nyuuzyou/wikis huggingface dataset:
Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence

== Theorem ==
There are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.
== Proof ==
Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in an arithmetic sequence.
Then:
Since $n$ is rational, we require $8 k + 17$ to be a square.
Since $8 k + 17$ is odd, if $8 k + 17$ is square, then $\sqrt {8 k + 17}$ is odd.
Write $\sqrt {8 k + 17} = 2 x + 1$.
Notice that:
Using the substitution $k = \dfrac {x^2 + x - 4} 2$:
Each $x$ with $k = \dfrac {x^2 + x - 4} 2 > 0$ give a value for $n$.
Therefore there are an infinite number of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence.
== Examples ==
== Also see ==
* Rows in Pascal's Triangle containing Numbers in Geometric Sequence
* Rows in Pascal's Triangle containing Numbers in Harmonic Sequence
== Historical Note ==
== Sources ==
* : $35$
* : $35$
Category:Pascal's Triangle
Category:Arithmetic Sequences
Category:Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence/Examples

== Examples of Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence ==
The first instances of rows of Pascal's triangle which contain $3$ integers in arithmetic sequence are as follows:
=== $7, 21, 35$ ===
=== $1001, 2002, 3003$ ===
=== $490 \,314, 817 \, 190, 1 \, 144 \, 066$ ===
Category:Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence/Examples/1001, 2002, 3003

== Example of Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence ==
The integers:
:$1001, 2002, 3003$
are in arithmetic sequence and appear in row $14$ of Pascal's triangle.
== Proof ==
We have:
thus demonstrating the common difference of $1001$.
Then we have that row $14$ of Pascal's triangle is:
:$1, 14, 91, 364, \color {red} {1001, 2002, 3003}, 3432, 3003, 2002, 1001, 364, 91, 14, 1$
== Sources ==
* : $35$
* : $35$
Category:Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence/Examples/490314, 817190, 1144066

== Example of Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence ==
The integers:
:$490 \, 314, 817 \, 190, 1 \, 144 \, 066$
are in arithmetic sequence and appear in row $23$ of Pascal's triangle.
== Proof ==
We have:
thus demonstrating the common difference of $326 \, 876$.
Then we have that row $23$ of Pascal's triangle is:
:$1, 23, 253, 1771, 8855, 33 \, 649, 100 \, 947,$
:$245 \, 157, \color {red} {490 \, 314, 817 \, 190, 1 \, 144 \, 066}, 1 \, 352 \, 078, 1 \, 352 \, 078,$
:$1 \, 144 \, 066, 817 \, 190, 490 \, 314, 245 \, 157, 100 \, 947, 33 \, 649, 8855, 1771, 253, 23, 1$
== Sources ==
* : $35$
* : $35$
Category:Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence/Examples/7, 21, 35

== Example of Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence ==
The integers:
:$7, 21, 35$
are in arithmetic sequence and appear in row $7$ of Pascal's triangle.
== Proof ==
We have:
thus demonstrating the common difference of $14$.
Then we have that row $7$ of Pascal's triangle is:
:$1, \color {red} {7, 21, 35}, 35, 21, 7, 1$
== Sources ==
* : $35$
* : $35$
Category:Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
Rows in Pascal's Triangle containing Numbers in Geometric Sequence

== Theorem ==
There exist no rows of Pascal's triangle which contain $3$ integers in geometric sequence.
== Proof ==
Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a geometric sequence.
Then:
Since $n \ge 0$, no row of Pascal's triangle contains $3$ integers in geometric sequence.
However, suppose one extends the definition of binomial coefficients to allow $n < 0$.
Then by Negated Upper Index of Binomial Coefficient, we have: 
:$\dbinom {-1} k = \paren {-1}^k$
which indeed forms a geometric sequence.
== Also see ==
* Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
* Rows in Pascal's Triangle containing Numbers in Harmonic Sequence
== Historical Note ==
== Sources ==
* : $35$
* : $35$
Category:Pascal's Triangle
Category:Geometric Sequences
Rows in Pascal's Triangle containing Numbers in Harmonic Sequence

== Theorem ==
There exist no rows of Pascal's triangle which contain $3$ integers in harmonic sequence.
== Proof ==
Suppose $\dbinom n k$, $\dbinom n {k + 1}$ and $\dbinom n {k + 2}$ are in a harmonic sequence.
Then:
This is a quadratic equation in $n$, so we can calculate its discriminant.
Notice that for each $k \ge 0$:
:$\paren {4 k + 3}^2 - 4 \paren {4 k^2 + 8 k + 4} = - \paren {8 k + 7} < 0$
By Solution to Quadratic Equation with Real Coefficients, there is no real solution for $n$.
Therefore there is no row of Pascal's triangle which contain $3$ integers in harmonic sequence.
== Also see ==
* Rows in Pascal's Triangle containing Numbers in Arithmetic Sequence
* Rows in Pascal's Triangle containing Numbers in Geometric Sequence
== Historical Note ==
== Sources ==
* : $35$
* : $35$
Category:Pascal's Triangle
Category:Harmonic Sequences
Rubik's Cube has 54 Facets

== Theorem ==
Let $S$ be the set of facets of Rubik's cube.
Then the cardinality of $S$ is given by:
:$\card S = 54$
That is:
:A Rubik's cube has $54$ facets.
== Proof ==
A cube, by definition, has $6$ faces.
Each face is subdivided into $9$ facets.
Hence there are $6 \times 9 = 54$ facets in total.
== Sources ==
* : Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson
Category:Rubik's Cube
Ruelle-Perron-Frobenius Theorem

== Theorem ==
Let $\struct {X_\mathbf A ^+, \sigma}$ be a one-sided shift of finite type.
Let $\map C {X_\mathbf A ^+} := \map C {X_\mathbf A ^+, \C}$ be the continuous mapping space.
Let $F_\theta ^+$ be the space of Lipschitz functions on $X_\mathbf A ^+$.
Let $f \in F_\theta ^+$ be a real-valued function.
Let $\LL_f : \map C {X_\mathbf A ^+} \to \map C {X_\mathbf A ^+}$ be the transfer operator.
Let $\beta $ be the spectral radius of $\LL_f$.
If $\mathbf A$ is irreducible, then:
:$(1): \quad \beta$ is a simple eigenvalue of $\LL_f$ with a strictly positive eigenfunction $h \in F_\theta ^+$, i.e.:
::::$\set {g \in \map C {X_\mathbf A ^+} : \LL_f g = \beta g} = \C \cdot h$
Furthermore, if $\mathbf A$ is irreducible and aperiodic, then:
:$(2): \quad$ There is an $r \in \openint 0 \beta$ such that:
::::$\map \sigma {\LL_f} \setminus \set \beta \subseteq \map B {0, r}$
:::where:
::::the denotes a closed disk
::::$\map \sigma {\LL_f}$ denotes the spectrum of $\LL_f : F_\theta ^+ \to F_\theta ^+$
:$(3): \quad$ There is a Borel probability measure $\mu$ such that:
::::$\ds \forall v \in \map C {X_\mathbf A ^+} : \int \LL_f v \rd \mu = \beta \int v \rd \mu$
:$(4): \quad \ds \forall v \in \map C {X_\mathbf A ^+} : \lim _{n \mathop \to \infty} \norm {\beta^{-n} \LL_f ^n v - h \int v \rd \mu}_\infty = 0$
:$(5): \quad \ds \int h \rd \mu = 1$
== Also known as ==
$(2)$ is said that $\LL_f : F_\theta ^+ \to F_\theta ^+$:
:has a spectral gap, or
:is quasi-compact.
== Also see ==
* Definition:Essential Spectral Radius of Bounded Linear Operator
* Definition:Quasi-Compact Operator
== Proof ==
Category:Ergodic Theory
Category:Functional Analysis
Ruffini-Horner Method/Examples/Arbitrary Example 1

== Examples of Use of the Ruffini-Horner method ==
Consider the polynomial equation:
:$\map f x = x^2 - x - 1 = 0$
We have that:
so we observe there is a root between $x = 1$ and $x = 2$.
Then:
We then identify a root between $x = 0.6$ and $x = 0.7$.
This leads to calculating:
:$\map {f_2} x = \map {f_1} {x - 0.6}$
Hence and so, until the required accuracy is achieved.
== Sources ==
* 
Category:Examples of Use of Ruffini-Horner Method
Rule for Extracting Random Variable from Conditional Expectation of Product

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\GG \subseteq \Sigma$ be a sub-$\sigma$-algebra.
Let $X$ and $Y$ be integrable random variables such that: 
:$X Y$ is integrable
and:
:$Y$ is $\GG$-measurable. 
Let $\expect {X \mid \GG}$ be a version of the conditional expectation of $X$ given $\GG$.
Let $\expect {X Y \mid \GG}$ be a version of the conditional expectation of $X Y$ given $\GG$.
Then: 
:$\expect {X Y \mid \GG} = Y \expect {X \mid \GG}$ almost everywhere.
== Proof ==
Let $X$ and $Y$ be non-negative random variables. 
We show first that the statement holds if $Y = \chi_A$ for some $A \in \GG$. 
We show that $Y \expect {X \mid \GG}$ is a version of the conditional expectation of $X Y$ given $\GG$.
We will then obtain the demand from Existence and Essential Uniqueness of Conditional Expectation Conditioned on Sigma-Algebra.
Then, for any $G \in \GG$ we have: 
So we have:
:$\expect {X Y \mid \GG} = Y \expect {X \mid \GG}$ almost everywhere
for all $Y$ of the form $\chi_A$ for $A \in \GG$. 
Now let $Y$ be a positive simple real-valued random variable that is $\GG$-measurable.
From Simple Function has Standard Representation, there exists:
:a finite sequence $a_1, \ldots, a_n$ of real numbers
:a partition $A_1, A_2, \ldots, A_n$ of $\GG$-measurable sets
with:
:$\ds Y = \ds \sum_{j \mathop = 1}^n a_j \chi_{A_j}$
We then have:
Now let $Y$ be a general non-negative random variable. 
Similarly, there exists an increasing sequence $\sequence {Y_n}_{n \mathop \in \N}$ of positive simple random variables such that: 
:$\ds \map Y \omega = \lim_{n \mathop \to \infty} \map {Y_n} \omega$
for $\omega \in \Omega$. 
Then $\sequence {X Y_n}_{n \mathop \in \N}$ is an increasing sequence of random variables.
So, by the Conditional Monotone Convergence Theorem, we have: 
:$\ds \lim_{n \mathop \to \infty} \expect {X Y_n \mid \GG} = \expect {X Y \mid \GG}$ almost everywhere. 
We also see that: 
:$\ds \lim_{n \mathop \to \infty} Y_n \expect {X \mid \GG} = Y \expect {X \mid \GG}$
Hence we have: 
:$\expect {X Y \mid \GG} = Y \expect {X \mid \GG}$ almost everywhere
when $X$ and $Y$ are non-negative. 
Now, suppose that $X$ and $Y$ are general integrable random variables.
Then we have: 
From Function Measurable iff Positive and Negative Parts Measurable, we have that:
:$Y^+$ and $Y^-$ are $\GG$-measurable.
Then we have, since the theorem is known to hold for non-negative random variables:
== Also known as ==
Applying this rule is often known as taking out what is known.
== Sources ==
* : $9.7$: Properties of conditional expectation: a list
Category:Conditional Expectation
Rule of Addition

== Sequent ==
The rule of addition is a valid argument in types of logic dealing with disjunctions $\lor$.
This includes propositional logic and predicate logic, and in particular natural deduction.
=== Proof Rule ===
=== Sequent Form ===
== Explanation ==
== Also known as ==
== Also see ==
* Definition:Disjunction
* Proof by Cases
Category:Disjunction
Category:Rule of Addition
Rule of Addition/Explanation

== Proof Rule ==
The Rule of Addition consists of two proof rules in one.
The first of the two can be expressed in natural language as:
:Given a statement, we may infer a disjunction where the given statement is the first of the disjuncts.
The second of the two can be expressed in natural language as:
:Given a statement, we may infer a disjunction where the given statement is the second of the disjuncts.
The statement being added may be any statement at all.
It does not matter what its truth value is.
That is: if $p$ is true, then $p \lor q$ is likewise true, whatever $q$ may be.
This may seem a bewildering and perhaps paradoxical axiom to admit. How can you deduce a valid argument from a statement form that can deliberately be used to include a statement whose truth value can be completely arbitrary? Or even blatantly false?
But consider the common (although admittedly rhetorical) figure of speech which goes:
:Reading Football Club are going up this season or I'm a monkey's uncle.
Category:Rule of Addition
Rule of Addition/Proof Rule

== Proof Rule ==
The rule of addition is a valid argument in types of logic dealing with disjunctions $\lor$.
This includes propositional logic and predicate logic, and in particular natural deduction.
As a proof rule it is expressed in either of the two forms:
:$(1): \quad$ If we can conclude $\phi$, then we may infer $\phi \lor \psi$.
:$(2): \quad$ If we can conclude $\psi$, then we may infer $\phi \lor \psi$.
It can be written:
:$\ds {\phi \over \phi \lor \psi} \lor_{i_1} \qquad \qquad {\psi \over \phi \lor \psi} \lor_{i_2}$
=== Tableau Form ===
== Explanation ==
== Also known as ==
== Also see ==
* This is a rule of inference of the following proof systems:
** Definition:Natural Deduction
== Technical Note ==
== Sources ==
* : $\S \text{II}.12$: Laws of sentential calculus
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$
* : Chapter $1$: The Propositional Calculus $1$: $3$ Conjunction and Disjunction
* : $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $8$
* : $\S 1.2.1$: Rules for natural deduction
Category:Rule of Addition
Category:Rules of Inference
Rule of Addition/Proof Rule/Tableau Form

== Proof Rule ==
Let $\phi$ and $\psi$ be two propositional formulas.
The Rule of Addition is invoked in a tableau proof for $\phi$ or $\psi$ in either of the two forms:
;Form 1:
Let $\phi$ be a propositional formula in a tableau proof.
;Form 2:
Let $\psi$ be a propositional formula in a tableau proof.
Category:Tableau Proofs
Rule of Addition/Proof Rule/Technical Note

== Technical Note on ==
When invoking the in a tableau proof, use the template:
:{{Addition|line|pool|statement|depend|1 or 2}}
or:
:{{Addition|line|pool|statement|depend|1 or 2|comment}}
where:
:line is the number of the line on the tableau proof where the is to be invoked
:pool is the pool of assumptions (comma-separated list)
:statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters
:depend is the line of the tableau proof upon which this line directly depends
:1 or 2 should hold 1 for Addition_1, and 2 for Addition_2
:comment is the (optional) comment that is to be displayed in the Notes column.
Category:Technical Notes
Rule of Addition/Sequent Form/Formulation 1/Form 1

== Theorem ==
== Proof 1 ==
== Proof by Truth Table ==
== Sources ==
* : $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables: Exercise $\text{I} \ \mathbf a.$
* : $3$: The Method of Deduction: $3.1$: Formal Proof of Validity: Rules of Inference: $9.$
Category:Rule of Addition
Rule of Addition/Sequent Form/Formulation 1/Form 1/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
$\begin{array}{|c||ccc|} \hline
p & p & \lor & q \\
\hline
\F & \F & \F & \F \\
\F & \F & \T & \T \\
\T & \T & \T & \F \\
\T & \T & \T & \T \\
\hline
\end{array}$
As can be seen, when $p$ is true so is $p \lor q$.
Category:Truth Table Proofs
Category:Rule of Addition
Rule of Addition/Sequent Form/Formulation 1/Form 2/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
$\begin{array}{|c||ccc|} \hline
q & p & \lor & q \\
\hline
\F & \F & \F & \F \\
\T & \F & \T & \T \\
\F & \T & \T & \F \\
\T & \T & \T & \T \\
\hline
\end{array}$
As can be seen, when $q$ is true so is $p \lor q$.
Category:Truth Table Proofs
Category:Rule of Addition
Rule of Addition/Sequent Form/Formulation 2/Form 1

== Theorem ==
:$\vdash p \implies \paren {p \lor q}$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 3.6$: Reference Formulae: $RF \, 6$
* : Chapter $2$: The Propositional Calculus $2$: $2$: Theorems and Derived Rules: Exercise $3 \ \text{(a)}$
Category:Rule of Addition
Rule of Addition/Sequent Form/Formulation 2/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives are $T$ for all boolean interpretations.
$\begin{array}{|c|c|ccccc|ccccc|} \hline
p & q & p & \implies & (p & \lor & q) & q & \implies & (p & \lor & q) \\
\hline
\F & \F & \F & \T & \F & \F & \F & \F & \T & \F & \F & \F \\
\F & \T & \F & \T & \F & \T & \T & \T & \T & \F & \T & \T \\
\T & \F & \T & \T & \T & \T & \F & \F & \T & \T & \T & \F \\
\T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\
\hline
\end{array}$
Category:Truth Table Proofs
Category:Rule of Addition
Rule of Addition/Sequent Form/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
$\begin{array}{|c|c||ccc|} \hline
p & q & p & \lor & q\\
\hline
\F & \F & \F & \F & \F \\
\F & \T & \F & \T & \T \\
\T & \F & \T & \T & \F \\
\T & \T & \T & \T & \T \\
\hline
\end{array}$
As can be seen, whenever either $p$ or $q$ (or both) are true, then so is $p \lor q$.
Category:Truth Table Proofs
Category:Rule of Addition
Rule of Association

== Theorem ==
== Conjunction ==
== Disjunction ==
== Also known as ==
The rule of association is also known as the associative law.
Note that this term is also used throughout mathematics in the context of addition and multiplication of numbers:
:Associative Law of Addition
:Associative Law of Multiplication
so it is wise to be aware of the context.
== Also see ==
* Rule of Commutation
== Sources ==
* 
Category:Conjunction
Category:Disjunction
Rule of Association/Conjunction/Formulation 1

== Theorem ==
:$p \land \left({q \land r}\right) \dashv \vdash \left({p \land q}\right) \land r$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S 1.2.1$: Rules for natural deduction: Exercise $1.2: \ 3$
* : $\S 2.3.3$
Category:Rule of Association
Rule of Association/Conjunction/Formulation 1/Proof 2

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccccc||ccccc|} \hline
p & \land & (q & \land & r) & (p & \land & q) & \land & r \\
\hline
F & F & F & F & F & F & F & F & F & F \\
F & F & F & F & T & F & F & F & F & T \\
F & F & T & F & F & F & F & T & F & F \\
F & F & T & T & T & F & F & T & F & T \\
T & F & F & F & F & T & F & F & F & F \\
T & F & F & F & T & T & F & F & F & T \\
T & F & T & F & F & T & T & T & F & F \\
T & T & T & T & T & T & T & T & T & T \\
\hline
\end{array}$
Category:Rule of Association
Category:Truth Table Proofs
Rule of Association/Conjunction/Formulation 2

== Theorem ==
:$\vdash \paren {p \land \paren {q \land r} } \iff \paren {\paren {p \land q} \land r}$
== Proof ==
== Sources ==
* : $\S \text{II}.13$: Symbolism of sentential calculus
* : $\S 3.6$: Reference Formulae: $RF \, 10$
* : $\S 4.7$: The Derivation of Formulae: $D \, 27$
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$: Theorem $\text{T25}$
* : $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $12.$
* : $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $11$
* : Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(v)}$
* : $\S 1.14$: Exercise $12 \ (2)$
Category:Rule of Association
Rule of Association/Disjunction/Formulation 1

== Theorem ==
:$p \lor \paren {q \lor r} \dashv \vdash \paren {p \lor q} \lor r$
== Proof 1 ==
== Proof by Truth Table ==
== Also see ==
* Rule of Commutation
== Sources ==
* : $\S 2.3.3$
Category:Rule of Association
Rule of Association/Disjunction/Formulation 1/Proof by Truth Table

== Theorem ==
== Proof ==
We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccccc||ccccc|} \hline
p & \lor & (q & \lor & r) & (p & \lor & q) & \lor & r \\
\hline
\F & \F & \F & \F & \F & \F & \F & \F & \F & \F \\
\F & \T & \F & \T & \T & \F & \F & \F & \T & \T \\
\F & \T & \T & \T & \F & \F & \T & \T & \T & \F \\
\F & \T & \T & \T & \T & \F & \T & \T & \T & \T \\
\T & \T & \F & \F & \F & \T & \T & \F & \T & \F \\
\T & \T & \F & \T & \T & \T & \T & \F & \T & \T \\
\T & \T & \T & \T & \F & \T & \T & \T & \T & \F \\
\T & \T & \T & \T & \T & \T & \T & \T & \T & \T \\
\hline
\end{array}$
Category:Rule of Association
Category:Truth Table Proofs
Rule of Association/Disjunction/Formulation 2

== Theorem ==
:$\vdash \paren {p \lor \paren {q \lor r} } \iff \paren {\paren {p \lor q} \lor r}$
=== Forward Implication ===
=== Reverse Implication ===
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\S \text{II}.13$: Symbolism of sentential calculus
* : $\S 3.6$: Reference Formulae: $RF \, 11$
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 5$: Theorem $\text{T54}$
* : $3$: The Method of Deduction: $3.2$: The Rule of Replacement: $12.$
* : $\S \text{II}$: The Logic of Statements $(2): \ 6$: Using logical equivalences: $11$
* : Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic: Exercise $(1) \ \text{(vi)}$
* : $\S 1.14$: Exercise $12 \ (3)$
* 
* 
Category:Rule of Association
Rule of Association/Disjunction/Formulation 2/Forward Implication

== Theorem ==
:$\vdash \paren {p \lor \paren {q \lor r} } \implies \paren {\paren {p \lor q} \lor r}$
== Proof ==
== Sources ==
* : $\S 4.7$: The Derivation of Formulae: $D \, 10$
Category:Rule of Association
Category:Hilbert Proof System Instance 2
Rule of Association/Disjunction/Formulation 2/Reverse Implication

== Theorem ==
:$\vdash \paren {p \lor \paren {q \lor r} } \impliedby \paren {\paren {p \lor q} \lor r}$
== Proof ==
By definition of $\impliedby$, we prove:
:$\vdash \paren {\paren {p \lor q} \lor r} \implies \paren {p \lor \paren {q \lor r} }$
 
 
 
== Sources ==
* : $\S 4.7$: The Derivation of Formulae: $D \, 5$ - $D \, 9$
Category:Rule of Association
Category:Hilbert Proof System Instance 2
Rule of Assumption

== Sequent ==
The rule of assumption is a valid argument in many proof systems, including natural deduction.
=== Proof Rule ===
=== Sequent Form ===
For structure-technical reasons, the rule of assumption is symbolised by the sequent:
In this form it is usually referred to as the Law of Identity.
=== Boolean Interpretation ===
== Explanation ==
== Also known as ==
== Also see ==
* Definition:Premise
* Law of Identity
* Rule of Implication
Category:Law of Identity
Rule of Assumption/Boolean Interpretation

== Definition ==
The truth value of a propositional formula $\mathbf A$ under a boolean interpretation $v$ is given by:
:$\map v {\mathbf A} = \begin{cases}
\T & : \map v {\mathbf A} = \T \\
\F & : \map v {\mathbf A} = \F
\end{cases}$
== Sources ==
* : $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 6 \ (1)$
* : $\S 1.5$: Semantics of Propositional Logic
Category:Law of Identity
Rule of Assumption/Explanation

== Proof Rule==
Let the Rule of Assumption be invoked.
It does not matter whether the assumption is true -- all we are concerned about is making sure that any conclusion based on the assumptions made is as the result of a valid argument.
The introduction of an assumption $\phi$ into an argument by means of the Rule of Assumption can be interpreted in natural language as:
:"Suppose it were true that $\phi$"
or:
:"What if $\phi$ were true?"
Category:Law of Identity
Rule of Assumption/Proof Rule

== Proof Rule ==
The rule of assumption is a valid argument in many proof systems, including natural deduction.
As a proof rule it is expressed in the form:
:An assumption $\phi$ may be introduced at any stage of an argument.
=== Tableau Form ===
== Explanation ==
== Also known as ==
== Also see ==
* This is a rule of inference of the following proof system:
** Definition:Natural Deduction
* Definition:Premise
* Law of Identity
== Technical Note ==
== Sources ==
* : $\text{I}$: 'NOT' and 'IF': $\S 3$
* : Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation
* : $\S 1.2.3$: Natural Deduction in summary
Category:Law of Identity
Category:Rules of Inference