberline {34.4}Further Conditional Probability}{396}{section.34.4}%
\contentsline {subsection}{\numberline {34.4.1}Probability Axiom in Conditional Probability}{396}{subsection.34.4.1}%
\contentsline {subsection}{\numberline {34.4.2}Multi-conditional Probability}{399}{subsection.34.4.2}%
\contentsline {subsection}{\numberline {34.4.3}Exercises}{400}{subsection.34.4.3}%
\contentsline {chapter}{\numberline {35}Random Variable and Discrete Distribution}{403}{chapter.35}%
\contentsline {section}{\numberline {35.1}Random Variable}{403}{section.35.1}%
\contentsline {subsection}{\numberline {35.1.1}Analysis of Random Variables}{405}{subsection.35.1.1}%
\contentsline {subsection}{\numberline {35.1.2}Discrete Random Variables and Discrete Distributions}{406}{subsection.35.1.2}%
\contentsline {subsection}{\numberline {35.1.3}Exercises}{408}{subsection.35.1.3}%
\contentsline {section}{\numberline {35.2}Expectation and Variance}{410}{section.35.2}%
\contentsline {subsection}{\numberline {35.2.1}Expectation of Discrete Random Variable}{411}{subsection.35.2.1}%
\contentsline {subsection}{\numberline {35.2.2}Expectation of Function and Linearity}{412}{subsection.35.2.2}%
\contentsline {subsection}{\numberline {35.2.3}Variance}{416}{subsection.35.2.3}%
\contentsline {subsection}{\numberline {35.2.4}PDF and CDF}{418}{subsection.35.2.4}%
\contentsline {subsection}{\numberline {35.2.5}Composition of Dsicrete Random Variable}{418}{subsection.35.2.5}%
\contentsline {subsection}{\numberline {35.2.6}Exercises}{419}{subsection.35.2.6}%
\contentsline {section}{\numberline {35.3}Common Discrete Distributions}{428}{section.35.3}%
\contentsline {subsection}{\numberline {35.3.1}Bernoulli and Binomial Distribution}{428}{subsection.35.3.1}%
\contentsline {subsection}{\numberline {35.3.2}Poisson Distribution}{432}{subsection.35.3.2}%
\contentsline {subsection}{\numberline {35.3.3}Geometric Distribution}{437}{subsection.35.3.3}%
\contentsline {subsection}{\numberline {35.3.4}Hypergeometric Distribution}{440}{subsection.35.3.4}%
\contentsline {subsection}{\numberline {35.3.5}Exercises}{444}{subsection.35.3.5}%
\contentsline {section}{\numberline {35.4}Other Discrete Distributions}{445}{section.35.4}%
\contentsline {subsection}{\numberline {35.4.1}Discrete Uniform Distribution}{445}{subsection.35.4.1}%
\contentsline {subsection}{\numberline {35.4.2}Negative Binomial Distribution}{445}{subsection.35.4.2}%
\contentsline {subsection}{\numberline {35.4.3}Zeta-Bernoulli Distribution}{445}{subsection.35.4.3}%
\contentsline {subsection}{\numberline {35.4.4}Logarithmic Series Distribution}{445}{subsection.35.4.4}%
\contentsline {subsection}{\numberline {35.4.5}Zipf's Distribution}{445}{subsection.35.4.5}%
\contentsline {subsection}{\numberline {35.4.6}Exercises}{445}{subsection.35.4.6}%
\contentsline {section}{\numberline {35.5}Properties of Random Variable, PDF, and CDF}{445}{section.35.5}%
\contentsline {subsection}{\numberline {35.5.1}Exercises}{445}{subsection.35.5.1}%
\contentsline {chapter}{\numberline {36}Continuous Distribution}{447}{chapter.36}%
\contentsline {chapter}{\numberline {37}Joint Cumulative Distribution}{449}{chapter.37}%
\contentsline {chapter}{\numberline {38}Limit Theory in Probability}{451}{chapter.38}%
\contentsline {chapter}{\numberline {39}Stochastic Process}{453}{chapter.39}%
\contentsline {part}{\numberline {VII}Statistics}{455}{part.7}%
\contentsline {chapter}{\numberline {40}Sampling and Parameters}{457}{chapter.40}%
\contentsline {chapter}{\numberline {41}Descriptive Statistics}{459}{chapter.41}%
\contentsline {chapter}{\numberline {42}Graphical Statistics}{461}{chapter.42}%
\contentsline {chapter}{\numberline {43}Statistical Inference}{463}{chapter.43}%
\contentsline {section}{\numberline {43.1}Parameter Estimation}{463}{section.43.1}%
\contentsline {section}{\numberline {43.2}Interval Estimation}{463}{section.43.2}%
\contentsline {section}{\numberline {43.3}Hypothesis Testing}{463}{section.43.3}%
\contentsline {section}{\numberline {43.4}Variance Inference}{463}{section.43.4}%
\contentsline {section}{\numberline {43.5}Bayesian Inference}{463}{section.43.5}%
\contentsline {chapter}{\numberline {44}Hypothesis Testing}{465}{chapter.44}%
\contentsline {chapter}{\numberline {45}Regression and Regressive Analysis}{467}{chapter.45}%
\contentsline {chapter}{\numberline {46}Basic Multi-variable Statistical analysis}{469}{chapter.46}%
\contentsline {part}{\numberline {VIII}Information Theory}{471}{part.8}%
\contentsline {chapter}{\numberline {47}Measuring of Information}{473}{chapter.47}%
\contentsline {chapter}{\numberline {48}Information Entropy}{475}{chapter.48}%
\contentsline {chapter}{\numberline {49}Joined Entropy and Conditional Entropy}{477}{chapter.49}%
\contentsline {chapter}{\numberline {50}Cross Entropy and Relative Entropy}{479}{chapter.50}%
\contentsline {chapter}{\numberline {51}Mutual Information}{481}{chapter.51}%
\contentsline {chapter}{\numberline {52}Differential Entropy}{483}{chapter.52}%
\contentsfinish