hcju/proofwikips huggingface dataset:
Axiom:Axiom of Replacement
== Axiom ==  For any function $f$ and subset $S$ of the domain of $f$, there is a set containing the image $\map f S$. More formally, let us express this as follows: Let $\map P {y, z}$ be a propositional function, which determines a function. That is, we have: :$\forall y: \exists x: \forall z: \paren {\map P {y, z} \iff x = z}$. Then we state as an axiom: :$\forall w: \exists x: \forall y: \paren {y \in w \implies \paren {\forall z: \paren {\map P {y, z} \implies z \in x} } }$  == Also presented as == The two above statements may be combined into a single (somewhat unwieldy) expression: :$\paren {\forall y: \exists x: \forall z: \paren {\map P {y, z} \implies x = z} } \implies \forall w: \exists x: \forall y: \paren {y \in w \implies \forall z: \paren {\map P {y, z} \implies z \in x} }$ == Also known as == The '''axiom of replacement''' is also known as the '''axiom of substitution'''. == Historical Note == {{:Axiom:Axiom of Replacement/Historical Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Zermelo Set Theory/Historical Note|next = Axiom:Axiom of Replacement/Historical Note}}: Chapter $1$: General Background: $\S 9$ Zermelo set theory * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Replacement cb6yglqcwu8vyjbv8t8n9hr3wisjls5
Axiom:Axiom of Foundation
== Axiom ==  For all non-empty sets, there is an element of the set that shares no element with the set. That is: :$\forall S: \paren {\paren {\exists x: x \in S} \implies \exists y \in S: \forall z \in S: \neg \paren {z \in y} }$ The antecedent states that $S$ is not empty.  == Also defined as == It can also be stated as: :For every non-empty set $S$, there exists an element $x \in S$ such that $x$ and $S$ are disjoint. :A set contains no infinitely descending (membership) sequence. :A set contains a (membership) minimal element. :The membership relation is a foundational relation on any non-empty set. == Also known as == The '''axiom of foundation''' is also known as the '''axiom of regularity'''. == Sources == * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Foundation tj1k4ptei585ohnkgatal3y7fl9um99
Axiom:Axiom of Choice
== Axiom == === Formulation 1 === {{:Axiom:Axiom of Choice/Formulation 1}} === Formulation 2 === {{:Axiom:Axiom of Choice/Formulation 2}} === Formulation 3 === {{:Axiom:Axiom of Choice/Formulation 3}} == Comment == Although it seems intuitively obvious ("surely you can just pick an element?"), when it comes to infinite sets of sets this axiom leads to non-intuitive results, notably the famous Banach-Tarski Paradox. For this reason, the '''Axiom of Choice''' (often abbreviated '''AoC''' or '''AC''') is often treated separately from the rest of the Zermelo-Fraenkel Axioms. Set theory based on the Zermelo-Fraenkel axioms is referred to ZF, while that based on the Z-F axioms including the '''AoC''' is referred to as ZFC. == Additional forms == The following are equivalent, in ZF, to the '''Axiom of Choice''': * Zorn's Lemma * Kuratowski's Lemma * Hausdorff Maximal Principle * Tukey's Lemma * Tychonoff's Theorem * Kelley's Theorem * Vector Space has Basis == Historical Note == {{:Axiom:Axiom of Choice/Historical Note}} == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Topology: An Introduction with Application to Topological Groups|1967|George McCarty|prev = Definition:Surjection/Definition 2|next = Definition:Commutative Diagram}}: Chapter $\text{I}$: Sets and Functions: Composition of Functions * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Surjection iff Right Inverse/Proof 1|next = Definition:Inverse Mapping/Definition 2}}: Appendix $\text{A}.7$: Inverses * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Continuum Hypothesis is Independent of ZFC/Historical Note|next = Axiom:Axiom of Choice/Historical Note}}: Chapter $1$: General Background: $\S 5$ The continuum problem * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} Choice Choice Choice
Axiom:Comprehension Principle
== Definition == The '''comprehension principle''' states:  :Given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$: ::$\set {x: \map P x}$  From the definition of a set: :''A set is any aggregation of objects, called elements, which can be precisely defined in some way or other.'' == Also known as == The '''comprehension principle''' can also be referred to as: :the '''abstraction principle''' :the '''axiom of abstraction''' :the '''unlimited abstraction principle''' == Also see == * Axiom:Axiom of Comprehension -- do not confuse that with this * Definition:Set Definition by Predicate == Historical Note == {{:Axiom:Comprehension Principle/Historical Note}} == Sources == * {{BookReference|Axiomatic Set Theory|1972|Patrick Suppes|ed = 2nd|edpage = Second Edition|prev = Definition:Free Variable|next = Axiom:Comprehension Principle/Historical Note}}: $\S 1.3$ Axiom Schema of Abstraction and Russell's Paradox * {{BookReference|Introduction to Graph Theory|1993|Richard J. Trudeau|prev = Square Root of 2 is Irrational/Classic Proof|next = Russell's Paradox}}: $2$. Graphs: Paradox * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Abstraction|next = Definition:Abstract Space|entry = abstraction|index = 2}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Abstraction|next = Definition:Abstract Space|entry = abstraction|index = 2}} * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Frege Set Theory|next = Definition:Set Definition by Predicate}}: Chapter $1$: General Background: $\S 7$ Frege set theory
Rule of Conjunction
== Sequent == The '''rule of conjunction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Conjunction/Proof Rule}} === Sequent Form === {{:Rule of Conjunction/Sequent Form}} == Explanation == {{:Rule of Conjunction/Explanation}} Thus a conjunction is added to a sequent. == Also known as == {{:Rule of Conjunction/Also known as}} == Also see == * Rule of Simplification
Rule of Simplification
== Sequent == The '''rule of simplification'''  is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Simplification/Proof Rule}} === Sequent Form === {{:Rule of Simplification/Sequent Form}} == Explanation == {{:Rule of Simplification/Explanation}} == Also known as == {{:Rule of Simplification/Also known as}} == Also see == * Rule of Conjunction
Rule of Addition
== Sequent == The '''rule of addition'''  is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Addition/Proof Rule}} === Sequent Form === {{:Rule of Addition/Sequent Form}} == Explanation == {{:Rule of Addition/Explanation}} == Also known as == {{:Rule of Addition/Also known as}} == Also see == * Proof by Cases
Proof by Cases
== Sequent == '''Proof by cases''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Proof by Cases/Proof Rule}} === Sequent Form === {{:Proof by Cases/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Formulation 1 === {{:Proof by Cases/Formulation 1}} === Formulation 2 === {{:Proof by Cases/Formulation 2}} === Formulation 3 === {{:Proof by Cases/Formulation 3}} == Explanation == {{:Proof by Cases/Explanation}} Thus a disjunction is '''eliminated''' from a sequent. == Also known as == {{:Proof by Cases/Also known as}} == Also see == * Rule of Addition * Principle of Dilemma
Modus Ponendo Ponens
== Proof Rule == The '''modus ponendo ponens''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Modus Ponendo Ponens/Proof Rule}} === Sequent Form === {{:Modus Ponendo Ponens/Sequent Form}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Modus Ponendo Ponens/Variant 1}} === Variant 2 === {{:Modus Ponendo Ponens/Variant 2}} === Variant 3 === {{:Modus Ponendo Ponens/Variant 3}} == Also known as == {{:Modus Ponendo Ponens/Also known as}} == Linguistic Note == {{:Modus Ponendo Ponens/Linguistic Note}} == Also see == The following are related argument forms: * Modus Ponendo Tollens * Modus Tollendo Ponens * Modus Tollendo Tollens == Sources == * {{BookReference|Principia Mathematica|1910|Alfred North Whitehead|author2 = Bertrand Russell|volume = 1|prev = Law of Identity/Formulation 2|next = Modus Ponendo Ponens/Also known as}}: Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Method of Truth Tables|next = Modus Tollendo Tollens}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Direct Proof|next = Definition:Distinction between Logical Implication and Conditional}}: Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic * {{BookReference|Mathematical Logic and Computability|1996|H. Jerome Keisler|author2 = Joel Robbin|prev = Four Color Theorem for Finite Maps implies Four Color Theorem for Infinite Maps|next = Reductio ad Absurdum/Variant 2}}: $\S 1.12$: Valid Arguments * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Rigidity Modulus|next = Modus Tollendo Ponens|entry = ''modus ponens''}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Modulus Sign|next = Modus Tollendo Ponens|entry = ''modus ponens''}}
Rule of Implication
== Proof Rule == The '''rule of implication''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Implication/Proof Rule}} === Sequent Form === {{:Rule of Implication/Sequent Form}} == Explanation == {{:Rule of Implication/Explanation}} == Also known as == {{:Rule of Implication/Also known as}} == Sources == * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Implication Equivalent to Negation of Conjunction with Negative/Formulation 1|next = Rule of Transposition/Formulation 2}}: $2.4$: Statement Forms
Proof by Contradiction
== Proof Rule == '''Proof by contradiction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Proof by Contradiction/Proof Rule}} === Sequent Form === {{:Proof by Contradiction/Sequent Form}} == Explanation == {{:Proof by Contradiction/Explanation}} == Also known as == {{:Proof by Contradiction/Also known as}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Proof by Contradiction/Variant 1}} === Variant 2 === {{:Proof by Contradiction/Variant 2}} === Variant 3 === {{:Proof by Contradiction/Variant 3}} == Also see == * Reductio ad Absurdum * Law of Excluded Middle == Sources == * {{BookReference|Adventures in Group Theory|2008|David Joyner|ed = 2nd|edpage = Second Edition|prev = Definition:Contrapositive Statement|next = Reductio ad Absurdum}}: Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if... * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Proof|next = Definition:Proper Divisor|entry = proof by contradiction}}
Rule of Explosion
== Proof Rule == The '''Rule of Explosion''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Rule of Explosion/Proof Rule}} === Sequent Form === {{:Rule of Explosion/Sequent Form}} == Explanation == {{:Rule of Explosion/Explanation}} == Variants == The following can be used as variants of this theorem: === Variant 1 === {{:Rule of Explosion/Variant 1}} === Variant 2 === {{:Rule of Explosion/Variant 2}} === Variant 3 === {{:Rule of Explosion/Variant 3}} == Also known as == {{:Rule of Explosion/Also known as}} == Also see == * False Statement implies Every Statement
Law of Excluded Middle
== Proof Rule == The '''law of (the) excluded middle''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Law of Excluded Middle/Proof Rule}} === Sequent Form === {{:Law of Excluded Middle/Sequent Form}} == Explanation == {{:Law of Excluded Middle/Explanation}} == Also known as == {{:Law of Excluded Middle/Also known as}} == Also see == * Principle of Non-Contradiction == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Biconditional|next = Definition:Tautology (Boolean Interpretations)}}: Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic * {{BookReference|Logic for Mathematicians|1988|Alan G. Hamilton|ed = 2nd|edpage = Second Edition|prev = Definition:Unsatisfiable (Boolean Interpretations)|next = Principle of Non-Contradiction}}: $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(a)}$ * {{BookReference|Mathematical Logic for Computer Science|1993|M. Ben-Ari|prev = Definition:Intuitionistic Logic|next = Definition:Temporal Logic}}: Chapter $1$: Introduction: $\S 1.4$: Non-standard logics * {{BookReference|Introduction to Graph Theory|1993|Richard J. Trudeau|prev = Russell's Paradox|next = Definition:Theory of Types}}: $2$. Graphs: Paradox
Modus Tollendo Tollens
== Proof Rule == The '''modus tollendo tollens''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Modus Tollendo Tollens/Proof Rule}} === Sequent Form === {{:Modus Tollendo Tollens/Sequent Form}} == Explanation == {{:Modus Tollendo Tollens/Explanation}} == Also known as == {{:Modus Tollendo Tollens/Also known as}} == Also see == The following are related argument forms: * Modus Ponendo Ponens * Modus Ponendo Tollens * Modus Tollendo Ponens The Rule of Transposition is conceptually similar, and can be derived from the MTT by a simple application of the Rule of Implication. These are classic fallacies: * Affirming the Consequent * Denying the Antecedent == Linguistic Note == {{:Modus Tollendo Tollens/Linguistic Note}} == Sources == * {{BookReference|Symbolic Logic|1973|Irving M. Copi|ed = 4th|edpage = Fourth Edition|prev = Modus Ponendo Ponens|next = Hypothetical Syllogism/Formulation 1}}: $2.3$: Argument Forms and Truth Tables * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Modus Tollendo Ponens|next = Definition:Mole|entry = ''modus tollens''}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Modus Tollendo Ponens|next = Definition:Mole|entry = ''modus tollens''}}
Tautology and Contradiction
== Theorems == === Contradiction is Negation of Tautology === {{:Contradiction is Negation of Tautology}} === Tautology is Negation of Contradiction === {{:Tautology is Negation of Contradiction}} === Conjunction with Tautology === {{:Conjunction with Tautology}} === Disjunction with Tautology === {{:Disjunction with Tautology}} === Conjunction with Contradiction === {{:Conjunction with Contradiction}} === Disjunction with Contradiction === {{:Disjunction with Contradiction}} == Comment == Note that the proofs of: * $\neg \bot \vdash \top$ * $\neg \top \vdash \bot$ * $p \vdash p \land \top$ * $p \lor \top \vdash \top$ rely (directly or indirectly) upon Law of Excluded Middle - and it can be seen that they are just another way of stating that truth. The propositions: :''If it's not false, it must be true'' and :''If it's not true, it must be false'' are indeed valid ''only'' in the context where there are only two truth values. From the intuitionistic perspective, these results do not hold. == Sources == * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Definition:Contingent Statement|next = Definition:Truth Value/Aristotelian Logic}}: $\S \text{I}: 3$: Logical Constants $(2)$
Reductio ad Absurdum
== Proof Rule == The '''reductio ad absurdum''' is a valid deduction sequent in propositional logic: === Proof Rule === {{:Reductio ad Absurdum/Proof Rule}} === Sequent Form === {{:Reductio ad Absurdum/Sequent Form}} == Explanation == {{:Reductio ad Absurdum/Explanation}} == Variants == The following forms can be used as variants of this theorem: === Variant 1 === {{:Reductio ad Absurdum/Variant 1}} === Variant 2 === {{:Reductio ad Absurdum/Variant 2}}  == Also see == * Clavius's Law * Proof by Contradiction, often treated as another aspect of the same thing. From the point of view of purely classical logic, this is acceptable. However, in the context of intuitionistic logic, it is '''essential''' to bear in mind that only the Proof by Contradiction is valid. This is because Proof by Contradiction starts with a positive assumption $\phi$. As a result, it does not depend on the Law of Excluded Middle. == Linguistic Note == {{:Reductio ad Absurdum/Linguistic Note}} == Sources == * {{BookReference|A Handbook of Terms used in Algebra and Analysis|1972|A.G. Howson|prev = Definition:Consistent Proof System|next = Definition:Independent Proof System}}: $\S 1$: Some mathematical language: Axiom systems * {{BookReference|Elementary Logic|1980|D.J. O'Connor|author2 = Betty Powell|prev = Biconditional as Disjunction of Conjunctions/Formulation 2/Proof 2|next = Definition:Natural Deduction}}: $\S \text{I}: 13$: A Short Cut to Truth-tables * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Definition:Converse Statement|next = Square Root of 2 is Irrational/Classic Proof}}: Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic * {{BookReference|Adventures in Group Theory|2008|David Joyner|ed = 2nd|edpage = Second Edition|prev = Proof by Contradiction|next = Definition:Conjunction/Truth Table}}: Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...
Conjunction and Implication
== Theorems == === Conjunction Equivalent to Negation of Implication of Negative === {{:Conjunction Equivalent to Negation of Implication of Negative}} === Implication Equivalent to Negation of Conjunction with Negative === {{:Implication Equivalent to Negation of Conjunction with Negative}} === Conjunction with Negative Equivalent to Negation of Implication === {{:Conjunction with Negative Equivalent to Negation of Implication}} === Modus Ponendo Tollens === {{:Modus Ponendo Tollens/Variant}} == Law of Excluded Middle == Note that the Modus Ponendo Tollens: :$\neg \left({p \land q}\right) \dashv \vdash p \implies \neg q$ can be proved in both directions without resorting to Law of Excluded Middle. All the others: * $p \land q \vdash \neg \left({p \implies \neg q}\right)$ * $p \implies q \vdash \neg \left({p \land \neg q}\right)$ * $p \land \neg q \vdash \neg \left({p \implies q}\right)$ are not reversible in intuitionistic logic. == Also see == * Negation of Conditional implies Antecedent * Negation of Conditional implies Negation of Consequent
Set Difference and Intersection form Partition/Corollary 2
== Corollary to Set Difference and Intersection form Partition ==  Let $\O \subsetneqq T \subsetneqq S$. Then: :$\set {T, \relcomp S T}$ is a partition of $S$.  == Proof == First we note that: :$\O \subsetneqq T \implies T \ne \O$ :$T \subsetneqq S \implies T \ne S$ from the definition of proper subset. From the definition of relative complement, we have: :$\relcomp S T = S \setminus T$. It follows from Set Difference with Self is Empty Set that: :$T \ne S \iff \relcomp S T \ne \O$ From Intersection with Relative Complement is Empty: :$T \cap \relcomp S T = \O$ That is, $T$ and $\relcomp S T$ are disjoint. From Union with Relative Complement: :$T \cup \relcomp S T = S$ That is, the union of $T$ and $\relcomp S T$ forms the whole set $S$. Thus all the conditions for a partition are satisfied.  == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Definition:Set Partition/Definition 1|next = Definition:Endorelation}}: $\S 6$. Indexed families; partitions; equivalence relations: Example $6.4$
Empty Intersection iff Subset of Complement
== Corollary to Intersection with Complement is Empty iff Subset ==  :$S \cap T = \O \iff S \subseteq \relcomp {} T$  where: :$S \cap T$ denotes the intersection of $S$ and $T$ :$\O$ denotes the empty set :$\complement$ denotes set complement :$\subseteq$ denotes subset. === Corollary === {{:Empty Intersection iff Subset of Complement/Corollary}} == Proof 1 == {{:Empty Intersection iff Subset of Complement/Proof 1}} == Proof 2 == {{:Empty Intersection iff Subset of Complement/Proof 2}} == Sources == * {{BookReference|Set Theory and Abstract Algebra|1975|T.S. Blyth|prev = Complement Union with Superset is Universe/Corollary|next = Intersection is Empty and Union is Universe if Sets are Complementary}}: $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Exercise $11 \ \text{(b)}$ * {{BookReference|Introduction to Topology|1975|Bert Mendelson|ed = 3rd|edpage = Third Edition|prev = Intersection with Subset is Subset|next = Complement Union with Superset is Universe/Corollary}}: Chapter $1$: Theory of Sets: $\S 3$: Set Operations: Union, Intersection and Complement: Exercise $1 \ \text{(c)}$
Image is Subset of Codomain/Corollary 1
== Corollary to Image is Subset of Codomain ==  Let $\mathcal R = S \times T$ be a relation. The image of $\mathcal R$ is a subset of the codomain of $\mathcal R$: :$\Img {\mathcal R} \subseteq T$  == Proof == {{begin-eqn}} {{eqn | l = \Img {\mathcal R}       | r = \mathcal R \sqbrk {\Dom {\mathcal R} }       | c = {{Defof|Image of Relation}} }} {{eqn | l = \Dom {\mathcal R}       | o = \subseteq       | r = \Dom {\mathcal R}       | c = Set is Subset of Itself }} {{eqn | ll= \leadsto       | l = \Img {\mathcal R}       | o = \subseteq       | r = T       | c = Image is Subset of Codomain }} {{end-eqn}}
Sundry Coset Results
 {{refactor|Once extension works with TOC, use it to fix headings}} == Theorems == Let $G$ be a group and let $H$ be a subgroup of $G$. Let $x, y \in G$. Let: : $x H$ denote the left coset of $H$ by $x$; : $H y$ denote the right coset of $H$ by $y$. Then the following results apply: === Element in Coset iff Product with Inverse in Subgroup === {{:Element in Coset iff Product with Inverse in Subgroup}} === Cosets are Equal iff Product with Inverse in Subgroup === {{:Cosets are Equal iff Product with Inverse in Subgroup}} === Cosets are Equal iff Element in Other Coset === {{:Cosets are Equal iff Element in Other Coset}} === Coset Equals Subgroup iff Element in Subgroup === {{:Coset Equals Subgroup iff Element in Subgroup}} === Elements in Same Coset iff Product with Inverse in Subgroup === {{:Elements in Same Coset iff Product with Inverse in Subgroup}} === Regular Representation on Subgroup is Bijection to Coset === {{:Regular Representation on Subgroup is Bijection to Coset}}