User:Jshflynn/Howson Handbook

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Chapter $1$: Some Mathematical Language

Connectives

Definition:Statement

Definition:Logical Connective

Definition:Logical Not

Definition:Disjunction

Definition:Conjunction

Definition:Conditional

Definition:Biconditional

Definition:Truth Table

Definition:Conditional/Antecedent

Definition:Conditional/Consequent

Definition:Equivalent Statements

Definition:Logical NAND


Axioms

Definition:Axiom

Definition:True

Definition:Rule of Inference

Definition:False

Modus Ponendo Ponens

Definition:Converse Statement

Definition:Inverse Statement

Definition:Contrapositive Statement


Variables and Quantifiers

Definition:Variable/Propositional Logic

Definition:Alphabetic Substitution

Definition:Contingent Statement

Definition:Predicate

Definition:Universal Quantifier

Definition:Existential Quantifier

Definition:Bound Occurrence

Definition:Free Occurrence


Equality

Definition:Equals


Axiom Systems

Definition:Formal System

Definition:Theorem (Formal Systems)

Definition:Lemma

Definition:Model (Logic)

Definition:Consistent (Logic)

Definition:Independent Statements


Chapter $2$: Sets and Functions

Sets

Sets

Definition:Set

Definition:Membership Relation

Definition:Element

Definition:Subset

Definition:Proper Subset

Definition:Set Complement

Definition:Empty Set

Definition:Singleton

Definition:Power Set

Definition:Set Intersection

Definition:Disjoint Sets

Definition:Set Union

Definition:Indexing Set

Definition:Index (Indexing Set)

Definition:Set Intersection/Family of Sets

Definition:Set Union/Family of Sets

Definition:Set Difference

Definition:Relative Complement

Definition:Symmetric Difference

Definition:Venn Diagram

Definition:Euler Diagram


Graphs and Functions

Definition:Ordered Pair

Definition:Cartesian Product

Definition:Relation

Definition:Truth Set

Definition:Mapping

Definition:Graph of Mapping

Definition:Domain of Mapping

Definition:Codomain of Mapping

Equality of Mappings

Definition:Image of Element under Mapping

Definition:Image of Mapping

Definition:Range of Relation

Definition:Constant Mapping

Definition:Set of All Mappings

Definition:Restriction of Mapping

Definition:Extension of Mapping


Some Special Types of Function

Definition:Surjection

Definition:Injection

Definition:Bijection

Definition:Permutation

Definition:Identity Mapping

Definition:Inclusion Mapping


Composition of Functions

Definition:Composite Mapping

Definition:Commutative Diagram

Definition:Preimage of Subset under Mapping

Definition:Left Inverse Mapping

Definition:Right Inverse Mapping

Definition:Inverse Mapping


Functions of Several Variables

Definition:Mapping/General Definition


Operations

Definition:Operation/Binary Operation

Definition:Operation/N-Ary Operation

Definition:Commutative Operation

Definition:Associative

Definition:Identity (Abstract Algebra)/Left Identity

Definition:Identity (Abstract Algebra)/Right Identity

Definition:Identity (Abstract Algebra)/Two-Sided Identity

Definition:Distributive Operation

Definition:Algebraic Structure

Definition:Closure (Abstract Algebra)/Algebraic Structure


Chapter $3$: Equivalence Relations and Quotient Sets

Binary Relations

Definition:Relation

Definition:Relation/General Definition

Definition:Reflexive Relation

Definition:Symmetric Relation

Definition:Transitive Relation

Definition:Equivalence Relation

Definition:Antisymmetric Relation/Definition 1

Definition:Partial Ordering

Definition:Ordered Set


Quotient Sets

Definition:Equivalence Relation Induced by Mapping

Definition:Equivalence Class

Definition:Quotient Set

Definition:Quotient Mapping

Definition:Set Partition/Definition 2


Chapter $4$: Number Systems $\text{I}$

Peano's Axioms

A Set-Theoretic Approach

The Rational Integers

The Rational Numbers

Chapter $5$: Groups $\text{I}$

Subgroups

If you would like to take over

I have to stop doing this for a week so I'm turning off the log in page for me. If anyone would like to take over, here you go:


The List

+ Chapter 4: Number Systems I

Definition:Natural Number

++ Peano's Axioms

Definition:Successor Mapping on Natural Numbers

Axiom:Peano's Axioms

Principle of Mathematical Induction

Definition by Induction of Natural Number Addition

Definition:Multiplication/Natural Numbers/Minimal Infinite Successor Set

Definition:Inequality

Definition:Trichotomy

Definition:Relation Compatible with Operation

Definition:Well-Ordering/Definition 1

Definition:Ordered Set

Definition:Totally Ordered Set

++ A Set-Theoretic Approach

Definition:Set Equivalence

Definition:Cardinality

Definition:Cardinal

Definition:Cardinal Number

Definition:Zero (Cardinal)

Definition:One (Cardinal)

Definition:Sum of Cardinals

Definition:Product of Cardinals

Definition:Cardinal Exponentiation

Definition:Finite Set

Definition:Infinite Set

Definition:Finite Cardinal

Definition:Infinite Cardinal

Natural Numbers as Cardinals

Definition:Ordinal/Transfinite

Definition:Countable Set/Definition 1

Definition:Uncountable Set

Continuum Hypothesis

++ The Rational Integers

Definition:Integer

Definition:Addition/Integers

Definition:Multiplication/Integers

Definition:Negative/Integer

Definition:Positive/Integer


++ The Rational Numbers

Definition:Rational Number

Definition:Addition/Rational Numbers

Definition:Multiplication/Rational Numbers


+ Chapter 5: Groups I

Definition:Group

Definition:Multiplicative Notation

Definition:Additive Notation

Definition:Identity (Abstract Algebra)/Two-Sided Identity

Definition:Inverse (Abstract Algebra)/Inverse

Definition:Semigroup

Definition:Monoid

Definition:Magma

Definition:Abelian Group

Definition:Finite Group

Definition:Order of Structure

Definition:Order of Group Element

Definition:Infinite Group

Definition:P-Group

Definition:Periodic Group

Definition:Aperiodic Group

Definition:Cyclic Group

Definition:Power of Element/Group

Definition:Generator of Group

Definition:Group of Permutations

Definition:Symmetric Group

Definition:Alternating Group

Definition:Symmetry Group

Definition:Dihedral Group

++ Subgroups

Definition:Subgroup

Definition:Proper Subgroup

Definition:Permutation Group

Definition:Coset/Left Coset

Definition:Coset/Right Coset

Definition:Index of Subgroup

Definition:Normal Subgroup/Definition 1


++ Chapter 6: Rings and Fields

Definition:Ring

Definition:Additive Group

Definition:Ring with Unity

Definition:Commutative Ring

Definition:Subring

Definition:Unit of Ring

Definition:Group of Units

Definition:Division Ring

Definition:Field

Definition:Subfield

Definition:Ordered Ring

Positive Elements of Ordered Ring

Definition:Integral Domain

Definition:Zero Divisor

Definition:Zero Divisor/Ring

Definition:Characteristic (Ring Theory)

Definition:Ideal (Ring Theory)

Definition:Left Ideal

Definition:Right Ideal

Definition:Maximal Ideal

Definition:Subset Product

Definition:Principal Ideal

Definition:Generator of Ideal

Definition:Principle Ideal Domain

Definition:Greatest Common Divisor/Integral Domain

Definition:GCD Domain

Definition:Coprime/GCD Domain

Definition:Lowest Common Multiple/Integral Domain

Definition:Irreducible (Ring Theory)

Definition:Associate/Commutative Ring

Definition:Unique Factorization Domain


+ Chapter 7: Homomorphisms and Quotient Algebras

Definition:Homomorphism (Abstract Algebra)

Definition:Group Homomorphism

Definition:Ring Homomorphism

Definition:Monomorphism (Abstract Algebra)

Definition:Epimorphism (Abstract Algebra)

Definition:Isomorphism (Abstract Algebra)

Definition:Endomorphism

Definition:Automorphism (Abstract Algebra)

Definition:Homomorphism (Abstract Algebra)/Image

Definition:Kernel of Group Homomorphism

Definition:Exact Sequence

Definition:Short Exact Sequence

Definition:Quotient Group

Definition:Quotient Ring

Definition:Quotient Field