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Mathematics
Mathematics
1
Foundations
1.1
Logic and Proof Techniques
1.1.1
Propositional Logic
1.1.1.1
Propositions and Logical Connectives
1.1.1.2
Truth Tables
1.1.1.3
Logical Equivalences
1.1.1.4
Validity of Arguments
1.1.2
Predicate Logic
1.1.2.1
Predicates
1.1.2.2
Quantifiers
1.1.2.3
Negating Quantifiers
1.1.2.4
Uniqueness Quantifier
1.1.3
Proof Techniques
1.1.3.1
Direct Proof
1.1.3.2
Proof by Contrapositive
1.1.3.3
Proof by Contradiction
1.1.3.4
Proof by Cases
1.1.3.5
Existence Proofs
1.1.3.6
Uniqueness Proofs
1.1.3.7
Mathematical Induction
1.1.3.8
Strong Induction
1.1.3.9
Well-Ordering Principle
1.2
Sets, Functions, and Relations
1.2.1
Sets
1.2.1.1
Set Notation
1.2.1.2
Set Operations
1.2.1.2.1
Union
1.2.1.2.2
Intersection
1.2.1.2.3
Set Difference
1.2.1.2.4
Complement
1.2.1.3
Subsets
1.2.1.4
Cartesian Products
1.2.1.5
Power Sets
1.2.1.6
Cardinality
1.2.2
Functions
1.2.2.1
Formal Definition
1.2.2.2
Domain, Codomain, Range
1.2.2.3
Injective, Surjective, Bijective
1.2.2.4
Composition
1.2.2.5
Inverse
1.2.2.6
Image and Preimage
1.2.3
Relations
1.2.3.1
What is a Relation?
1.2.3.2
Properties of Relations
1.2.3.3
Equivalence Relations
1.2.3.4
Partial Orders
1.2.3.5
Total Orders
1.3
Algebra
1.3.1
Real Number System
1.3.1.1
Types of Numbers
1.3.1.2
Properties of Real Numbers
1.3.1.3
Absolute Value
1.3.2
Equations and Inequalities
1.3.2.1
Linear Equations
1.3.2.2
Linear Inequalities
1.3.2.3
Absolute Value Equations
1.3.2.4
Absolute Value Inequalities
1.3.2.5
Systems of Linear Equations
1.3.3
Polynomials
1.3.3.1
Polynomial Operations
1.3.3.2
Factoring
1.3.3.3
Roots and Factor Theorem
1.3.3.4
Polynomial Division
1.3.3.5
Rational Root Theorem
1.3.4
Rational Expressions
1.3.4.1
Simplification
1.3.4.2
Operations
1.3.4.3
Complex Fractions
1.3.5
Exponents and Radicals
1.3.5.1
Laws of Exponents
1.3.5.2
Simplifying Radicals
1.3.5.3
Rational Exponents
1.3.6
Quadratics
1.3.6.1
Factoring and Completing the Square
1.3.6.2
Quadratic Formula
1.3.6.3
Discriminant and Nature of Roots
1.3.7
Logarithms and Exponentials
1.3.7.1
Introduction to Logarithms
1.3.7.2
Logarithm Properties
1.3.7.3
Solving Exponential and Logarithmic Equations
1.3.8
Functions
1.3.8.1
Definition, Domain, Range
1.3.8.2
Interval Notation
1.3.8.3
Function Notation
1.3.8.4
The Difference Quotient
1.3.8.5
Piecewise Functions
1.3.8.6
Even and Odd Functions
1.3.8.7
Arithmetic Operations on Functions
1.3.8.8
Function Composition
1.3.8.9
Inverse Functions
1.3.8.10
Translations
1.3.8.11
Reflections
1.3.8.12
Stretches and Compressions
1.3.8.13
Common Functions
Cryptography
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