hcju/proofwikips huggingface dataset:
L'Hôpital's Rule/Corollary 2
== Corollary to L'Hôpital's Rule == Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Suppose that $\forall x \in \openint a b: \map {g'} x \ne 0$.  Suppose that $\map f x \to \infty$ and $\map g x \to \infty$ as $x \to a^+$. Then: :$\displaystyle \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$ provided that the second limit exists.  == Proof == Let $\displaystyle \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x} = L$. Let $\sequence {x_n}$ be a sequence such that: :$\quad x_n \in \openint a b$ for all $n \in \N$ and $\displaystyle \lim_{n \mathop \to \infty} x_n = a$ From Intermediate Value Theorem for Derivatives and the definition of limit of real function, it follows that: :$\displaystyle \lim_{n \mathop \to \infty} \map g {x_n} = \infty$ and $\sequence {\map g {x_n} }$ is strictly increasing. Consider the range $\closedint {x_{n - 1} } {x_n} \subset \openint a b$ where $n \ge 2$. By Cauchy Mean Value Theorem, there exists $c_n \in \openint {x_{n - 1} } {x_n}$ such that: :$\displaystyle \frac {\map f {x_n} - \map f {x_{n - 1} } } {\map g {x_n} - \map g {x_{n - 1} } } = \frac {\map {f'} {c_n} } {\map {g'} {c_n} }$ From the above and the Squeeze Theorem for Real Sequences: :$\displaystyle \lim_{n \mathop \to \infty} c_n = a$ and: :$\displaystyle \lim_{n \mathop \to \infty} \frac {\map f {x_n} - \map f {x_{n - 1} } } {\map g {x_n} - \map g {x_{n - 1} } } = \lim_{n \mathop \to \infty} \frac {\map {f'} {c_n} } {\map {g'} {c_n} } = L$ So, by Stolz-Cesàro Theorem: :$\displaystyle \lim_{n \mathop \to \infty} \frac {\map f {x_n} } {\map g {x_n} } = L$ From the definition of limit of real function, we deduce that: :$\displaystyle \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = L = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$  == Remarks == * The proof does not actually use the assumption $\displaystyle \lim_{x \mathop \to a^+} \map f x = \infty$. {{refactor|Add these as separate corollaries}} * Cases $x \to b^-$, $x \to \pm \infty$ and $\map g x \to -\infty$ can be proven similarly. {{Namedfor|Guillaume de l'Hôpital|cat = L'Hôpital}} == Historical Note == {{:L'Hôpital's Rule/Historical Note}}
Law of Inverses (Modulo Arithmetic)/Corollary 1
== Corollary to Law of Inverses (Modulo Arithmetic) ==  Let $m, n \in \Z$ such that: :$m \perp n$ that is, such that $m$ and $n$ are coprime. Then: :$\exists n' \in \Z: n n' \equiv 1 \pmod m$  == Proof == By definition of coprime: :$m \perp n \iff \gcd \set {m, n} = 1$ The result follows directly from Law of Inverses (Modulo Arithmetic).  == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Chinese Remainder Theorem|next = Common Factor Cancelling in Congruence/Corollary 1}}: $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $19$
Continuous Image of Compact Space is Compact/Corollary 2
== Corollary to Continuous Image of Compact Space is Compact ==  A continuous mapping from a compact topological space to a metric space is bounded.  == Proof == Follows from Continuous Image of Compact Space is Compact and Compact Subspace of Metric Space is Bounded.  == Sources == * {{BookReference|Principles of Mathematical Analysis|1953|Walter Rudin|prev = Continuous Image of Compact Space is Compact|next = Continuous Image of Compact Space is Compact/Corollary 3/Proof 1}}: $4.15$ * {{BookReference|Introduction to Metric and Topological Spaces|1975|W.A. Sutherland|prev=Continuous Image of Compact Space is Compact/Corollary 1|next=Continuous Image of Compact Space is Compact/Corollary 3/Proof 1}}: $5.5$: Continuous maps on compact spaces: Corollary $5.5.3$
Continuous Image of Compact Space is Compact/Corollary 3
== Corollary to Continuous Image of Compact Space is Compact ==  Let $S$ be a compact topological space. Let $f: S \to \R$ be a continuous real-valued function. Then $f$ attains its bounds on $S$.  == Proof 1 == {{:Continuous Image of Compact Space is Compact/Corollary 3/Proof 1}} == Proof 2 == {{:Continuous Image of Compact Space is Compact/Corollary 3/Proof 2}}
Area of Triangle in Terms of Two Sides and Angle
== Corollary to Area of Triangle in Terms of Side and Altitude ==  The area of a triangle $ABC$ is given by: :$\dfrac 1 2 a b \sin C$ where: : $a, b$ are two of the sides : $C$ is the angle of the vertex opposite its other side $c$.  == Proof 1 == {{:Area of Triangle in Terms of Two Sides and Angle/Proof 1}} == Proof 2 == {{:Area of Triangle in Terms of Two Sides and Angle/Proof 2}} == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Area of Triangle in Terms of Side and Altitude|next = Heron's Formula}}: $\S 4$: Geometric Formulas: $4.5$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Area of Triangle in Terms of Side and Altitude|next = Law of Sines|entry = triangle|subentry = i}}
Derivative of Cosine Function/Corollary
== Corollary to Derivative of Cosine Function ==  :$\map {\dfrac \d {\d x} } {\cos a x} = -a \sin a x$  == Proof == {{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\cos x}       | r = -\sin x       | c = Derivative of $\cos x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\cos a x}       | r = -a \sin a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}}  == Also see == * Derivative of $\sin a x$ * Derivative of $\tan a x$ * Derivative of $\cot a x$ * Derivative of $\sec a x$ * Derivative of $\csc a x$
Derivative of Sine Function/Corollary
== Corollary to Derivative of Sine Function ==  :$\map {\dfrac \d {\d x} } {\sin a x} = a \cos a x$  == Proof == {{begin-eqn}} {{eqn | l = \map {\dfrac \d {\d x} } {\sin x}       | r = \cos x       | c = Derivative of $\sin x$ }} {{eqn | ll= \leadsto       | l = \map {\dfrac \d {\d x} } {\sin a x}       | r = a \cos a x       | c = Derivative of Function of Constant Multiple }} {{end-eqn}}  == Also see == * Derivative of $\cos a x$ * Derivative of $\tan a x$ * Derivative of $\cot a x$ * Derivative of $\sec a x$ * Derivative of $\csc a x$
Cosine of Sum/Corollary
== Corollary to Cosine of Sum ==  :$\map \cos {a - b} = \cos a \cos b + \sin a \sin b$  where $\sin$ denotes the sine and $\cos$ denotes the cosine. == Proof == {{begin-eqn}} {{eqn | l = \map \cos {a - b}       | r = \cos a \, \map \cos {-b} - \sin a \, \map \sin {-b}       | c = Cosine of Sum }} {{eqn | r = \cos a \cos b - \sin a \, \map \sin {-b}       | c = Cosine Function is Even }} {{eqn | r = \cos a \cos b + \sin a \sin b       | c = Sine Function is Odd }} {{end-eqn}}  == Historical Note == {{:Cosine of Sum/Historical Note}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Sine of Difference|next = Sum of Squares of Sine and Cosine/Proof 1}}: $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Cosine of Sum|next = Tangent of Sum}}: $\S 5$: Trigonometric Functions: $5.35$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Cosine of Sum|next = Tangent of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Cosine of Sum/Mistake|next = Tangent of Sum|entry = compound angle formulae|subentry = in trigonometry}}
Union of Connected Sets with Non-Empty Intersections is Connected/Corollary
== Corollary to Union of Connected Sets with Non-Empty Intersections is Connected ==  Let $T = \left({S, \tau}\right)$ be a topological space. Let $I$ be an indexing set. Let $\mathcal A = \left \langle{A_\alpha}\right \rangle_{\alpha \mathop \in I}$ be an indexed family of subsets of $S$, all connected in $T$. Let $B$ be a connected set of $T$ such that: : $\forall C \in \mathcal A: B \cap C \ne \varnothing$ Then $\displaystyle B \cup \bigcup \mathcal A$ is connected.  == Proof 1 == {{:Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 1}} == Proof 2 == {{:Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 2}} == Proof 3 == {{:Union of Connected Sets with Non-Empty Intersections is Connected/Corollary/Proof 3}}
Sine of Sum/Corollary
== Corollary to Sine of Sum ==  :$\map \sin {a - b} = \sin a \cos b - \cos a \sin b$  where $\sin$ denotes the sine and $\cos$ denotes the cosine. == Proof == {{begin-eqn}} {{eqn | l = \map \sin {a - b}       | r = \sin a \, \map \cos {-b} + \cos a \, \map \sin {-b}       | c = Sine of Sum }} {{eqn | r = \sin a \cos b + \cos a \, \map \sin {-b}       | c = Cosine Function is Even }} {{eqn | r = \sin a \cos b - \cos a \sin b       | c = Sine Function is Odd }} {{end-eqn}}  == Historical Note == {{:Sine of Sum/Historical Note}} == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Cosine of Sum|next = Cosine of Difference}}: $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$ * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Sine of Sum|next = Cosine of Sum}}: $\S 5$: Trigonometric Functions: $5.34$ * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Sine of Sum|next = Cosine of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $4$ * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Sine of Sum|next = Cosine of Sum|entry = compound angle formulae|subentry = in trigonometry}}
Ideal of Unit is Whole Ring/Corollary
== Corollary to Ideal of Unit is Whole Ring ==  Let $\struct {R, +, \circ}$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ contains the unity of $R$, then $J = R$.  == Proof == Follows directly from Ideal of Unit is Whole Ring and Unity is Unit.  == Sources == * {{BookReference|Introduction to Abstract Algebra|1969|C.R.J. Clapham|prev = Ideals of Ring of Integers Modulo m/Examples/Order 12|next = Field has 2 Ideals}}: Chapter $5$: Rings: $\S 21$. Ideals: Theorem $35$ * {{BookReference|Rings, Modules and Linear Algebra|1970|B. Hartley|author2 = T.O. Hawkes|prev = Subring of Integers is Ideal|next = Ideals of Field}}: $\S 2$: Exercise $1$ * {{BookReference|An Introduction to Abstract Algebra|1978|Thomas A. Whitelaw|prev = Ideal of Unit is Whole Ring|next = Field has 2 Ideals}}: $\S 58.2$ Ideals
Sum of Geometric Sequence/Corollary 1
== Corollary to Sum of Geometric Sequence ==  Let $a, a r, a r^2, \ldots, a r^{n-1}$ be a geometric sequence. Then: :$\displaystyle \sum_{j \mathop = 0}^{n - 1} a r^j = \frac {a \paren {r^n - 1} } {r - 1}$  {{:Euclid:Proposition/IX/35}} == Proof 1 == {{:Sum of Geometric Sequence/Corollary 1/Proof 1}} == Proof 2 == {{:Sum of Geometric Sequence/Corollary 1/Proof 2}} == Also presented as == This result is also seen presented as: :$\displaystyle \sum_{j \mathop = 0}^{n - 1} a r^j = \frac {a \paren {1 - r^n} } {1 - r}$ which is usually more manageable when $r < 1$. {{Euclid Note|35|IX}} == Sources == * {{BookReference|Euclid: The Thirteen Books of The Elements: Volume 2|1926|ed = 2nd|edpage = Second Edition|Sir Thomas L. Heath|prev = Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd|next = Theorem of Even Perfect Numbers/Sufficient Condition}}: Book $\text{IX}$. Propositions * {{BookReference|Handbook of Mathematical Functions|1964|Milton Abramowitz|author2 = Irene A. Stegun|prev = Sum of Arithmetic Sequence|next = Sum of Infinite Geometric Sequence/Corollary 2}}: $3.1.10$: Sum of Geometric Progression to $n$ Terms * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Definition:Geometric Series|next = Sum of Infinite Geometric Sequence/Corollary 2}}: $\S 19$: Geometric Series: $19.4$ * {{BookReference|Elementary Number Theory|1980|David M. Burton|ed = revised|edpage = Revised Printing|prev = Sum of Sequence of Cubes/Proof by Induction|next = Difference of Two Powers/Proof by Induction}}: Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $2$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Geometric Series|next = Sum of Infinite Geometric Sequence/Corollary 2|entry = geometric series}} * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Common Ratio of Geometric Sequence|next = Definition:Geometric Series|entry = geometric progression (geometric sequence)}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Common Ratio of Geometric Sequence|next = Definition:Geometric Series|entry = geometric progression (geometric sequence)}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Geometric Series|next = Sum of Infinite Geometric Sequence/Corollary 2|entry = geometric series}}
Upper and Lower Bounds of Integral/Corollary
== Corollary to Upper and Lower Bounds of Integral ==  Let $f$ be a real function which is continuous on the closed interval $\closedint a b$. Suppose that $\forall t \in \closedint a b: \size {\map f t} < \kappa$. Then: :$\ds \forall \xi, x \in \closedint a b: \size {\int_x^\xi \map f t \rd t} < \kappa \size {x - \xi}$  == Proof == Follows directly from Upper and Lower Bounds of Integral.  == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Integral of Constant/Definite|next = Definite Integral of Function plus Constant}}: $\S 13.6$
Double Negation/Double Negation Introduction
== Proof Rule ==  The rule of '''double negation introduction''' is a valid deduction sequent in propositional logic. === Proof Rule === {{:Double Negation/Double Negation Introduction/Proof Rule}} === Sequent Form === {{:Double Negation/Double Negation Introduction/Sequent Form}}
User:Dfeuer/CRG2
User:Dfeuer/CRG2 2 22614 125790 2013-01-11T19:57:12Z Dfeuer 1672 Dfeuer moved page User:Dfeuer/CRG2 to Properties of Relation Compatible with Group Operation/CRG2 wikitext text/x-wiki #REDIRECT Properties of Relation Compatible with Group Operation/CRG2 brgxg8uewnap1es2qrv2q92h4j7pmhw
Biconditional is Commutative/Formulation 1
== Theorems == The biconditional operator is commutative:  :$p \iff q \dashv \vdash q \iff p$  == Proof 1 == {{:Biconditional is Commutative/Formulation 1/Proof 1}} == Proof 2 == {{:Biconditional is Commutative/Formulation 1/Proof 2}} == Proof 3 == {{:Biconditional is Commutative/Formulation 1/Proof 3}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition|prev=Rule of Commutation/Conjunction/Formulation 1|next=Exclusive Or is Commutative}}: $\S 2.3.3$
De Morgan's Laws (Logic)/Disjunction of Negations
== Theorems == ==== Formulation 1 ==== {{:De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1}} ==== Formulation 2 ==== {{:De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 2}} == Also see == * De Morgan's Laws (Set Theory) for a set theoretic application. * De Morgan's Laws (Predicate Logic) for a predicate logic application. == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Conjunction implies Disjunction|next = Definition:Subset}}: $\S 1.2$: Some Remarks on the Use of the Connectives ''and'', ''or'', ''implies'': Exercise $4$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|next = De Morgan's Laws (Logic)/Conjunction of Negations|entry = De Morgan's Laws}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|next = De Morgan's Laws (Logic)/Conjunction of Negations|entry = De Morgan's Laws}}
Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Forward Implication
== Definition ==  :$p \land \left({q \lor r}\right) \vdash \left({p \land q}\right) \lor \left({p \land r}\right)$  == Proof == {{BeginTableau|p \land \left({q \lor r}\right) \vdash \left({p \land q}\right) \lor \left({p \land r}\right)}} {{Premise|1|p \land \left({q \lor r}\right)}} {{Simplification|2|1|p|1|1}} {{Simplification|3|1|q \lor r|1|2}} {{Assumption|4|q}} {{Conjunction|5|1, 4|p \land q|2|4}} {{Addition|6|1, 4|\left({p \land q}\right) \lor \left({p \land r}\right)|5|1}} {{Assumption|7|r}} {{Conjunction|8|1, 7|p \land r|2|7}} {{Addition|9|1, 7|\left({p \land q}\right) \lor \left({p \land r}\right)|8|2}} {{ProofByCases|10|1|\left({p \land q}\right) \lor \left({p \land r}\right)|3|4|6|7|9}}   == Sources == * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Rule of Association/Disjunction/Formulation 1/Proof 1|next = Rule of Simplification/Sequent Form/Formulation 2/Proof 1/Form 1}}: $\S 1.2.1$: Rules for natural deduction: Example $1.18$
Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication
== Definition ==  :$\left({p \land q}\right) \lor \left({p \land r}\right) \vdash p \land \left({q \lor r}\right)$  == Proof == {{BeginTableau|\left({p \land q}\right) \lor \left({p \land r}\right) \vdash p \land \left({q \lor r}\right)}} {{Premise|1|\left({p \land q}\right) \lor \left({p \land r}\right)}} {{Assumption|2|p \land q}} {{Simplification|3|2|p|2|1}} {{Assumption|4|p \land r}} {{Simplification|5|4|p|2|1}} {{ProofByCases|6|1|p|1|2|3|4|5}} {{Assumption|7|p \land q}} {{Simplification|8|7|q|7|2}} {{Addition|9|7|q \lor r|7|1}} {{Assumption|10|p \land r}} {{Simplification|11|10|r|10|2}} {{Addition|12|10|q \lor r|11|2}} {{ProofByCases|13|1|q \lor r|1|7|9|10|12}} {{Conjunction|14|1|p \land \left({q \lor r}\right)|6|13}}   == Sources == * {{BookReference|Logic in Computer Science: Modelling and reasoning about systems|2000|Michael R.A. Huth|author2 = Mark D. Ryan|prev = Absorption Laws (Logic)/Disjunction Absorbs Conjunction/Forward Implication|next = Definition:Contradiction}}: $\S 1.2.1$: Rules for natural deduction: Exercises $1.4: \ 2 \ \text{(q)}$
Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1
== Definition == The conjunction operator is left distributive over the disjunction operator:  :$p \land \paren {q \lor r} \dashv \vdash \paren {p \land q} \lor \paren {p \land r}$  This can be expressed as two separate theorems: === Forward Implication === {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Forward Implication}} === Reverse Implication === {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Reverse Implication}} == Proof == {{:Rule of Distribution/Conjunction Distributes over Disjunction/Left Distributive/Formulation 1/Proof}} == Sources == * {{BookReference|Beginning Logic|1965|E.J. Lemmon|prev = Rule of Idempotence/Conjunction/Formulation 1|next = Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1}}: $\S 1.5$: Further Proofs: Résumé of Rules: Exercise $1 \ \text{(c)}$ * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Rule of Distribution/Disjunction Distributes over Conjunction/Left Distributive/Formulation 1|next = Rule of Material Equivalence/Formulation 1}}: $\S 2.3.3$ * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Completeness Theorem for Semantic Tableaus|next = De Morgan's Laws (Logic)/Disjunction/Formulation 1}}: $\S 2.10$: Exercise $2.3$
Conjunction Equivalent to Negation of Implication of Negative
== Theorems ==  ==== Formulation 1 ==== {{:Conjunction Equivalent to Negation of Implication of Negative/Formulation 1}} ==== Formulation 2 ==== {{:Conjunction Equivalent to Negation of Implication of Negative/Formulation 2}}
Operations of Boolean Algebra are Idempotent
== Definition == Let $\left({S, \vee, \wedge}\right)$ be a Boolean algebra. Then: :$\forall x \in S: x \wedge x = x = x \vee x$ That is, both $\vee$ and $\wedge$ are idempotent operations. == Proof == Let $x \in S$. Then: {{begin-eqn}} {{eqn | l = x       | r = x \vee \bot       | c = as $\bot$ is the identity of $\vee$ }} {{eqn | r = x \vee \left({x \wedge \neg x}\right)       | c = as $x \wedge \neg x = \bot$ }} {{eqn | r = \left({x \vee x}\right) \wedge \left({x \vee \neg x}\right)       | c = both $\vee$ and $*$ distribute over the other }} {{eqn | r = \left({x \vee x}\right) \wedge \top       | c = as $x \vee \neg x = \top$ }} {{eqn | r = x \vee x       | c =  }} {{end-eqn}} So $x = x \vee x$. {{qed|lemma}} The result $x = x \wedge x$ follows from Duality Principle (Boolean Algebras).  == Sources == * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Idempotent Elements for Integer Multiplication|next = Duality Principle (Boolean Algebras)}}: $\S 1.5$: Theorem $1.13$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Duality Principle (Boolean Algebras)|next = Operations of Boolean Algebra are Associative|entry = Boolean algebra}} * {{BookReference|Introduction to Boolean Algebras|2008|Paul Halmos|author2 = Steven Givant|prev = Complement of Complement (Boolean Algebras)|next = De Morgan's Laws (Boolean Algebras)}}: $\S 2$: Exercise $2$ * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Duality Principle (Boolean Algebras)|next = Operations of Boolean Algebra are Associative|entry = Boolean algebra}}
Method of Truth Tables/Proof of Interderivability
== Proof Technique ==  Suppose we have two propositional formulas $P$ and $Q$ and we wish to see whether $P \dashv \vdash Q$ or not. :Example: Let $P$ be $p \uparrow q$ and let $Q$ be $\neg \left({p \land q}\right)$. There are two things we can do. === Express two statements as a single WFF === We express $P \dashv \vdash Q$ as a single WFF $\left({P \iff Q}\right)$ and perform the method of truth tables on that. Demonstrating this with the example given: $\begin{array}{cc||cccccccc} p & q & (p & \uparrow & q) & \iff & \neg & (p & \land & q) \\ \hline F & F & F & T & F & T & T & F & F & F \\ F & T & F & T & T & T & T & F & F & T \\ T & F & T & T & F & T & T & T & F & F \\ T & T & T & F & T & T & F & T & T & T \\ \end{array}$ As can be seen, the column under the main connective $\iff$ of $\left({P \iff Q}\right)$ is all $T$, so $\left({\left({p \uparrow q}\right) \iff \neg \left({p \land q}\right)}\right)$ is a tautology. Hence from Equivalences are Interderivable, $\left({\left({p \uparrow q}\right) \dashv \vdash \neg \left({p \land q}\right)}\right)$ and the two formulas are interderivable.  === Compare two WFFs in the same table === Alternatively, we can place the two WFFs side by side in the same truth table: $\begin{array}{cc||ccc||cccc} p & q & (p & \uparrow & q) & \neg & (p & \land & q) \\ \hline F & F & F & T & F & T & F & F & F \\ F & T & F & T & T & T & F & F & T \\ T & F & T & T & F & T & T & F & F \\ T & T & T & F & T & F & T & T & T \\ \end{array}$ This time, we need to make sure that the truth values in the columns under the main connectives of both formulae are the same. Note that this is exactly the same as putting a $\iff$ between the two and making a WFF out of the pair of them.