id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-2200 | Halting Problem is Not Algorithmically Decidable | Let $H: \N^2 \to \N$ be the function given by:
:$\map H {m, n} = 1$ if $m$ codes a URM program which halts with input $n$
:$\map H {m, n} = 0$ otherwise.
Then $H$ is not recursive. | We perform a proof by Cantor's Diagonal Argument.
{{AimForCont}} $H$ is recursive.
Consider the universal URM computable function $\Phi_1: \N^2 \to \N$.
Let $f: \N \to \N$ be the function given by:
:$\map f n = \begin{cases}
\map {\Phi_1} {n, n} & : \map H {n, n} = 1 \\
0 & : \text{otherwise}
\end{cases}$
As $H$ is rec... | Let $H: \N^2 \to \N$ be the [[Definition:Function|function]] given by:
:$\map H {m, n} = 1$ if $m$ [[Unique Code for URM Program|codes]] a [[Definition:URM Program|URM program]] which [[Definition:Unlimited Register Machine/Program/Termination|halts]] with [[Definition:Unlimited Register Machine#Input|input]] $n$
:$\ma... | We perform a proof by [[Cantor's Diagonal Argument]].
{{AimForCont}} $H$ is [[Definition:Recursive Function|recursive]].
Consider the [[Universal URM Computable Functions|universal URM computable function]] $\Phi_1: \N^2 \to \N$.
Let $f: \N \to \N$ be the [[Definition:Function|function]] given by:
:$\map f n = \beg... | Halting Problem is Not Algorithmically Decidable | https://proofwiki.org/wiki/Halting_Problem_is_Not_Algorithmically_Decidable | https://proofwiki.org/wiki/Halting_Problem_is_Not_Algorithmically_Decidable | [
"URM Programs",
"Recursion Theory"
] | [
"Definition:Function",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Unlimited Register Machine/Program/Termination",
"Definition:Unlimited Register Machine",
"Definition:Recursive/Function"
] | [
"Cantor's Diagonal Argument",
"Definition:Recursive/Function",
"Universal URM Computable Functions",
"Definition:Function",
"Definition:Recursive/Function",
"Definition:Recursive/Relation",
"Universal URM Computable Functions",
"Definition:Recursive/Function",
"URM Computable Function is Recursive",... |
proofwiki-2201 | Set of Total Functions is Not Recursive | The set $\operatorname{Tot}$ of natural numbers which code URM programs which compute total functions of one variable is not recursive. | We perform a proof by Cantor's Diagonal Argument.
Suppose the contrary, that $\operatorname{Tot}$ is a recursive set.
First we define a recursive function $h$ which enumerates the code numbers of URM programs which compute total functions of one variable.
The program of this sort with the smallest code is:
{|
|-
! alig... | The set $\operatorname{Tot}$ of [[Definition:Natural Numbers|natural numbers]] which [[Unique Code for URM Program|code]] [[Definition:URM Program|URM programs]] which compute [[Definition:Total Function|total functions]] of one variable is not [[Definition:Recursive Set|recursive]]. | We perform a proof by [[Cantor's Diagonal Argument]].
Suppose the contrary, that $\operatorname{Tot}$ is a [[Definition:Recursive Set|recursive set]].
First we define a [[Definition:Recursive Function|recursive function]] $h$ which enumerates the [[Unique Code for URM Program|code numbers]] of [[Definition:URM Progr... | Set of Total Functions is Not Recursive | https://proofwiki.org/wiki/Set_of_Total_Functions_is_Not_Recursive | https://proofwiki.org/wiki/Set_of_Total_Functions_is_Not_Recursive | [
"URM Programs",
"Recursion Theory"
] | [
"Definition:Natural Numbers",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Total Function",
"Definition:Recursive/Set"
] | [
"Cantor's Diagonal Argument",
"Definition:Recursive/Set",
"Definition:Recursive/Function",
"Unique Code for URM Program",
"Definition:Unlimited Register Machine/Program",
"Definition:Total Function",
"Definition:Basic Primitive Recursive Function",
"Definition:Recursive/Function",
"Definition:Minimi... |
proofwiki-2202 | Infinitely Many Programs for URM Computable Function | Let $g: \N^k \to \N$ be a URM computable function.
Then there is an infinite number of URM programs which compute $g$. | As $g$ is URM computable, there exists a URM program which computes it.
Let $Q$ be such a program.
Let $n \in \N$.
Increment the <tt>Jump</tt>s in $Q$ by $n$ lines<ref>To '''increment the <tt>Jump</tt>s by $r$''' for any normalized URM program is done by changing all <tt>Jump</tt>s of the form $J \left({m, n, q}\right)... | Let $g: \N^k \to \N$ be a [[Definition:URM Computability|URM computable function]].
Then there is an [[Definition:Infinite|infinite number]] of [[Definition:URM Program|URM programs]] which compute $g$. | As $g$ is [[Definition:URM Computability|URM computable]], there exists a [[Definition:URM Program|URM program]] which computes it.
Let $Q$ be such a program.
Let $n \in \N$.
Increment the <tt>Jump</tt>s in $Q$ by $n$ lines<ref>To '''increment the <tt>Jump</tt>s by $r$''' for any [[Normalized URM Program|normalized ... | Infinitely Many Programs for URM Computable Function | https://proofwiki.org/wiki/Infinitely_Many_Programs_for_URM_Computable_Function | https://proofwiki.org/wiki/Infinitely_Many_Programs_for_URM_Computable_Function | [
"URM Programs"
] | [
"Definition:URM Computability",
"Definition:Infinite",
"Definition:Unlimited Register Machine/Program"
] | [
"Definition:URM Computability",
"Definition:Unlimited Register Machine/Program",
"Normalized URM Program",
"Definition:Unlimited Register Machine/Program",
"Single Instruction URM Programs",
"Definition:Unlimited Register Machine/Program",
"Infinite if Injection from Natural Numbers"
] |
proofwiki-2203 | Bounds of GCD for Sum and Difference Congruent Squares | Let $x, y, n$ be integers.
Let:
:$x \not \equiv \pm y \pmod n$
and:
:$x^2 \equiv y^2 \pmod n$
where $a \equiv b \pmod n$ denotes that $a$ is congruent to $b$ modulo $n$.
Then:
:$1 < \gcd \set {x - y, n} < n$
and:
:$1 < \gcd \set {x + y, n} < n$
where $\gcd \set {a, b}$ is the GCD of $a$ and $b$. | {{begin-eqn}}
{{eqn | l = x^2
| o = \equiv
| r = y^2
| rr= \pmod n
| c =
}}
{{eqn | ll= \leadsto
| l = n
| o = \divides
| r = \paren {x^2 - y^2}
| c =
}}
{{eqn | ll= \leadsto
| l = n
| o = \divides
| r = \paren {x + y} \paren {x - y}
| c =
}}
{{... | Let $x, y, n$ be [[Definition:Integer|integers]].
Let:
:$x \not \equiv \pm y \pmod n$
and:
:$x^2 \equiv y^2 \pmod n$
where $a \equiv b \pmod n$ denotes that $a$ is [[Definition:Congruence Modulo Integer|congruent to $b$ modulo $n$]].
Then:
:$1 < \gcd \set {x - y, n} < n$
and:
:$1 < \gcd \set {x + y, n} < n$
where $\... | {{begin-eqn}}
{{eqn | l = x^2
| o = \equiv
| r = y^2
| rr= \pmod n
| c =
}}
{{eqn | ll= \leadsto
| l = n
| o = \divides
| r = \paren {x^2 - y^2}
| c =
}}
{{eqn | ll= \leadsto
| l = n
| o = \divides
| r = \paren {x + y} \paren {x - y}
| c =
}}
{{... | Bounds of GCD for Sum and Difference Congruent Squares | https://proofwiki.org/wiki/Bounds_of_GCD_for_Sum_and_Difference_Congruent_Squares | https://proofwiki.org/wiki/Bounds_of_GCD_for_Sum_and_Difference_Congruent_Squares | [
"Greatest Common Divisor"
] | [
"Definition:Integer",
"Definition:Congruence (Number Theory)/Integers",
"Definition:Greatest Common Divisor/Integers"
] | [
"Definition:Prime Factor",
"Category:Greatest Common Divisor"
] |
proofwiki-2204 | Triangle Angle-Side-Angle Congruence | If two triangles have:
:two angles equal to two angles, respectively
:the sides between the two angles equal
then the remaining angles are equal, and the remaining sides equal the respective sides.
That is to say, if two pairs of angles and the included sides are equal, then the triangles are congruent. | :400px
Let $\angle ABC = \angle DEF$, $\angle BCA = \angle EFD$, and $BC = EF$.
{{AimForCont}} that $AB \ne DE$.
If this is the case, one of the two must be greater.
{{WLOG}}, we let $AB > DE$.
We construct a point $G$ on $AB$ such that $BG = ED$.
Using Euclid's first postulate, we construct the segment $CG$.
Now, sinc... | If two [[Definition:Triangle (Geometry)|triangles]] have:
:two [[Definition:Angle|angles]] equal to two [[Definition:Angle|angles]], respectively
:the [[Definition:Side of Polygon|sides]] between the two [[Definition:Angle|angles]] equal
then the remaining [[Definition:Angle|angles]] are equal, and the remaining [[Def... | :[[File:Euclid-I-26-1.png|400px]]
Let $\angle ABC = \angle DEF$, $\angle BCA = \angle EFD$, and $BC = EF$.
{{AimForCont}} that $AB \ne DE$.
If this is the case, one of the two must be greater.
{{WLOG}}, we let $AB > DE$.
We [[Construction of Equal Straight Lines from Unequal|construct a point]] $G$ on $AB$ such th... | Triangle Angle-Side-Angle Congruence | https://proofwiki.org/wiki/Triangle_Angle-Side-Angle_Congruence | https://proofwiki.org/wiki/Triangle_Angle-Side-Angle_Congruence | [
"Triangle Angle-Side-Angle Congruence",
"Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Triangle (Geometry)",
"Definition:Congru... | [
"File:Euclid-I-26-1.png",
"Construction of Equal Straight Lines from Unequal",
"Axiom:Euclid's First Postulate",
"Triangle Side-Angle-Side Congruence",
"Axiom:Euclid's Common Notions",
"Proof by Contradiction",
"Triangle Side-Angle-Side Congruence"
] |
proofwiki-2205 | Triangle Side-Angle-Angle Congruence | If two triangles have:
:two angles equal to two angles, respectively
:the sides opposite one pair of equal angles equal
then the remaining angles are equal, and the remaining sides equal the respective sides.
That is to say, if two pairs of angles and a pair of opposite sides are equal, then the triangles are congruent... | :360px
Let:
:$\angle ABC = \angle DEF$
:$\angle BCA = \angle EFD$
:$AB = DE$
{{AimForCont}} that $BC \ne EF$.
If this is the case, one of the two must be greater.
{{WLOG}}, let $BC > EF$.
We construct a point $H$ on $BC$ such that $BH = EF$, and then we construct the segment $AH$.
Now, since we have:
:$BH = EF$
:$\angl... | If two [[Definition:Triangle (Geometry)|triangles]] have:
:two [[Definition:Angle|angles]] equal to two [[Definition:Angle|angles]], respectively
:the [[Definition:Side of Polygon|sides]] opposite one pair of equal [[Definition:Angle|angles]] equal
then the remaining [[Definition:Angle|angles]] are equal, and the rema... | :[[File:Euclid-I-26-2.png|360px]]
Let:
:$\angle ABC = \angle DEF$
:$\angle BCA = \angle EFD$
:$AB = DE$
{{AimForCont}} that $BC \ne EF$.
If this is the case, one of the two must be greater.
{{WLOG}}, let $BC > EF$.
We [[Construction of Equal Straight Lines from Unequal|construct a point]] $H$ on $BC$ such that $BH... | Triangle Side-Angle-Angle Congruence | https://proofwiki.org/wiki/Triangle_Side-Angle-Angle_Congruence | https://proofwiki.org/wiki/Triangle_Side-Angle-Angle_Congruence | [
"Triangle Side-Angle-Angle Congruence",
"Triangles",
"Congruence (Geometry)"
] | [
"Definition:Triangle (Geometry)",
"Definition:Angle",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Angle",
"Definition:Polygon/Side",
"Definition:Polygon/Side",
"Definition:Angle",
"Definition:Triangle (Geometry)/Opposite",
"Definition:Polygon/Side",
"Definiti... | [
"File:Euclid-I-26-2.png",
"Construction of Equal Straight Lines from Unequal",
"Axiom:Euclid's First Postulate",
"Triangle Side-Angle-Side Congruence",
"External Angle of Triangle is Greater than Internal Opposite",
"Proof by Contradiction",
"Triangle Side-Angle-Side Congruence"
] |
proofwiki-2206 | Equal Alternate Angles implies Parallel Lines | Given two infinite straight lines which are cut by a transversal, if the alternate angles are equal, then the lines are parallel.
{{:Euclid:Proposition/I/27}} | :400px
Let $AB$ and $CD$ be two infinite straight lines.
Let $EF$ be a transversal that cuts them.
Let at least one pair of alternate angles be equal.
{{WLOG}}, let $\angle AHJ = \angle HJD$.
{{AimForCont}} that $AB$ and $CD$ are not parallel.
Then they meet at some point $G$.
{{WLOG}}, let $G$ be on the same side as $... | Given two [[Definition:Infinite Straight Line|infinite straight lines]] which are cut by a [[Definition:Transversal (Geometry)|transversal]], if the [[Definition:Alternate Angles of Transversal|alternate angles]] are equal, then the lines are [[Definition:Parallel Lines|parallel]].
{{:Euclid:Proposition/I/27}} | :[[File:Transversal.png|400px]]
Let $AB$ and $CD$ be two [[Definition:Infinite Straight Line|infinite straight lines]].
Let $EF$ be a [[Definition:Transversal (Geometry)|transversal]] that cuts them.
Let at least one pair of [[Definition:Alternate Angles of Transversal|alternate angles]] be equal.
{{WLOG}}, let $\a... | Equal Alternate Angles implies Parallel Lines | https://proofwiki.org/wiki/Equal_Alternate_Angles_implies_Parallel_Lines | https://proofwiki.org/wiki/Equal_Alternate_Angles_implies_Parallel_Lines | [
"Transversals (Geometry)",
"Parallel Lines"
] | [
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Alternate Angles",
"Definition:Parallel (Geometry)/Lines"
] | [
"File:Transversal.png",
"Definition:Line/Infinite Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Transversal (Geometry)/Alternate Angles",
"Definition:Parallel (Geometry)/Lines",
"Definition:Polygon/External Angle",
"External Angle of Triangle is Greater than Internal Opposite",
"Def... |
proofwiki-2207 | Parallelism is Transitive Relation | Parallelism between straight lines is a transitive relation.
{{:Euclid:Proposition/I/30}} | :300px
Let the straight lines $AB$ and $CD$ both be parallel to the straight line $EF$.
Let the straight line $GK$ be a transversal that cuts the parallel lines $AB$ and $EF$.
By Parallelism implies Equal Alternate Angles:
:$\angle AGK = \angle GHF$
By Playfair's Axiom, there is only one line that passes through $H$ th... | [[Definition:Parallel Lines|Parallelism]] between [[Definition:Straight Line|straight lines]] is a [[Definition:Transitive Relation|transitive relation]].
{{:Euclid:Proposition/I/30}} | :[[File:Euclid-I-30.png|300px]]
Let the [[Definition:Straight Line|straight lines]] $AB$ and $CD$ both be [[Definition:Parallel Lines|parallel]] to the [[Definition:Straight Line|straight line]] $EF$.
Let the [[Definition:Straight Line|straight line]] $GK$ be a [[Definition:Transversal (Geometry)|transversal]] that ... | Parallelism is Transitive Relation | https://proofwiki.org/wiki/Parallelism_is_Transitive_Relation | https://proofwiki.org/wiki/Parallelism_is_Transitive_Relation | [
"Parallel Lines",
"Examples of Transitive Relations"
] | [
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Transitive Relation"
] | [
"File:Euclid-I-30.png",
"Definition:Line/Straight Line",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line",
"Definition:Line/Straight Line",
"Definition:Transversal (Geometry)",
"Definition:Parallel (Geometry)/Lines",
"Parallelism implies Equal Alternate Angles",
"Axiom:Playfai... |
proofwiki-2208 | Sum from 1 to n of 1 over r(r+1) | :$\ds \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$ | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \forall n \ge 1: \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$
=== Basis for the Induction ===
$\map P 1$ is true, as this just says $\dfrac 1 2 = \dfrac 1 2$.
This is our basis for the induction.
=== Induction... | :$\ds \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \forall n \ge 1: \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$
=== Basis for the Induction ===
$\map P 1$ is true, as this just says $\dfra... | Sum from 1 to n of 1 over r(r+1)/Proof 1 | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1) | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1)/Proof_1 | [
"Sum from 1 to n of 1 over r(r+1)",
"Sums of Sequences",
"Reciprocals"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Sum from 1 to n of 1 over r(r+1)/Proof 1",
"Principle of Mathematical Induction"
] |
proofwiki-2209 | Sum from 1 to n of 1 over r(r+1) | :$\ds \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$ | From Partial Fractions Expansion: $\dfrac 1 {x \paren {x + 1} }$:
:$\dfrac 1 {r \paren {r + 1} } = \dfrac 1 r - \dfrac 1 {r + 1}$
and that $\ds \sum_{r \mathop = 1}^n \paren {\frac 1 r - \frac 1 {r + 1} }$ is a telescoping series.
Therefore:
{{begin-eqn}}
{{eqn | l = \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} }
... | :$\ds \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$ | From [[Partial Fractions Expansion/Examples/1 over x(x+1)|Partial Fractions Expansion: $\dfrac 1 {x \paren {x + 1} }$]]:
:$\dfrac 1 {r \paren {r + 1} } = \dfrac 1 r - \dfrac 1 {r + 1}$
and that $\ds \sum_{r \mathop = 1}^n \paren {\frac 1 r - \frac 1 {r + 1} }$ is a [[Definition:Telescoping Series|telescoping series]]. ... | Sum from 1 to n of 1 over r(r+1)/Proof 2 | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1) | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1)/Proof_2 | [
"Sum from 1 to n of 1 over r(r+1)",
"Sums of Sequences",
"Reciprocals"
] | [] | [
"Partial Fractions Expansion/Examples/1 over x(x+1)",
"Definition:Telescoping Series",
"Telescoping Series/Example 1"
] |
proofwiki-2210 | Sum from 1 to n of 1 over r(r+1) | :$\ds \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$ | Observe that:
{{begin-eqn}}
{{eqn | l = \int_r^{r + 1} {\dfrac {\rd x} {x^2} }
| r = \intlimits {\dfrac {-1} x} r {r + 1}
| c = Primitive of Power
}}
{{eqn | r = \dfrac 1 r - \dfrac 1 {r + 1}
}}
{{eqn | r = \dfrac 1 {r \paren {r + 1} }
| c =
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \sum_{r... | :$\ds \sum_{r \mathop = 1}^n \frac 1 {r \paren {r + 1} } = \frac n {n + 1}$ | Observe that:
{{begin-eqn}}
{{eqn | l = \int_r^{r + 1} {\dfrac {\rd x} {x^2} }
| r = \intlimits {\dfrac {-1} x} r {r + 1}
| c = [[Primitive of Power]]
}}
{{eqn | r = \dfrac 1 r - \dfrac 1 {r + 1}
}}
{{eqn | r = \dfrac 1 {r \paren {r + 1} }
| c =
}}
{{end-eqn}}
Therefore:
{{begin-eqn}}
{{eqn | l = \s... | Sum from 1 to n of 1 over r(r+1)/Proof 3 | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1) | https://proofwiki.org/wiki/Sum_from_1_to_n_of_1_over_r(r+1)/Proof_3 | [
"Sum from 1 to n of 1 over r(r+1)",
"Sums of Sequences",
"Reciprocals"
] | [] | [
"Primitive of Power",
"Primitive of Power"
] |
proofwiki-2211 | Sum of Sequence of Fibonacci Numbers | :$\ds \forall n \in \Z_{\ge 0}: \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$ | Proof by induction:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = F_0
| r = 0
| c =
}}
{{eqn | r = 1 - 1
| c =
}}
{{eqn | r = F_2 - 1
| c =
}}
{{end-eqn}}
which is seen to hold... | :$\ds \forall n \in \Z_{\ge 0}: \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 0}^n F_j = F_{n + 2} - 1$
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = F_0
| r = 0
| c =
}}
{{eqn | r = 1 - 1
| c =
}... | Sum of Sequence of Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Sums of Sequences",
"Proofs by Induction"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-2212 | Sum of Sequence of Odd Index Fibonacci Numbers | {{begin-eqn}}
{{eqn | q = \forall n \ge 1
| l = \sum_{j \mathop = 1}^n F_{2 j - 1}
| r = F_1 + F_3 + F_5 + \cdots + F_{2 n - 1}
| c =
}}
{{eqn | r = F_{2 n}
| c =
}}
{{end-eqn}} | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 1}^n F_{2 j - 1} = F_{2 n}$ | {{begin-eqn}}
{{eqn | q = \forall n \ge 1
| l = \sum_{j \mathop = 1}^n F_{2 j - 1}
| r = F_1 + F_3 + F_5 + \cdots + F_{2 n - 1}
| c =
}}
{{eqn | r = F_{2 n}
| c =
}}
{{end-eqn}} | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 1}^n F_{2 j - 1} = F_{2 n}$ | Sum of Sequence of Odd Index Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Index_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Odd_Index_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Sums of Sequences",
"Proofs by Induction"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-2213 | Sum of Sequence of Even Index Fibonacci Numbers | {{begin-eqn}}
{{eqn | q = \forall n \ge 1
| l = \sum_{j \mathop = 1}^n F_{2 j}
| r = F_2 + F_4 + F_6 + \cdots + F_{2 n}
| c =
}}
{{eqn | r = F_{2 n + 1} - 1
| c =
}}
{{end-eqn}} | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 1}^n F_{2 j} = F_{2 n + 1} - 1$ | {{begin-eqn}}
{{eqn | q = \forall n \ge 1
| l = \sum_{j \mathop = 1}^n F_{2 j}
| r = F_2 + F_4 + F_6 + \cdots + F_{2 n}
| c =
}}
{{eqn | r = F_{2 n + 1} - 1
| c =
}}
{{end-eqn}} | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 1}^n F_{2 j} = F_{2 n + 1} - 1$ | Sum of Sequence of Even Index Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Even_Index_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Even_Index_Fibonacci_Numbers | [
"Fibonacci Numbers",
"Sums of Sequences",
"Proofs by Induction"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-2214 | Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers | :$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$ | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$ | :$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 1}^{2 n - 1} F_j F_{j + 1} = {F_{2 n} }^2$ | Sum of Odd Sequence of Products of Consecutive Fibonacci Numbers | https://proofwiki.org/wiki/Sum_of_Odd_Sequence_of_Products_of_Consecutive_Fibonacci_Numbers | https://proofwiki.org/wiki/Sum_of_Odd_Sequence_of_Products_of_Consecutive_Fibonacci_Numbers | [
"Sums of Sequences",
"Fibonacci Numbers",
"Proofs by Induction"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-2215 | Lucas Number as Sum of Fibonacci Numbers | Let $L_k$ be the $k$th Lucas number, defined as:
:$L_n = \begin{cases}
2 & : n = 0 \\
1 & : n = 1 \\
L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
Then:
:$L_n = F_{n - 1} + F_{n + 1}$ | Proof by induction:
For all $n \in \N_{>0}$, let $\map P n$ be the proposition:
:$L_n = F_{n - 1} + F_{n + 1}$ | Let $L_k$ be the $k$th [[Definition:Lucas Number/Definition 1|Lucas number]], defined as:
:$L_n = \begin{cases}
2 & : n = 0 \\
1 & : n = 1 \\
L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$
Then:
:$L_n = F_{n - 1} + F_{n + 1}$ | Proof by [[Second Principle of Mathematical Induction|induction]]:
For all $n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$L_n = F_{n - 1} + F_{n + 1}$ | Lucas Number as Sum of Fibonacci Numbers | https://proofwiki.org/wiki/Lucas_Number_as_Sum_of_Fibonacci_Numbers | https://proofwiki.org/wiki/Lucas_Number_as_Sum_of_Fibonacci_Numbers | [
"Lucas Numbers",
"Fibonacci Numbers"
] | [
"Definition:Lucas Number/Definition 1"
] | [
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction",
"Second Principle of Mathematical Induction"
] |
proofwiki-2216 | Sum of Sequence of Product of Lucas Numbers with Powers of 2 | Let $L_k$ be the $k$th Lucas number.
Let $F_k$ be the $k$th Fibonacci number.
Then:
:$\ds \forall n \in \N_{>0}: \sum_{j \mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$
That is:
:$2^0 L_1 + 2^1 L_2 + 2^2 L_3 + \cdots + 2^{n - 1} L^n = 2^n F_{n + 1} - 1$ | Proof by induction:
For all $\forall n \in \N_{>0}$, let $\map P n$ be the proposition:
:$\ds \sum_{j \mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$ | Let $L_k$ be the $k$th [[Definition:Lucas Number|Lucas number]].
Let $F_k$ be the $k$th [[Definition:Fibonacci Number|Fibonacci number]].
Then:
:$\ds \forall n \in \N_{>0}: \sum_{j \mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$
That is:
:$2^0 L_1 + 2^1 L_2 + 2^2 L_3 + \cdots + 2^{n - 1} L^n = 2^n F_{n + 1} - 1$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $\forall n \in \N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{j \mathop = 1}^n 2^{j - 1} L_j = 2^n F_{n + 1} - 1$ | Sum of Sequence of Product of Lucas Numbers with Powers of 2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Product_of_Lucas_Numbers_with_Powers_of_2 | https://proofwiki.org/wiki/Sum_of_Sequence_of_Product_of_Lucas_Numbers_with_Powers_of_2 | [
"Lucas Numbers",
"Fibonacci Numbers"
] | [
"Definition:Lucas Number",
"Definition:Fibonacci Number"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-2217 | Sum of Odd Positive Powers | Let $n \in \N$ be an odd positive integer.
Let $x, y \in \Z_{>0}$ be (strictly) positive integers.
Then $x + y$ is a divisor of $x^n + y^n$. | Given that $n \in \N$ be odd, it can be expressed in the form:
:$n = 2 m + 1$
where $m \in \N$.
The proof proceeds by strong induction.
For all $m \in \N$, let $\map P m$ be the proposition:
: $x^{2 m + 1} + y^{2 m + 1} = \paren {x + y} \paren {x^{2 m} + \cdots + y^{2 m} }$
$\map P 0$ is the case:
: $x + y = x + y$
whi... | Let $n \in \N$ be an [[Definition:Odd Integer|odd positive integer]].
Let $x, y \in \Z_{>0}$ be [[Definition:Strictly Positive Integer|(strictly) positive integers]].
Then $x + y$ is a [[Definition:Divisor of Integer|divisor]] of $x^n + y^n$. | Given that $n \in \N$ be [[Definition:Odd Integer|odd]], it can be expressed in the form:
:$n = 2 m + 1$
where $m \in \N$.
The proof proceeds by [[Second Principle of Mathematical Induction|strong induction]].
For all $m \in \N$, let $\map P m$ be the [[Definition:Proposition|proposition]]:
: $x^{2 m + 1} + y^{2 m +... | Sum of Odd Positive Powers | https://proofwiki.org/wiki/Sum_of_Odd_Positive_Powers | https://proofwiki.org/wiki/Sum_of_Odd_Positive_Powers | [
"Number Theory"
] | [
"Definition:Odd Integer",
"Definition:Strictly Positive/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Odd Integer",
"Second Principle of Mathematical Induction",
"Definition:Proposition",
"Second Principle of Mathematical Induction"
] |
proofwiki-2218 | Infinite Sequence Property of Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Then $\RR$ is a well-founded relation {{iff}} there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: \paren {a_{n + 1} \mathrel \RR a_n} \text { and } \paren {a_{n + 1} \ne a_n}$ | === Reverse Implication ===
{{:Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 1}}{{qed|lemma}} | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Then $\RR$ is a [[Definition:Well-Founded Relation|well-founded relation]] {{iff}} there exists no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such that:
:$\forall n... | === [[Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 1|Reverse Implication]] ===
{{:Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 1}}{{qed|lemma}} | Infinite Sequence Property of Well-Founded Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Well-Founded_Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Well-Founded_Relation | [
"Well-Founded Relations",
"Infinite Sequence Property of Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 1"
] |
proofwiki-2219 | Infinite Sequence Property of Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Then $\RR$ is a well-founded relation {{iff}} there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: \paren {a_{n + 1} \mathrel \RR a_n} \text { and } \paren {a_{n + 1} \ne a_n}$ | Suppose $\RR$ is not a well-founded relation.
So by definition there exists a non-empty subset $T$ of $S$ which has no minimal element.
Let $a \in T$.
Since $a$ is not minimal in $T$, we can find $b \in T: \paren {b \mathrel \RR a} \text { and } \paren {b \ne a}$.
This holds for all $a \in T$.
Hence the restriction $\R... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Then $\RR$ is a [[Definition:Well-Founded Relation|well-founded relation]] {{iff}} there exists no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such that:
:$\forall n... | Suppose $\RR$ is not a [[Definition:Well-Founded Relation|well-founded relation]].
So by definition there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] $T$ of $S$ which has no [[Definition:Minimal Element|minimal element]].
Let $a \in T$.
Since $a$ is not [[Definition:Minimal Element|m... | Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Well-Founded_Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Well-Founded_Relation/Reverse_Implication/Proof_1 | [
"Well-Founded Relations",
"Infinite Sequence Property of Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Definition:Well-Founded Relation",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Restriction/Relation",
"Definition:Right-Total Relation",
"Definition:Endorelation",
"Axiom:Axiom of Dependent Choice/Right-Total",
"Definiti... |
proofwiki-2220 | Infinite Sequence Property of Well-Founded Relation | Let $\struct {S, \RR}$ be a relational structure.
Then $\RR$ is a well-founded relation {{iff}} there exists no infinite sequence $\sequence {a_n}$ of elements of $S$ such that:
:$\forall n \in \N: \paren {a_{n + 1} \mathrel \RR a_n} \text { and } \paren {a_{n + 1} \ne a_n}$ | Suppose $\RR$ is not a well-founded relation.
Hence there exists $T \subseteq S$ such that $T$ has no minimal element under $\RR$.
Let $a_0 \in T$.
We have that $a_0$ is not minimal in $T$.
So:
:$\exists a_1 \in T: \paren {a_1 \mathrel \RR a_0} \text { and } a_1 \ne a_0$
Similarly, $a_1$ is not minimal in $T$.
So:
:$\e... | Let $\struct {S, \RR}$ be a [[Definition:Relational Structure|relational structure]].
Then $\RR$ is a [[Definition:Well-Founded Relation|well-founded relation]] {{iff}} there exists no [[Definition:Infinite Sequence|infinite sequence]] $\sequence {a_n}$ of [[Definition:Element|elements]] of $S$ such that:
:$\forall n... | Suppose $\RR$ is not a [[Definition:Well-Founded Relation|well-founded relation]].
Hence there exists $T \subseteq S$ such that $T$ has no [[Definition:Minimal Element|minimal element]] under $\RR$.
Let $a_0 \in T$.
We have that $a_0$ is not [[Definition:Minimal Element|minimal]] in $T$.
So:
:$\exists a_1 \in T: \... | Infinite Sequence Property of Well-Founded Relation/Reverse Implication/Proof 2 | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Well-Founded_Relation | https://proofwiki.org/wiki/Infinite_Sequence_Property_of_Well-Founded_Relation/Reverse_Implication/Proof_2 | [
"Well-Founded Relations",
"Infinite Sequence Property of Well-Founded Relation"
] | [
"Definition:Relational Structure",
"Definition:Well-Founded Relation",
"Definition:Sequence/Infinite Sequence",
"Definition:Element"
] | [
"Definition:Well-Founded Relation",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Axiom:Axiom of Dependent Choice/Right-Total",
"Definition:Minimal/Element",
"Definition:Right-Total Relation",
"Axiom:Axiom of Dependent Choice/Right-Total",
"Definition:Seq... |
proofwiki-2221 | Subset Relation on Power Set is Partial Ordering | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\struct {\powerset S, \subseteq}$ be the relational structure defined on $\powerset S$ by the subset relation $\subseteq$.
Then $\struct {\powerset S, \subseteq}$ is an ordered set.
The ordering $\subseteq$ is partial {{iff}} $S$ is neither empty nor a s... | From Subset Relation is Ordering, we have that $\subseteq$ is an ordering on any set of subsets of a given set.
Suppose $S$ is neither a singleton nor the empty set.
Then $\exists a, b \in S$ such that $a \ne b$.
Then $\set a \in \powerset S$ and $\set b \in \powerset S$.
However, $\set a \nsubseteq \set b$ and $\set b... | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Let $\struct {\powerset S, \subseteq}$ be the [[Definition:Relational Structure|relational structure]] defined on $\powerset S$ by the [[Definition:Subset Relation|subset relation]] $\subseteq$.
Then $\struct {\... | From [[Subset Relation is Ordering]], we have that $\subseteq$ is an [[Definition:Ordering|ordering]] on any [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of a given [[Definition:Set|set]].
Suppose $S$ is neither a [[Definition:Singleton|singleton]] nor the [[Definition:Empty Set|empty set]].
Then ... | Subset Relation on Power Set is Partial Ordering | https://proofwiki.org/wiki/Subset_Relation_on_Power_Set_is_Partial_Ordering | https://proofwiki.org/wiki/Subset_Relation_on_Power_Set_is_Partial_Ordering | [
"Partial Orderings",
"Power Set",
"Subset Relation"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Relational Structure",
"Definition:Subset Relation",
"Definition:Ordered Set",
"Definition:Partial Ordering",
"Definition:Empty Set",
"Definition:Singleton",
"Definition:Total Ordering"
] | [
"Subset Relation is Ordering",
"Definition:Ordering",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Set",
"Definition:Singleton",
"Definition:Empty Set",
"Definition:Partial Ordering",
"Empty Set is Subset of All Sets",
"Definition:Total Ordering",
"Definition:Singleton",
"Empty S... |
proofwiki-2222 | Ordering is Equivalent to Subset Relation | Let $\struct {S, \preceq}$ be an ordered set.
Then there exists a set $\mathbb S$ of subsets of $S$ such that:
:$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$
where:
:$\struct {\mathbb S, \subseteq}$ is the relational structure consisting of $\mathbb S$ and the subset relation
:$\cong$ denotes order isomor... | From Subset Relation is Ordering, we have that $\struct {\mathbb S, \subseteq}$ is an ordered set.
Then let $T$ be defined as:
:$T := \set {a^\prec: a \in S}$
Let the mapping $\phi: S \to T$ be defined as:
:$\map \phi a = a^\prec$
We are to show that $\phi$ is an order isomorphism.
$\phi$ is clearly surjective, as ever... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then there exists a [[Definition:Set|set]] $\mathbb S$ of [[Definition:Subset|subsets]] of $S$ such that:
:$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$
where:
:$\struct {\mathbb S, \subseteq}$ is the [[Definition:Relational Structu... | From [[Subset Relation is Ordering]], we have that $\struct {\mathbb S, \subseteq}$ is an [[Definition:Ordered Set|ordered set]].
Then let $T$ be defined as:
:$T := \set {a^\prec: a \in S}$
Let the [[Definition:Mapping|mapping]] $\phi: S \to T$ be defined as:
:$\map \phi a = a^\prec$
We are to show that $\phi$ is ... | Ordering is Equivalent to Subset Relation/Proof 1 | https://proofwiki.org/wiki/Ordering_is_Equivalent_to_Subset_Relation | https://proofwiki.org/wiki/Ordering_is_Equivalent_to_Subset_Relation/Proof_1 | [
"Orderings",
"Subset Relation",
"Representation Theorems",
"Ordering is Equivalent to Subset Relation"
] | [
"Definition:Ordered Set",
"Definition:Set",
"Definition:Subset",
"Definition:Relational Structure",
"Definition:Subset Relation",
"Definition:Order Isomorphism",
"Definition:Ordering",
"Definition:Set",
"Definition:Unique",
"Definition:Set",
"Definition:Subset",
"Definition:Set",
"Definition... | [
"Subset Relation is Ordering",
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Order Isomorphism",
"Definition:Surjection",
"Definition:Ordering",
"Definition:Antisymmetric Relation",
"Definition:Injection",
"Definition:Bijection",
"Definition:Ordering",
"Definition:Transitive Relati... |
proofwiki-2223 | Ordering is Equivalent to Subset Relation | Let $\struct {S, \preceq}$ be an ordered set.
Then there exists a set $\mathbb S$ of subsets of $S$ such that:
:$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$
where:
:$\struct {\mathbb S, \subseteq}$ is the relational structure consisting of $\mathbb S$ and the subset relation
:$\cong$ denotes order isomor... | First a lemma:
=== Lemma ===
{{:Ordering is Equivalent to Subset Relation/Lemma}}{{qed|lemma}}
From Subset Relation is Ordering, we have that $\struct {\mathbb S, \subseteq}$ is an ordered set.
We are to show that $\phi$ is an order isomorphism.
$\phi$ is clearly surjective, as every $a^\preceq$ is defined from some $a... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Then there exists a [[Definition:Set|set]] $\mathbb S$ of [[Definition:Subset|subsets]] of $S$ such that:
:$\struct {S, \preceq} \cong \struct {\mathbb S, \subseteq}$
where:
:$\struct {\mathbb S, \subseteq}$ is the [[Definition:Relational Structu... | First a [[Definition:Lemma|lemma]]:
=== [[Ordering is Equivalent to Subset Relation/Lemma|Lemma]] ===
{{:Ordering is Equivalent to Subset Relation/Lemma}}{{qed|lemma}}
From [[Subset Relation is Ordering]], we have that $\struct {\mathbb S, \subseteq}$ is an [[Definition:Ordered Set|ordered set]].
We are to show tha... | Ordering is Equivalent to Subset Relation/Proof 2 | https://proofwiki.org/wiki/Ordering_is_Equivalent_to_Subset_Relation | https://proofwiki.org/wiki/Ordering_is_Equivalent_to_Subset_Relation/Proof_2 | [
"Orderings",
"Subset Relation",
"Representation Theorems",
"Ordering is Equivalent to Subset Relation"
] | [
"Definition:Ordered Set",
"Definition:Set",
"Definition:Subset",
"Definition:Relational Structure",
"Definition:Subset Relation",
"Definition:Order Isomorphism",
"Definition:Ordering",
"Definition:Set",
"Definition:Unique",
"Definition:Set",
"Definition:Subset",
"Definition:Set",
"Definition... | [
"Definition:Lemma",
"Ordering is Equivalent to Subset Relation/Lemma",
"Subset Relation is Ordering",
"Definition:Ordered Set",
"Definition:Order Isomorphism/Definition 2",
"Definition:Surjection",
"Ordering is Equivalent to Subset Relation/Lemma",
"Definition:Increasing/Mapping",
"Definition:Subset... |
proofwiki-2224 | Subset Relation is Ordering | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Let $\mathbb S \subseteq \powerset S$ be any subset of $\powerset S$, that is, an arbitrary set of subsets of $S$.
Then $\subseteq$ is an ordering on $\mathbb S$. | To establish that $\subseteq$ is an ordering, we need to show that it is reflexive, antisymmetric and transitive.
So, checking in turn each of the criteria for an ordering: | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
Let $\mathbb S \subseteq \powerset S$ be any [[Definition:Subset|subset]] of $\powerset S$, that is, an arbitrary [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$.
Then $\subseteq$ is an [[Definiti... | To establish that $\subseteq$ is an [[Definition:Ordering|ordering]], we need to show that it is [[Definition:Reflexive Relation|reflexive]], [[Definition:Antisymmetric Relation|antisymmetric]] and [[Definition:Transitive Relation|transitive]].
So, checking in turn each of the criteria for an [[Definition:Ordering|ord... | Subset Relation is Ordering | https://proofwiki.org/wiki/Subset_Relation_is_Ordering | https://proofwiki.org/wiki/Subset_Relation_is_Ordering | [
"Subset Relation",
"Orderings",
"Subset Relation is Ordering"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Subset",
"Definition:Set",
"Definition:Subset",
"Definition:Ordering"
] | [
"Definition:Ordering",
"Definition:Reflexive Relation",
"Definition:Antisymmetric Relation",
"Definition:Transitive Relation",
"Definition:Ordering",
"Definition:Reflexive Relation",
"Definition:Antisymmetric Relation",
"Definition:Transitive Relation",
"Definition:Ordering"
] |
proofwiki-2225 | Principle of Mathematical Induction/Well-Ordered Set | Let $\struct {S, \preceq}$ be a well-ordered set.
Let $T \subseteq S$ be a subset of $S$ such that:
:$\forall s \in S: \paren {\forall t \in S: t \prec s \implies t \in T} \implies s \in T$
Then $T = S$. | {{AimForCont}} that $T \ne S$.
From Set Difference is Subset, $S \setminus T \subset S$.
From Set Difference with Proper Subset, $S \setminus T \ne \O$.
By the definition of a well-ordered set, there exists a smallest element $s$ of $S \setminus T$.
As $s \in S$, it follows from the definition of $T$ that:
:$\forall t ... | Let $\struct {S, \preceq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
Let $T \subseteq S$ be a [[Definition:Subset|subset]] of $S$ such that:
:$\forall s \in S: \paren {\forall t \in S: t \prec s \implies t \in T} \implies s \in T$
Then $T = S$. | {{AimForCont}} that $T \ne S$.
From [[Set Difference is Subset]], $S \setminus T \subset S$.
From [[Set Difference with Proper Subset]], $S \setminus T \ne \O$.
By the definition of a [[Definition:Well-Ordered Set|well-ordered set]], [[Definition:Existential Quantifier|there exists]] a [[Definition:Smallest Element... | Principle of Mathematical Induction/Well-Ordered Set | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Well-Ordered_Set | https://proofwiki.org/wiki/Principle_of_Mathematical_Induction/Well-Ordered_Set | [
"Well-Orderings",
"Principle of Mathematical Induction"
] | [
"Definition:Well-Ordered Set",
"Definition:Subset"
] | [
"Set Difference is Subset",
"Set Difference with Proper Subset",
"Definition:Well-Ordered Set",
"Definition:Existential Quantifier",
"Definition:Smallest Element",
"Definition:By Hypothesis",
"Proof by Contradiction"
] |
proofwiki-2226 | Order Isomorphism from Woset onto Subset | Let $\struct {S, \preceq}$ be a woset.
Let $T \subseteq S$.
Let $f: S \to T$ be an order isomorphism.
Then $\forall x \in S: x \preceq \map f x$. | Let $T = \set {x \in S: \map f x \prec x}$.
We are to show that $T = \O$.
{{AimForCont}} that $T \ne \O$.
Then as $\struct {S, \preceq}$ is a woset, by definition $T$ has a minimal element: call it $x_0$.
Since $x_0 \in T$, we have $\map f {x_0} \prec x_0$.
So, let $x_1 = \map f {x_0}$.
$f$ is an order isomorphism, so ... | Let $\struct {S, \preceq}$ be a [[Definition:Woset|woset]].
Let $T \subseteq S$.
Let $f: S \to T$ be an [[Definition:Order Isomorphism|order isomorphism]].
Then $\forall x \in S: x \preceq \map f x$. | Let $T = \set {x \in S: \map f x \prec x}$.
We are to show that $T = \O$.
{{AimForCont}} that $T \ne \O$.
Then as $\struct {S, \preceq}$ is a [[Definition:Woset|woset]], by definition $T$ has a [[Definition:Minimal Element|minimal element]]: call it $x_0$.
Since $x_0 \in T$, we have $\map f {x_0} \prec x_0$.
So, ... | Order Isomorphism from Woset onto Subset | https://proofwiki.org/wiki/Order_Isomorphism_from_Woset_onto_Subset | https://proofwiki.org/wiki/Order_Isomorphism_from_Woset_onto_Subset | [
"Well-Orderings",
"Order Isomorphisms"
] | [
"Definition:Well-Ordered Set",
"Definition:Order Isomorphism"
] | [
"Definition:Well-Ordered Set",
"Definition:Minimal/Element",
"Definition:Order Isomorphism",
"Definition:Minimal/Element",
"Definition:Contradiction"
] |
proofwiki-2227 | Order Isomorphism between Wosets is Unique | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be wosets.
Let $\struct {S_1, \preceq_1} \cong \struct {S_2, \preceq_2}$, that is, let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be order isomorphic.
Then there is exactly one mapping $f: S_1 \to S_2$ such that $f$ is an order isomorphism. | Let $f: S_1 \to S_2$ and $g: S_1 \to S_2$ both be order isomorphisms.
By Inverse of Order Isomorphism is Order Isomorphism, the inverse $f^{-1}$ is also an order isomorphism.
Let $h = f^{-1} \circ g$ be the composition of $f^{-1}$ and $g$, which, by Composite of Order Isomorphisms is Order Isomorphism, is itself an ord... | Let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be [[Definition:Woset|wosets]].
Let $\struct {S_1, \preceq_1} \cong \struct {S_2, \preceq_2}$, that is, let $\struct {S_1, \preceq_1}$ and $\struct {S_2, \preceq_2}$ be [[Definition:Order Isomorphism|order isomorphic]].
Then there is exactly one [[Definit... | Let $f: S_1 \to S_2$ and $g: S_1 \to S_2$ both be [[Definition:Order Isomorphism|order isomorphisms]].
By [[Inverse of Order Isomorphism is Order Isomorphism]], the [[Definition:Inverse Mapping|inverse]] $f^{-1}$ is also an [[Definition:Order Isomorphism|order isomorphism]].
Let $h = f^{-1} \circ g$ be the [[Definiti... | Order Isomorphism between Wosets is Unique | https://proofwiki.org/wiki/Order_Isomorphism_between_Wosets_is_Unique | https://proofwiki.org/wiki/Order_Isomorphism_between_Wosets_is_Unique | [
"Order Isomorphisms",
"Well-Orderings"
] | [
"Definition:Well-Ordered Set",
"Definition:Order Isomorphism",
"Definition:Mapping",
"Definition:Order Isomorphism"
] | [
"Definition:Order Isomorphism",
"Inverse of Order Isomorphism is Order Isomorphism",
"Definition:Inverse Mapping",
"Definition:Order Isomorphism",
"Definition:Composition of Mappings",
"Composite of Order Isomorphisms is Order Isomorphism",
"Definition:Order Isomorphism",
"Order Isomorphism from Woset... |
proofwiki-2228 | Woset is Isomorphic to Set of its Initial Segments | Let $\struct {S, \preceq}$ be a well-ordered set.
Let:
:$A = \set {a^\prec: a \in S}$
where $a^\prec$ is the strict lower closure of $S$ determined by $a$.
Then:
:$\struct {S, \preceq} \cong \struct {A, \subseteq}$
where $\cong$ denotes order isomorphism. | Define $f: S \to A$ as:
:$\forall a \in S: \map f a = a^\prec$
where $a^\prec$ is the initial segment determined by $a$. | Let $\struct {S, \preceq}$ be a [[Definition:Well-Ordered Set|well-ordered set]].
Let:
:$A = \set {a^\prec: a \in S}$
where $a^\prec$ is the [[Definition:Strict Lower Closure of Element|strict lower closure]] of $S$ determined by $a$.
Then:
:$\struct {S, \preceq} \cong \struct {A, \subseteq}$
where $\cong$ denotes [... | Define $f: S \to A$ as:
:$\forall a \in S: \map f a = a^\prec$
where $a^\prec$ is the [[Definition:Initial Segment|initial segment]] determined by $a$. | Woset is Isomorphic to Set of its Initial Segments | https://proofwiki.org/wiki/Woset_is_Isomorphic_to_Set_of_its_Initial_Segments | https://proofwiki.org/wiki/Woset_is_Isomorphic_to_Set_of_its_Initial_Segments | [
"Well-Orderings",
"Order Isomorphisms"
] | [
"Definition:Well-Ordered Set",
"Definition:Strict Lower Closure/Element",
"Definition:Order Isomorphism"
] | [
"Definition:Initial Segment",
"Definition:Initial Segment",
"Definition:Initial Segment"
] |
proofwiki-2229 | Minimal Element of an Ordinal | The minimal element of any nonempty ordinal is the empty set.
That is, if $S$ is a nonempty ordinal, $\bigcap S = \O$ | Let $S$ be an ordinal.
Let the minimal element of $S$ be $s_0$.
This exists by dint of an ordinal being a woset.
From Ordering on Ordinal is Subset Relation, $S$ is well-ordered by $\subseteq$.
So, by definition of an ordinal:
:$s_0 = \set {s \in S: s \subset s_0}$
But as $s_0$ is minimal, there ''are'' no elements of ... | The [[Definition:Minimal Element|minimal element]] of any [[Definition:Non-Empty Set|nonempty]] [[Definition:Ordinal|ordinal]] is the [[Definition:Empty Set|empty set]].
That is, if $S$ is a [[Definition:Non-Empty Set|nonempty]] [[Definition:Ordinal|ordinal]], $\bigcap S = \O$ | Let $S$ be an [[Definition:Ordinal|ordinal]].
Let the [[Definition:Minimal Element|minimal element]] of $S$ be $s_0$.
This exists by dint of an [[Definition:Ordinal|ordinal]] being a [[Definition:Woset|woset]].
From [[Ordering on Ordinal is Subset Relation]], $S$ is [[Definition:Well-Ordering|well-ordered]] by $\sub... | Minimal Element of an Ordinal | https://proofwiki.org/wiki/Minimal_Element_of_an_Ordinal | https://proofwiki.org/wiki/Minimal_Element_of_an_Ordinal | [
"Ordinals"
] | [
"Definition:Minimal/Element",
"Definition:Non-Empty Set",
"Definition:Ordinal",
"Definition:Empty Set",
"Definition:Non-Empty Set",
"Definition:Ordinal"
] | [
"Definition:Ordinal",
"Definition:Minimal/Element",
"Definition:Ordinal",
"Definition:Well-Ordered Set",
"Ordering on Ordinal is Subset Relation",
"Definition:Well-Ordering",
"Definition:Ordinal",
"Definition:Minimal/Element",
"Definition:Element",
"Definition:Subset"
] |
proofwiki-2230 | Ordering on Ordinal is Subset Relation | Let $\struct {S, \prec}$ be an ordinal.
Then $\forall x, y \in S:$
:$x \in y \iff x \prec y \iff S_x \subsetneqq S_y \iff x \subsetneqq y$
where $S_x$ and $S_y$ are the initial segments of $S$ determined by $x$ and $y$ respectively.
Thus there is no need to specify what the ordering on an ordinal is -- it is always the... | The first equivalence is an immediate consequence of Equivalence of Definitions of Ordinal.
The second equivalence holds for any well-ordered set by Woset Isomorphic to Set of its Sections.
The third equivalence holds by definition of an ordinal.
It follows from Ordering is Equivalent to Subset Relation and Order Isomo... | Let $\struct {S, \prec}$ be an [[Definition:Ordinal|ordinal]].
Then $\forall x, y \in S:$
:$x \in y \iff x \prec y \iff S_x \subsetneqq S_y \iff x \subsetneqq y$
where $S_x$ and $S_y$ are the [[Definition:Initial Segment|initial segments]] of $S$ determined by $x$ and $y$ respectively.
Thus there is no need to spec... | The first equivalence is an immediate consequence of [[Equivalence of Definitions of Ordinal]].
The second equivalence holds for any [[Definition:Well-Ordered Set|well-ordered set]] by [[Woset Isomorphic to Set of its Sections]].
The third equivalence holds by definition of an [[Definition:Ordinal|ordinal]].
It fol... | Ordering on Ordinal is Subset Relation | https://proofwiki.org/wiki/Ordering_on_Ordinal_is_Subset_Relation | https://proofwiki.org/wiki/Ordering_on_Ordinal_is_Subset_Relation | [
"Ordinals",
"Orderings",
"Subset Relation"
] | [
"Definition:Ordinal",
"Definition:Initial Segment",
"Definition:Ordering",
"Definition:Ordinal",
"Definition:Subset Relation"
] | [
"Equivalence of Definitions of Ordinal",
"Definition:Well-Ordered Set",
"Woset is Isomorphic to Set of its Initial Segments",
"Definition:Ordinal",
"Ordering is Equivalent to Subset Relation",
"Order Isomorphism between Wosets is Unique",
"Definition:Ordering"
] |
proofwiki-2231 | Initial Segment of Ordinal is Ordinal | Let $S$ be an ordinal.
Let $a \in S$.
Then the initial segment $S_a = a$ of $S$ determined by $a$ is also an ordinal.
In other words, every element of a (non-empty) ordinal is also an ordinal. | By Subset of Well-Ordered Set is Well-Ordered, $S_a$ is well-ordered.
Suppose that $b \in S_a$.
From Ordering on Ordinal is Subset Relation, and the definition of an initial segment, it follows that $b \subset a$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {S_a}_b
| r = \set {x \in S_a: x \subset b}
| c = {{Def... | Let $S$ be an [[Definition:Ordinal|ordinal]].
Let $a \in S$.
Then the [[Definition:Initial Segment|initial segment]] $S_a = a$ of $S$ determined by $a$ is also an [[Definition:Ordinal|ordinal]].
In other words, every [[Definition:Element|element]] of a ([[Definition:Non-Empty Set|non-empty]]) [[Definition:Ordinal|... | By [[Subset of Well-Ordered Set is Well-Ordered]], $S_a$ is [[Definition:Well-Ordered Set|well-ordered]].
Suppose that $b \in S_a$.
From [[Ordering on Ordinal is Subset Relation]], and the definition of an [[Definition:Initial Segment|initial segment]], it follows that $b \subset a$.
Then:
{{begin-eqn}}
{{eqn | l... | Initial Segment of Ordinal is Ordinal | https://proofwiki.org/wiki/Initial_Segment_of_Ordinal_is_Ordinal | https://proofwiki.org/wiki/Initial_Segment_of_Ordinal_is_Ordinal | [
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Initial Segment",
"Definition:Ordinal",
"Definition:Element",
"Definition:Non-Empty Set",
"Definition:Ordinal",
"Definition:Ordinal"
] | [
"Subset of Well-Ordered Set is Well-Ordered",
"Definition:Well-Ordered Set",
"Ordering on Ordinal is Subset Relation",
"Definition:Initial Segment",
"Definition:Ordinal",
"Definition:Ordinal"
] |
proofwiki-2232 | Ordinal Subset of Ordinal is Initial Segment | Let $S$ be an ordinal.
Let $T \subset S$ also be an ordinal.
Then $\exists a \in S: T = S_a$, where $S_a$ is the initial segment of $S$ determined by $a$.
That is, $T = S_a = a \in S$. | Recall that the Ordering on Ordinal is Subset Relation.
Let $a$ be the minimal element of $S \setminus T$.
By definition of minimal element:
:$\forall x \in S: x \subset a$
Hence $x \notin S \setminus T$.
By definition of set difference:
:$x \in T$
Thus by definition of initial segment:
:$S_a \subseteq T$
Now let $b \i... | Let $S$ be an [[Definition:Ordinal|ordinal]].
Let $T \subset S$ also be an [[Definition:Ordinal|ordinal]].
Then $\exists a \in S: T = S_a$, where $S_a$ is the [[Definition:Initial Segment|initial segment]] of $S$ determined by $a$.
That is, $T = S_a = a \in S$. | Recall that the [[Ordering on Ordinal is Subset Relation]].
Let $a$ be the [[Definition:Minimal Element|minimal element]] of $S \setminus T$.
By definition of [[Definition:Minimal Element|minimal element]]:
:$\forall x \in S: x \subset a$
Hence $x \notin S \setminus T$.
By definition of [[Definition:Set Difference... | Ordinal Subset of Ordinal is Initial Segment | https://proofwiki.org/wiki/Ordinal_Subset_of_Ordinal_is_Initial_Segment | https://proofwiki.org/wiki/Ordinal_Subset_of_Ordinal_is_Initial_Segment | [
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Ordinal",
"Definition:Initial Segment"
] | [
"Ordering on Ordinal is Subset Relation",
"Definition:Minimal/Element",
"Definition:Minimal/Element",
"Definition:Set Difference",
"Definition:Initial Segment",
"Definition:Ordinal",
"Definition:Set Difference",
"Ordering on Ordinal is Subset Relation",
"Definition:Well-Ordering",
"Definition:Tota... |
proofwiki-2233 | Intersection of Two Ordinals is Ordinal | Let $S$ and $T$ be ordinals.
Then $S \cap T$ is an ordinal. | Because $S$ and $T$ are ordinals, {{afortiori}} they are (strictly) well-ordered by the subset relation.
Let $a \in S \cap T$.
Then the initial segments $S_a$ and $T_a$ are such that:
:$S_a = a = T_a$
That is:
:$\set {x \in S: x \subset a} = a = \set {y \in T: y \subset a}$
So:
:$a = \set {z \in S \cap T: z \subset a} ... | Let $S$ and $T$ be [[Definition:Ordinal|ordinals]].
Then $S \cap T$ is an [[Definition:Ordinal|ordinal]]. | Because $S$ and $T$ are [[Definition:Ordinal|ordinals]], {{afortiori}} they are [[Definition:Strictly Well-Ordered Set|(strictly) well-ordered]] by the [[Definition:Subset Relation|subset relation]].
Let $a \in S \cap T$.
Then the [[Definition:Initial Segment|initial segments]] $S_a$ and $T_a$ are such that:
:$S_a = ... | Intersection of Two Ordinals is Ordinal | https://proofwiki.org/wiki/Intersection_of_Two_Ordinals_is_Ordinal | https://proofwiki.org/wiki/Intersection_of_Two_Ordinals_is_Ordinal | [
"Ordinals",
"Set Intersection"
] | [
"Definition:Ordinal",
"Definition:Ordinal"
] | [
"Definition:Ordinal",
"Definition:Strictly Well-Ordered Set",
"Definition:Subset Relation",
"Definition:Initial Segment",
"Definition:Initial Segment",
"Initial Segment of Ordinal is Ordinal"
] |
proofwiki-2234 | Relation between Two Ordinals | Let $S$ and $T$ be ordinals.
Then either $S \subseteq T$ or $T \subseteq S$. | {{improve|Worth removing the technical complexity to make it more accessible<br/>A better approach may be to write a second proof rather than change this one.}}
{{AimForCont}} the claim is false.
That is, by De Morgan's laws: Conjunction of Negations:
:$\paren {\neg \paren {S \subseteq T} } \land \paren {\neg \paren {T... | Let $S$ and $T$ be [[Definition:Ordinal|ordinals]].
Then either $S \subseteq T$ [[Definition:Disjunction|or]] $T \subseteq S$. | {{improve|Worth removing the technical complexity to make it more accessible<br/>A better approach may be to write a second proof rather than change this one.}}
{{AimForCont}} the claim is [[Definition:False|false]].
That is, by [[De Morgan's Laws (Logic)/Conjunction of Negations|De Morgan's laws: Conjunction of Nega... | Relation between Two Ordinals | https://proofwiki.org/wiki/Relation_between_Two_Ordinals | https://proofwiki.org/wiki/Relation_between_Two_Ordinals | [
"Relation between Two Ordinals",
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Disjunction"
] | [
"Definition:False",
"De Morgan's Laws (Logic)/Conjunction of Negations",
"Definition:Logical Not",
"Definition:Conjunction",
"Intersection is Subset",
"Intersection with Subset is Subset",
"Intersection of Two Ordinals is Ordinal",
"Definition:Ordinal",
"Transitive Set is Proper Subset of Ordinal if... |
proofwiki-2235 | Relation between Two Ordinals | Let $S$ and $T$ be ordinals.
Then either $S \subseteq T$ or $T \subseteq S$. | By Ordinal Membership is Trichotomy, either $S \in T$ or $T \in S$.
By definition, every element of an ordinal is an initial segment.
Hence the result.
{{qed}} | Let $S$ and $T$ be [[Definition:Ordinal|ordinals]].
Then either $S \subseteq T$ [[Definition:Disjunction|or]] $T \subseteq S$. | By [[Ordinal Membership is Trichotomy]], either $S \in T$ or $T \in S$.
By definition, every [[Definition:Element|element]] of an [[Definition:Ordinal|ordinal]] is an [[Definition:Initial Segment|initial segment]].
Hence the result.
{{qed}} | Relation between Two Ordinals/Corollary/Proof 1 | https://proofwiki.org/wiki/Relation_between_Two_Ordinals | https://proofwiki.org/wiki/Relation_between_Two_Ordinals/Corollary/Proof_1 | [
"Relation between Two Ordinals",
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Disjunction"
] | [
"Ordinal Membership is Trichotomy",
"Definition:Element",
"Definition:Ordinal",
"Definition:Initial Segment"
] |
proofwiki-2236 | Relation between Two Ordinals | Let $S$ and $T$ be ordinals.
Then either $S \subseteq T$ or $T \subseteq S$. | If either $S \subset T$ or $T \subset S$ then we invoke Ordinal Subset of Ordinal is Initial Segment, and the proof is complete.
{{AimForCont}} $S \not \subset T$ and $T \not \subset S$.
Now from Intersection is Subset, we have $S \cap T \subset T$ and $S \cap T \subset S$.
By Intersection of Two Ordinals is Ordinal, $... | Let $S$ and $T$ be [[Definition:Ordinal|ordinals]].
Then either $S \subseteq T$ [[Definition:Disjunction|or]] $T \subseteq S$. | If either $S \subset T$ or $T \subset S$ then we invoke [[Ordinal Subset of Ordinal is Initial Segment]], and the proof is complete.
{{AimForCont}} $S \not \subset T$ and $T \not \subset S$.
Now from [[Intersection is Subset]], we have $S \cap T \subset T$ and $S \cap T \subset S$.
By [[Intersection of Two Ordinals... | Relation between Two Ordinals/Corollary/Proof 2 | https://proofwiki.org/wiki/Relation_between_Two_Ordinals | https://proofwiki.org/wiki/Relation_between_Two_Ordinals/Corollary/Proof_2 | [
"Relation between Two Ordinals",
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Disjunction"
] | [
"Ordinal Subset of Ordinal is Initial Segment",
"Intersection is Subset",
"Intersection of Two Ordinals is Ordinal",
"Definition:Ordinal",
"Ordinal Subset of Ordinal is Initial Segment",
"Definition:Contradiction",
"Ordinal Subset of Ordinal is Initial Segment"
] |
proofwiki-2237 | Relation between Two Ordinals | Let $S$ and $T$ be ordinals.
Then either $S \subseteq T$ or $T \subseteq S$. | We have that $S \ne T$
Therefore, from Relation between Two Ordinals either $S \subset T$ or $T \subset S$.
By Ordering on Ordinal is Subset Relation or Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, either $S \in T$ or $T \in S$.
By definition, every element of an ordinal is an initial segment; hen... | Let $S$ and $T$ be [[Definition:Ordinal|ordinals]].
Then either $S \subseteq T$ [[Definition:Disjunction|or]] $T \subseteq S$. | We have that $S \ne T$
Therefore, from [[Relation between Two Ordinals]] either $S \subset T$ [[Definition:Disjunction|or]] $T \subset S$.
By [[Ordering on Ordinal is Subset Relation]] or [[Transitive Set is Proper Subset of Ordinal iff Element of Ordinal]], either $S \in T$ or $T \in S$.
By definition, every [[Defi... | Relation between Two Ordinals/Corollary/Proof 3 | https://proofwiki.org/wiki/Relation_between_Two_Ordinals | https://proofwiki.org/wiki/Relation_between_Two_Ordinals/Corollary/Proof_3 | [
"Relation between Two Ordinals",
"Ordinals"
] | [
"Definition:Ordinal",
"Definition:Disjunction"
] | [
"Relation between Two Ordinals",
"Definition:Disjunction",
"Ordering on Ordinal is Subset Relation",
"Transitive Set is Proper Subset of Ordinal iff Element of Ordinal",
"Definition:Element",
"Definition:Ordinal",
"Definition:Initial Segment"
] |
proofwiki-2238 | Isomorphic Ordinals are Equal | Let $A$ and $B$ be ordinals that are order isomorphic.
Then $A = B$. | Let $S \cong T$.
{{AimForCont}} that $S \ne T$.
Then from {{Corollary|Relation between Two Ordinals}}, either:
:$S$ is an initial segment of $T$
or:
:$T$ is an initial segment of $S$.
But as $S \cong T$, from Well-Ordered Class is not Isomorphic to Initial Segment, neither $S$ nor $T$ can be an initial segment of the o... | Let $A$ and $B$ be [[Definition:Ordinal|ordinals]] that are [[Definition:Order Isomorphism|order isomorphic]].
Then $A = B$. | Let $S \cong T$.
{{AimForCont}} that $S \ne T$.
Then from {{Corollary|Relation between Two Ordinals}}, either:
:$S$ is an [[Definition:Initial Segment|initial segment]] of $T$
or:
:$T$ is an [[Definition:Initial Segment|initial segment]] of $S$.
But as $S \cong T$, from [[Well-Ordered Class is not Isomorphic to Init... | Isomorphic Ordinals are Equal/Proof 1 | https://proofwiki.org/wiki/Isomorphic_Ordinals_are_Equal | https://proofwiki.org/wiki/Isomorphic_Ordinals_are_Equal/Proof_1 | [
"Ordinals",
"Order Isomorphisms",
"Isomorphic Ordinals are Equal"
] | [
"Definition:Ordinal",
"Definition:Order Isomorphism"
] | [
"Definition:Initial Segment",
"Definition:Initial Segment",
"Well-Ordered Class is not Isomorphic to Initial Segment",
"Definition:Initial Segment",
"Proof by Contradiction"
] |
proofwiki-2239 | Isomorphic Ordinals are Equal | Let $A$ and $B$ be ordinals that are order isomorphic.
Then $A = B$. | From Well-Ordered Class is not Isomorphic to Initial Segment, neither $A$ nor $B$ can be an initial segment of the other.
By definition, every element of an ordinal is an initial segment of it.
Hence, neither $A$ nor $B$ can be an element of the other.
By Ordinal Membership is Trichotomy, it follows that $A = B$.
{{qed... | Let $A$ and $B$ be [[Definition:Ordinal|ordinals]] that are [[Definition:Order Isomorphism|order isomorphic]].
Then $A = B$. | From [[Well-Ordered Class is not Isomorphic to Initial Segment]], neither $A$ nor $B$ can be an [[Definition:Initial Segment|initial segment]] of the other.
By definition, every [[Definition:Element|element]] of an [[Definition:Ordinal|ordinal]] is an [[Definition:Initial Segment|initial segment]] of it.
Hence, neith... | Isomorphic Ordinals are Equal/Proof 2 | https://proofwiki.org/wiki/Isomorphic_Ordinals_are_Equal | https://proofwiki.org/wiki/Isomorphic_Ordinals_are_Equal/Proof_2 | [
"Ordinals",
"Order Isomorphisms",
"Isomorphic Ordinals are Equal"
] | [
"Definition:Ordinal",
"Definition:Order Isomorphism"
] | [
"Well-Ordered Class is not Isomorphic to Initial Segment",
"Definition:Initial Segment",
"Definition:Element",
"Definition:Ordinal",
"Definition:Initial Segment",
"Definition:Element",
"Ordinal Membership is Trichotomy"
] |
proofwiki-2240 | Condition for Woset to be Isomorphic to Ordinal | Let $\struct {S, \preceq}$ be a woset.
Let $\struct {S, \preceq}$ be such that $\forall a \in S$, the initial segment $S_a$ of $S$ determined by $a$ is order isomorphic to some ordinal.
Then $\struct {S, \preceq}$ itself is order isomorphic to an ordinal. | For each $a \in S$, let $g_a: S_a \to \map Z a$ be an order isomorphism from $S_a$ to an ordinal $\map Z a$.
By Isomorphic Ordinals are Equal and Order Isomorphism between Wosets is Unique, both $\map Z a$ and $g_a$ are unique.
So this defines a mapping $Z$ on $S$.
Let the image of $Z$ be $W$:
:$W = \set {\map Z a: a \... | Let $\struct {S, \preceq}$ be a [[Definition:Woset|woset]].
Let $\struct {S, \preceq}$ be such that $\forall a \in S$, the [[Definition:Initial Segment|initial segment]] $S_a$ of $S$ determined by $a$ is [[Definition:Order Isomorphism|order isomorphic]] to some [[Definition:Ordinal|ordinal]].
Then $\struct {S, \prec... | For each $a \in S$, let $g_a: S_a \to \map Z a$ be an [[Definition:Order Isomorphism|order isomorphism]] from $S_a$ to an [[Definition:Ordinal|ordinal]] $\map Z a$.
By [[Isomorphic Ordinals are Equal]] and [[Order Isomorphism between Wosets is Unique]], both $\map Z a$ and $g_a$ are [[Definition:Unique|unique]].
So t... | Condition for Woset to be Isomorphic to Ordinal | https://proofwiki.org/wiki/Condition_for_Woset_to_be_Isomorphic_to_Ordinal | https://proofwiki.org/wiki/Condition_for_Woset_to_be_Isomorphic_to_Ordinal | [
"Ordinals"
] | [
"Definition:Well-Ordered Set",
"Definition:Initial Segment",
"Definition:Order Isomorphism",
"Definition:Ordinal",
"Definition:Order Isomorphism",
"Definition:Ordinal"
] | [
"Definition:Order Isomorphism",
"Definition:Ordinal",
"Isomorphic Ordinals are Equal",
"Order Isomorphism between Wosets is Unique",
"Definition:Unique",
"Definition:Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Initial Segment of Ordinal is Ordinal",
"Definition:Order Isomorphism",
... |
proofwiki-2241 | Counting Theorem | Every well-ordered set is order isomorphic to a unique ordinal. | === Existence ===
Let $\struct {S, \preceq}$ be a woset.
From Condition for Woset to be Isomorphic to Ordinal, it is enough to show that for every $a \in S$, the initial segment $S_a$ of $S$ determined by $a$ is order isomorphic to some ordinal.
Let:
:$E = \set {a \in S: S_a \text{ is not isomorphic to an ordinal} }$
W... | Every [[Definition:Well-Ordered Set|well-ordered set]] is [[Definition:Order Isomorphism|order isomorphic]] to a unique [[Definition:Ordinal|ordinal]]. | === Existence ===
Let $\struct {S, \preceq}$ be a [[Definition:Woset|woset]].
From [[Condition for Woset to be Isomorphic to Ordinal]], it is enough to show that for every $a \in S$, the [[Definition:Initial Segment|initial segment]] $S_a$ of $S$ determined by $a$ is [[Definition:Order Isomorphism|order isomorphic]] ... | Counting Theorem/Proof 1 | https://proofwiki.org/wiki/Counting_Theorem | https://proofwiki.org/wiki/Counting_Theorem/Proof_1 | [
"Counting Theorem",
"Ordinals",
"Well-Orderings",
"Named Theorems"
] | [
"Definition:Well-Ordered Set",
"Definition:Order Isomorphism",
"Definition:Ordinal"
] | [
"Definition:Well-Ordered Set",
"Condition for Woset to be Isomorphic to Ordinal",
"Definition:Initial Segment",
"Definition:Order Isomorphism",
"Definition:Ordinal",
"Definition:Minimal/Element",
"Definition:Well-Ordered Set",
"Definition:Order Isomorphism",
"Definition:Ordinal",
"Definition:Ordin... |
proofwiki-2242 | Counting Theorem | Every well-ordered set is order isomorphic to a unique ordinal. | Let $A$ be a properly well-ordered class.
Let $\On$ denote the class of all ordinals.
By the Comparability Theorem, either:
:$A$ is order isomorphic to a lower section of $\On$
or:
:$\On$ is order isomorphic to a lower section of $A$.
Let $A$ be a set.
From Well-Ordering on Set is Proper Well-Ordering, $A$ is a properl... | Every [[Definition:Well-Ordered Set|well-ordered set]] is [[Definition:Order Isomorphism|order isomorphic]] to a unique [[Definition:Ordinal|ordinal]]. | Let $A$ be a [[Definition:Proper Well-Ordering|properly well-ordered class]].
Let $\On$ denote the [[Definition:Class of All Ordinals|class of all ordinals]].
By the [[Comparability Theorem]], either:
:$A$ is [[Definition:Order Isomorphism on Well-Orderings (Class Theory)|order isomorphic]] to a [[Definition:Lower Se... | Counting Theorem/Proof 2 | https://proofwiki.org/wiki/Counting_Theorem | https://proofwiki.org/wiki/Counting_Theorem/Proof_2 | [
"Counting Theorem",
"Ordinals",
"Well-Orderings",
"Named Theorems"
] | [
"Definition:Well-Ordered Set",
"Definition:Order Isomorphism",
"Definition:Ordinal"
] | [
"Definition:Proper Well-Ordering",
"Definition:Class of All Ordinals",
"Fundamental Theorem of Well-Ordering",
"Definition:Order Isomorphism/Well-Orderings/Class Theory",
"Definition:Lower Section/Class Theory",
"Definition:Order Isomorphism/Well-Orderings/Class Theory",
"Definition:Lower Section/Class ... |
proofwiki-2243 | Product of Incidence Matrix of BIBD with its Transpose | Let $A$ be the block incidence matrix for a BIBD with parameters $v, k, \lambda$.
Then:
:$A^\intercal \cdot A = \sqbrk {a_{ij} } = \paren {r - \lambda} I_v + \lambda J_v$
where:
:$A$ is $v \times b$
:$A^\intercal$ is the transpose of $A$
:$J_v$ is the all $v \times v$ $1$'s matrix
:$I_v$ is the $v \times v$ identity ma... | Let row $i$ of $A$ be multiplied by column $i$ of $A^\intercal$.
This is the same as multiplying row $i$ of $A$ by row $i$ of $A$.
Each row of $A$ has $r$ entries (since any point must be in $r$ blocks).
Then:
:$\sqbrk {a_{ii} } = r = \sum $ of the all the $1$'s in row $i$
This completes the main diagonal.
Let row $i$ ... | Let $A$ be the [[Definition:Block Incidence Matrix of BIBD|block incidence matrix]] for a [[Definition:Balanced Incomplete Block Design|BIBD]] with parameters $v, k, \lambda$.
Then:
:$A^\intercal \cdot A = \sqbrk {a_{ij} } = \paren {r - \lambda} I_v + \lambda J_v$
where:
:$A$ is $v \times b$
:$A^\intercal$ is the [[D... | Let [[Definition:Row of Matrix|row]] $i$ of $A$ be multiplied by [[Definition:Column of Matrix|column]] $i$ of $A^\intercal$.
This is the same as multiplying [[Definition:Row of Matrix|row]] $i$ of $A$ by [[Definition:Row of Matrix|row]] $i$ of $A$.
Each [[Definition:Row of Matrix|row]] of $A$ has $r$ entries (since ... | Product of Incidence Matrix of BIBD with its Transpose | https://proofwiki.org/wiki/Product_of_Incidence_Matrix_of_BIBD_with_its_Transpose | https://proofwiki.org/wiki/Product_of_Incidence_Matrix_of_BIBD_with_its_Transpose | [
"Balanced Incomplete Block Designs"
] | [
"Definition:Block Incidence Matrix of BIBD",
"Definition:Balanced Incomplete Block Design",
"Definition:Transpose of Matrix",
"Definition:Ones Matrix",
"Definition:Unit Matrix"
] | [
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Matrix/Row",
"Definition:Point (Design Theory)",
"Definition:Randomized Block",
"Definition:Matrix/Row",
"Definition:Matrix/Diagonal/Main",
"Definition:Matrix/Row",
"Definition:Matr... |
proofwiki-2244 | Fisher's Inequality | For a BIBD $\struct {v, k, \lambda}$, the number of blocks $b$ must be greater than or equal to the number of points $v$:
:$ b \ge v$ | Let $A$ be the incidence matrix.
By Product of Incidence Matrix of BIBD with its Transpose, we have that:
:$A^\intercal \cdot A = \begin{bmatrix}
r & \lambda & \cdots & \lambda \\
\lambda & r & \cdots & \lambda \\
\vdots & \vdots & \ddots & \vdots \\
\lambda & \lambda & \cdots & r \\
\end{bmatrix}$
From Necessary Cond... | For a [[Definition:Balanced Incomplete Block Design|BIBD]] $\struct {v, k, \lambda}$, the number of [[Definition:Block (Block Design)|blocks]] $b$ must be greater than or equal to the number of [[Definition:Point (Design Theory)|points]] $v$:
:$ b \ge v$ | Let $A$ be the [[Definition:Block Incidence Matrix of BIBD|incidence matrix]].
By [[Product of Incidence Matrix of BIBD with its Transpose]], we have that:
:$A^\intercal \cdot A = \begin{bmatrix}
r & \lambda & \cdots & \lambda \\
\lambda & r & \cdots & \lambda \\
\vdots & \vdots & \ddots & \vdots \\
\lambda & \lambd... | Fisher's Inequality | https://proofwiki.org/wiki/Fisher's_Inequality | https://proofwiki.org/wiki/Fisher's_Inequality | [
"Balanced Incomplete Block Designs"
] | [
"Definition:Balanced Incomplete Block Design",
"Definition:Randomized Block",
"Definition:Point (Design Theory)"
] | [
"Definition:Block Incidence Matrix of BIBD",
"Product of Incidence Matrix of BIBD with its Transpose",
"Necessary Condition for Existence of BIBD",
"Definition:Combinatorial Matrix",
"Determinant of Combinatorial Matrix",
"Necessary Condition for Existence of BIBD",
"Definition:Balanced Incomplete Block... |
proofwiki-2245 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | {{BeginTableau|p \implies \paren {q \implies r} \vdash \paren {p \implies q} \implies \paren {p \implies r} }}
{{Premise|1|p \implies \paren {q \implies r} }}
{{Assumption|2|p \implies q}}
{{Assumption|3|p}}
{{ModusPonens|4|1, 3|q \implies r|1|3}}
{{ModusPonens|5|2, 3|q|2|3}}
{{ModusPonens|6|1, 2, 3|r|4|5}}
{{Implicati... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | {{BeginTableau|p \implies \paren {q \implies r} \vdash \paren {p \implies q} \implies \paren {p \implies r} }}
{{Premise|1|p \implies \paren {q \implies r} }}
{{Assumption|2|p \implies q}}
{{Assumption|3|p}}
{{ModusPonens|4|1, 3|q \implies r|1|3}}
{{ModusPonens|5|2, 3|q|2|3}}
{{ModusPonens|6|1, 2, 3|r|4|5}}
{{Implicati... | Self-Distributive Law for Conditional/Formulation 1/Forward Implication/Proof | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_1/Forward_Implication/Proof | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [] |
proofwiki-2246 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | We apply the Method of Truth Tables to the proposition:
$p \implies \paren {q \implies r} \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccccc||ccccccc|} \hline
p & \impl... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | We apply the [[Method of Truth Tables]] to the proposition:
$p \implies \paren {q \implies r} \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] mat... | Self-Distributive Law for Conditional/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_1/Proof_by_Truth_Table | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-2247 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | {{BeginTableau|\paren {p \implies q} \implies \paren {p \implies r} \vdash p \implies \paren {q \implies r} }}
{{Premise|1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{Assumption|2|p}}
{{Assumption|3|q}}
{{SequentIntro|4|3|p \implies q|3|True Statement is implied by Every Statement}}
{{ModusPonens|5|1, 3|p... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | {{BeginTableau|\paren {p \implies q} \implies \paren {p \implies r} \vdash p \implies \paren {q \implies r} }}
{{Premise|1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{Assumption|2|p}}
{{Assumption|3|q}}
{{SequentIntro|4|3|p \implies q|3|[[True Statement is implied by Every Statement]]}}
{{ModusPonens|5|1,... | Self-Distributive Law for Conditional/Formulation 1/Reverse Implication/Proof | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_1/Reverse_Implication/Proof | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [
"True Statement is implied by Every Statement"
] |
proofwiki-2248 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | {{BeginTableau|\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption |1|p \implies \paren {q \implies r} }}
{{SequentIntro|2|1|\paren {p \implies q} \implies \paren {p \implies r}|1|Self-Distributive Law for Conditional: Formulation 1}}
{{Imp... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | {{BeginTableau|\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption |1|p \implies \paren {q \implies r} }}
{{SequentIntro|2|1|\paren {p \implies q} \implies \paren {p \implies r}|1|[[Self-Distributive Law for Conditional/Formulation 1/Forwar... | Self-Distributive Law for Conditional/Formulation 2/Forward Implication/Proof 1 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication/Proof_1 | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [
"Self-Distributive Law for Conditional/Formulation 1/Forward Implication"
] |
proofwiki-2249 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | {{BeginTableau|\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption |1|p \implies \paren {q \implies r} }}
{{Assumption |2|p \implies q}}
{{Assumption |3|p}}
{{ModusPonens |4|1,3|q \implies r|1|3}}
{{ModusPonens |5|2,3|q|2|3}}
{{ModusPonen... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | {{BeginTableau|\paren {p \implies \paren {q \implies r} } \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption |1|p \implies \paren {q \implies r} }}
{{Assumption |2|p \implies q}}
{{Assumption |3|p}}
{{ModusPonens |4|1,3|q \implies r|1|3}}
{{ModusPonens |5|2,3|q|2|3}}
{{ModusPonen... | Self-Distributive Law for Conditional/Formulation 2/Forward Implication/Proof 2 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication/Proof_2 | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [] |
proofwiki-2250 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | We apply the Method of Truth Tables.
As can be seen by inspection, the truth value under the main connective is true for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc|c|ccccccc|}
\hline
(p & \implies & (q & \implies & r)) & \implies & ((p & \implies & q) & \implies & (p & \implies & r)) \\
\hline
\F & \T ... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth value]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|c... | Self-Distributive Law for Conditional/Formulation 2/Forward Implication/Proof by Truth Table | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Forward_Implication/Proof_by_Truth_Table | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-2251 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption |1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{SequentIntro |2|1|p \implies \paren {q \implies r}|1|Self-Distributive Law for Conditional: Formulation... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption |1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{SequentIntro |2|1|p \implies \paren {q \implies r}|1|[[Self-Distributive Law for Conditional/Formulatio... | Self-Distributive Law for Conditional/Formulation 2/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Reverse_Implication/Proof_1 | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [
"Self-Distributive Law for Conditional/Formulation 1/Reverse Implication"
] |
proofwiki-2252 | Self-Distributive Law for Conditional | The following is known as the Self-Distributive Law:
=== Formulation 1 ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== Formulation 2 ===
{{:Self-Distributive Law for Conditional/Formulation 2}} | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption|1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{Assumption|2|p}}
{{Assumption|3|q}}
{{SequentIntro|4|3|p \implies q|3|True Statement is implied by Every S... | The following is known as the [[Definition:Self Distributive|Self-Distributive]] Law:
=== [[Self-Distributive Law for Conditional/Formulation 1|Formulation 1]] ===
{{:Self-Distributive Law for Conditional/Formulation 1}}
=== [[Self-Distributive Law for Conditional/Formulation 2|Formulation 2]] ===
{{:Self-Distributi... | {{BeginTableau |\vdash \paren {\paren {p \implies q} \implies \paren {p \implies r} } \implies \paren {p \implies \paren {q \implies r} } }}
{{Assumption|1|\paren {p \implies q} \implies \paren {p \implies r} }}
{{Assumption|2|p}}
{{Assumption|3|q}}
{{SequentIntro|4|3|p \implies q|3|[[True Statement is implied by Every... | Self-Distributive Law for Conditional/Formulation 2/Reverse Implication/Proof 2 | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional | https://proofwiki.org/wiki/Self-Distributive_Law_for_Conditional/Formulation_2/Reverse_Implication/Proof_2 | [
"Self-Distributive Law for Conditional",
"Conditional",
"Examples of Self-Distributive Operations"
] | [
"Definition:Self-Distributive Operation",
"Self-Distributive Law for Conditional/Formulation 1",
"Self-Distributive Law for Conditional/Formulation 2"
] | [
"True Statement is implied by Every Statement"
] |
proofwiki-2253 | Disjunction of Conditional and Converse | :$\vdash \paren {p \implies q} \lor \paren {q \implies p}$ | {{BeginTableau|\vdash \paren {p \implies q} \lor \paren {q \implies p} }}
{{ExcludedMiddle|1|p \lor \neg p}}
{{Assumption|2|p}}
{{SequentIntro|3|2|q \implies p|2|True Statement is implied by Every Statement}}
{{Addition|4|2|\paren {p \implies q} \lor \paren {q \implies p}|2|2}}
{{Assumption|5|\neg p}}
{{SequentIntro|6|... | :$\vdash \paren {p \implies q} \lor \paren {q \implies p}$ | {{BeginTableau|\vdash \paren {p \implies q} \lor \paren {q \implies p} }}
{{ExcludedMiddle|1|p \lor \neg p}}
{{Assumption|2|p}}
{{SequentIntro|3|2|q \implies p|2|[[True Statement is implied by Every Statement]]}}
{{Addition|4|2|\paren {p \implies q} \lor \paren {q \implies p}|2|2}}
{{Assumption|5|\neg p}}
{{SequentIntr... | Disjunction of Conditional and Converse/Proof 1 | https://proofwiki.org/wiki/Disjunction_of_Conditional_and_Converse | https://proofwiki.org/wiki/Disjunction_of_Conditional_and_Converse/Proof_1 | [
"Disjunction of Conditional and Converse",
"Conditional",
"Disjunction"
] | [] | [
"True Statement is implied by Every Statement",
"False Statement implies Every Statement"
] |
proofwiki-2254 | Disjunction of Conditional and Converse | :$\vdash \paren {p \implies q} \lor \paren {q \implies p}$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations, proving a tautology.
$\begin{array}{|ccccccc|} \hline
(p & \implies & q) & \lor & (q & \implies & p) \\
\hline
\F & \T & \F & \T & \F & \T & \F \\
\F ... | :$\vdash \paren {p \implies q} \lor \paren {q \implies p}$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]], proving... | Disjunction of Conditional and Converse/Proof by Truth Table | https://proofwiki.org/wiki/Disjunction_of_Conditional_and_Converse | https://proofwiki.org/wiki/Disjunction_of_Conditional_and_Converse/Proof_by_Truth_Table | [
"Disjunction of Conditional and Converse",
"Conditional",
"Disjunction"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation",
"Definition:Tautology/Formal Semantics/Boolean Interpretations"
] |
proofwiki-2255 | Characteristics of Eulerian Graph | A finite (undirected) graph is Eulerian {{iff}} it is connected and each vertex is even. | Suppose that an (undirected) graph $G$ is connected and its vertices all have even degree.
If there is more than one vertex in $G$, then each vertex must have degree greater than $0$.
Begin at a vertex $v$.
From Graph with Even Vertices Partitions into Cycles, we know that $v$ will be on at least one cycle.
Since $G$ i... | A [[Definition:Finite Graph|finite]] [[Definition:Undirected Graph|(undirected) graph]] is [[Definition:Eulerian Graph|Eulerian]] {{iff}} it is [[Definition:Connected Graph|connected]] and each [[Definition:Vertex of Graph|vertex]] is [[Definition:Even Vertex of Graph|even]]. | Suppose that an [[Definition:Undirected Graph|(undirected) graph]] $G$ is [[Definition:Connected Graph|connected]] and its [[Definition:Vertex of Graph|vertices]] all have [[Definition:Even Vertex of Graph|even degree]].
If there is more than one [[Definition:Vertex of Graph|vertex]] in $G$, then each [[Definition:Ver... | Characteristics of Eulerian Graph/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Characteristics_of_Eulerian_Graph | https://proofwiki.org/wiki/Characteristics_of_Eulerian_Graph/Sufficient_Condition/Proof_1 | [
"Characteristics of Eulerian Graph",
"Eulerian Graphs"
] | [
"Definition:Finite Graph",
"Definition:Undirected Graph",
"Definition:Eulerian Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Even Vertex of Graph"
] | [
"Definition:Undirected Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Even Vertex of Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/V... |
proofwiki-2256 | Characteristics of Eulerian Graph | A finite (undirected) graph is Eulerian {{iff}} it is connected and each vertex is even. | Suppose that an (undirected) graph $G$ is connected and its vertices all have even degree.
From Graph with Even Vertices Partitions into Cycles, we can split $G$ into a number of cycles $\mathbb S = C_1, C_2, \ldots, C_k$.
Start at any vertex $v$ on cycle $C_1$ and traverse its edges until we encounter a vertex of anot... | A [[Definition:Finite Graph|finite]] [[Definition:Undirected Graph|(undirected) graph]] is [[Definition:Eulerian Graph|Eulerian]] {{iff}} it is [[Definition:Connected Graph|connected]] and each [[Definition:Vertex of Graph|vertex]] is [[Definition:Even Vertex of Graph|even]]. | Suppose that an [[Definition:Undirected Graph|(undirected) graph]] $G$ is [[Definition:Connected Graph|connected]] and its [[Definition:Vertex of Graph|vertices]] all have [[Definition:Even Vertex of Graph|even degree]].
From [[Graph with Even Vertices Partitions into Cycles]], we can split $G$ into a number of [[Defi... | Characteristics of Eulerian Graph/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Characteristics_of_Eulerian_Graph | https://proofwiki.org/wiki/Characteristics_of_Eulerian_Graph/Sufficient_Condition/Proof_2 | [
"Characteristics of Eulerian Graph",
"Eulerian Graphs"
] | [
"Definition:Finite Graph",
"Definition:Undirected Graph",
"Definition:Eulerian Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Even Vertex of Graph"
] | [
"Definition:Undirected Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Even Vertex of Graph",
"Graph with Even Vertices Partitions into Cycles",
"Definition:Cycle (Graph Theory)",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Cycle (Gr... |
proofwiki-2257 | Commutativity of Incidence Matrix with its Transpose for Symmetric Design | Let $A$ be the incidence matrix of a symmetric design.
Then:
:$A A^\intercal = A^\intercal A$
where $A^\intercal$ is the transpose of $A$. | First note, we have:
:$(1): \quad A J = J A = k J$, so $A^\intercal J = \paren {J A}^\intercal = \paren {k J}^\intercal = k J$, and likewise $J A^\intercal = k J$
:$(2): \quad J^2 = v J$
:$(3): \quad$ If a design is symmetric, then $A A^\intercal = \paren {r - \lambda} I + \lambda J = \paren {k - \lambda} I + \lambda J... | Let $A$ be the [[Definition:Block Incidence Matrix of BIBD|incidence matrix]] of a [[Definition:Symmetric Design|symmetric design]].
Then:
:$A A^\intercal = A^\intercal A$
where $A^\intercal$ is the [[Definition:Transpose of Matrix|transpose]] of $A$. | First note, we have:
:$(1): \quad A J = J A = k J$, so $A^\intercal J = \paren {J A}^\intercal = \paren {k J}^\intercal = k J$, and likewise $J A^\intercal = k J$
:$(2): \quad J^2 = v J$
:$(3): \quad$ If a design is symmetric, then $A A^\intercal = \paren {r - \lambda} I + \lambda J = \paren {k - \lambda} I + \lambda J... | Commutativity of Incidence Matrix with its Transpose for Symmetric Design | https://proofwiki.org/wiki/Commutativity_of_Incidence_Matrix_with_its_Transpose_for_Symmetric_Design | https://proofwiki.org/wiki/Commutativity_of_Incidence_Matrix_with_its_Transpose_for_Symmetric_Design | [
"Design Theory"
] | [
"Definition:Block Incidence Matrix of BIBD",
"Definition:Symmetric Design",
"Definition:Transpose of Matrix"
] | [
"Definition:Commutative/Elements",
"Category:Design Theory"
] |
proofwiki-2258 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc||cccc|} \hline
\neg & p & \land & \neg & q & \neg & (p & \lor & q) \\
\hline
\T & \F & \T & \T & \F & \T & \F & \F & \F \\
\T & \F & \F & \... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|ccccc||cccc|} \hline
... | De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Conjunction_of_Negations/Formulation_1/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-2259 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{BeginTableau|\vdash \paren {\neg p \land \neg q} \iff \paren {\neg \paren {p \lor q} } }}
{{Assumption|1|\neg p \land \neg q}}
{{SequentIntro|2|1|\neg \paren {p \lor q}|1|De Morgan's Laws (Logic): Disjunction of Negations: Formulation 1}}
{{Implication|3||\paren {\neg p \land \neg q} \implies \paren {\neg \paren {p \... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{BeginTableau|\vdash \paren {\neg p \land \neg q} \iff \paren {\neg \paren {p \lor q} } }}
{{Assumption|1|\neg p \land \neg q}}
{{SequentIntro|2|1|\neg \paren {p \lor q}|1|[[De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Forward Implication|De Morgan's Laws (Logic): Disjunction of Negations: Formulati... | De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 2/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Conjunction_of_Negations/Formulation_2/Proof_1 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Forward Implication",
"De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Reverse Implication"
] |
proofwiki-2260 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc|c|cccc|} \hline
\neg & p & \land & \neg & q & \iff & \neg & (p & \lor & q) \\
\hline
\T & \F & \T & \T & \F & \T & \T & \F & \F & \F \\
\T... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|... | De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Conjunction_of_Negations/Formulation_2/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-2261 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | 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}{|ccc||cccccc|} \hline
p & \land & q & \neg & (\neg & p & \lor & \neg & q) \\
\hline
\F & \F & \F & \F & \T & \F & \T & \T & \F \\
\F & \F & \T & \F & \T &... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc||cccccc|} \hline
p & \land... | De Morgan's Laws (Logic)/Conjunction/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Conjunction/Formulation_1/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-2262 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{BeginTableau|\vdash \paren {p \land q} \iff \paren {\neg \paren {\neg p \lor \neg q} } }}
{{Assumption|1|p \land q}}
{{SequentIntro|2|1|\neg \paren {\neg p \lor \neg q}|1|De Morgan's Laws (Logic): Conjunction: Formulation 1}}
{{Implication|3||\paren {p \land q} \implies \paren {\neg \paren {\neg p \lor \neg q} }|1|2}... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{BeginTableau|\vdash \paren {p \land q} \iff \paren {\neg \paren {\neg p \lor \neg q} } }}
{{Assumption|1|p \land q}}
{{SequentIntro|2|1|\neg \paren {\neg p \lor \neg q}|1|[[De Morgan's Laws (Logic)/Conjunction/Formulation 1/Forward Implication|De Morgan's Laws (Logic): Conjunction: Formulation 1]]}}
{{Implication|3||... | De Morgan's Laws (Logic)/Conjunction/Formulation 2/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Conjunction/Formulation_2/Proof_1 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Conjunction/Formulation 1/Forward Implication",
"De Morgan's Laws (Logic)/Conjunction/Formulation 1/Reverse Implication"
] |
proofwiki-2263 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective are true for all boolean interpretations.
$\begin{array}{|ccc|c|cccccc|} \hline
(p & \land & q) & \iff & (\neg & (\neg & p & \lor & \neg & q)) \\
\hline
\F & \F & \F & \T & \F & \T & \F & \T & \T & \F \\
\F & ... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] are [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc|c|cc... | De Morgan's Laws (Logic)/Conjunction/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Conjunction/Formulation_2/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-2264 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc||cccc|} \hline
\neg & p & \lor & \neg & q & \neg & (p & \land & q) \\
\hline
\T & \F & \T & \T & \F & \T & \F & \F & \F \\
\T & \F & \T & \... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|ccccc||cccc|} \hline
... | De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction_of_Negations/Formulation_1/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-2265 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{BeginTableau|\neg \paren {p \land q} \vdash \neg p \lor \neg q}}
{{Premise|1|\neg \paren {p \land q} }}
{{Assumption|2|\neg \paren {\neg p \lor \neg q} }}
{{Assumption|3|\neg p}}
{{Addition|4|3|\neg p \lor \neg q|3|1}}
{{NonContradiction|5|2, 3|4|2}}
{{Reductio|6|2|p|3|5}}
{{Assumption|7|\neg q}}
{{Addition|8|7|\neg ... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{BeginTableau|\neg \paren {p \land q} \vdash \neg p \lor \neg q}}
{{Premise|1|\neg \paren {p \land q} }}
{{Assumption|2|\neg \paren {\neg p \lor \neg q} }}
{{Assumption|3|\neg p}}
{{Addition|4|3|\neg p \lor \neg q|3|1}}
{{NonContradiction|5|2, 3|4|2}}
{{Reductio|6|2|p|3|5}}
{{Assumption|7|\neg q}}
{{Addition|8|7|\neg ... | De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction_of_Negations/Formulation_1/Reverse_Implication/Proof_1 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [] |
proofwiki-2266 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{BeginTableau|\vdash \paren {\neg p \lor \neg q} \iff \paren {\neg \paren {p \land q} } }}
{{Assumption|1|\neg p \lor \neg q}}
{{SequentIntro|2|1|\neg \paren {p \land q}|1|De Morgan's Laws (Logic): Disjunction of Negations: Formulation 1}}
{{Implication|3||\paren {\neg p \lor \neg q} \implies \paren {\neg \paren {p \l... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{BeginTableau|\vdash \paren {\neg p \lor \neg q} \iff \paren {\neg \paren {p \land q} } }}
{{Assumption|1|\neg p \lor \neg q}}
{{SequentIntro|2|1|\neg \paren {p \land q}|1|[[De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Forward Implication|De Morgan's Laws (Logic): Disjunction of Negations: Formulati... | De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 2/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction_of_Negations/Formulation_2/Proof_1 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Forward Implication",
"De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Reverse Implication"
] |
proofwiki-2267 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective is true for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc|c|cccc|} \hline
\neg & p & \lor & \neg & q & \iff & \neg & (p & \land & q) \\
\hline
\T & \F & \T & \T & \F & \T & \T & \F & \F & \F \\
\T... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|... | De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction_of_Negations/Formulation_2/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-2268 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | 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}{|ccc||cccccc|} \hline
p & \lor & q & \neg & (\neg & p & \land & \neg & q) \\
\hline
\F & \F & \F & \F & \T & \F & \T & \T & \F \\
\F & \T & \T & \T & \T & ... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|ccc||cccccc|} \hline
p & \lor ... | De Morgan's Laws (Logic)/Disjunction/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction/Formulation_1/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-2269 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{BeginTableau|\vdash \paren {p \lor q} \iff \paren {\neg \paren {\neg p \land \neg q} } }}
{{Assumption|1|p \lor q}}
{{SequentIntro|2|1|\neg \paren {\neg p \land \neg q}|1|De Morgan's Laws (Logic): Disjunction: Formulation 1}}
{{Implication|3||\paren {p \lor q} \implies \paren {\neg \paren {\neg p \land \neg q} }|1|2}... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{BeginTableau|\vdash \paren {p \lor q} \iff \paren {\neg \paren {\neg p \land \neg q} } }}
{{Assumption|1|p \lor q}}
{{SequentIntro|2|1|\neg \paren {\neg p \land \neg q}|1|[[De Morgan's Laws (Logic)/Disjunction/Formulation 1/Forward Implication|De Morgan's Laws (Logic): Disjunction: Formulation 1]]}}
{{Implication|3||... | De Morgan's Laws (Logic)/Disjunction/Formulation 2/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction/Formulation_2/Proof_1 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Disjunction/Formulation 1/Forward Implication",
"De Morgan's Laws (Logic)/Disjunction/Formulation 1/Reverse Implication"
] |
proofwiki-2270 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connective are true for all boolean interpretations.
:<nowiki>$\begin{array}{|ccc|c|cccccc|} \hline
(p & \lor & q) & \iff & (\neg & (\neg & p & \land & \neg & q)) \\
\hline
\F & \F & \F & \T & \F & \T & \F & \T & \T & \F ... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] are [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{... | De Morgan's Laws (Logic)/Disjunction/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Logic)/Disjunction/Formulation_2/Proof_by_Truth_Table | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-2271 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | Let the cardinality $\card I$ of the indexing set $I$ be $n$.
Then by the definition of cardinality, it follows that $I \cong \N^*_n$ and we can express the proposition:
:$\ds S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \mathop \in I} \paren {S \setminus T_i}$
as:
:$\ds S \setminus \bigcap_{i \mathop = 1}^n ... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | Let the [[Definition:Cardinality|cardinality]] $\card I$ of the [[Definition:Indexing Set|indexing set]] $I$ be $n$.
Then by the definition of [[Definition:Cardinality|cardinality]], it follows that $I \cong \N^*_n$ and we can express the proposition:
:$\ds S \setminus \bigcap_{i \mathop \in I} T_i = \bigcup_{i \math... | De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Intersection/Proof | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Proof_by_Induction/Difference_with_Intersection/Proof | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Definition:Cardinality",
"Definition:Indexing Set",
"Definition:Cardinality",
"Principle of Mathematical Induction",
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"In... |
proofwiki-2272 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | Let the cardinality $\size I$ of the indexing set $I$ be $n$.
Then by the definition of cardinality, it follows that $I \cong \N^*_n$ and we can express the proposition:
:$\ds S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \mathop \in I} \paren {S \setminus T_i}$
as:
:$\ds S \setminus \bigcup_{i \mathop = 1}^n ... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | Let the [[Definition:Cardinality|cardinality]] $\size I$ of the [[Definition:Indexing Set|indexing set]] $I$ be $n$.
Then by the definition of [[Definition:Cardinality|cardinality]], it follows that $I \cong \N^*_n$ and we can express the proposition:
:$\ds S \setminus \bigcup_{i \mathop \in I} T_i = \bigcap_{i \math... | De Morgan's Laws (Set Theory)/Proof by Induction/Difference with Union/Proof | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Proof_by_Induction/Difference_with_Union/Proof | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Definition:Cardinality",
"Definition:Indexing Set",
"Definition:Cardinality",
"Principle of Mathematical Induction",
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Union",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Union is ... |
proofwiki-2273 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | Let $x \in S$ througout.
{{begin-eqn}}
{{eqn | o =
| r = x \in \relcomp S {T_1 \cup T_2}
}}
{{eqn | o = \leadsto
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Relative Complement}}
}}
{{eqn | o = \leadsto
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eqn... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | Let $x \in S$ througout.
{{begin-eqn}}
{{eqn | o =
| r = x \in \relcomp S {T_1 \cup T_2}
}}
{{eqn | o = \leadsto
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Relative Complement}}
}}
{{eqn | o = \leadsto
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eq... | De Morgan's Laws (Set Theory)/Relative Complement/Complement of Union/Proof 2 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Relative_Complement/Complement_of_Union/Proof_2 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Conjunction of Negations",
"De Morgan's Laws (Logic)/Conjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-2274 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{begin-eqn}}
{{eqn | l = \overline {T_1 \cap T_2}
| r = \mathbb U \setminus \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \paren {\mathbb U \setminus T_1} \cup \paren {\mathbb U \setminus T_2}
| c = De Morgan's Laws: Difference with Intersection
}}
{{eqn | r = \overline {T_1} \c... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{begin-eqn}}
{{eqn | l = \overline {T_1 \cap T_2}
| r = \mathbb U \setminus \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | r = \paren {\mathbb U \setminus T_1} \cup \paren {\mathbb U \setminus T_2}
| c = [[De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection|De M... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 1 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_1 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection"
] |
proofwiki-2275 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cap T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \land x \in T_2}
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \l... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cap T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cap T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \land x \in T_2}
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \l... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 2 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_2 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-2276 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{begin-eqn}}
{{eqn | l = \map \complement {\map \complement A \cup \map \complement B}
| r = \map \complement {\map \complement A} \cap \map \complement {\map \complement B}
| c = De Morgan's Laws: Complement of Union
}}
{{eqn | r = A \cap B
| c = Complement of Complement
}}
{{eqn | ll= \leadstoandfr... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{begin-eqn}}
{{eqn | l = \map \complement {\map \complement A \cup \map \complement B}
| r = \map \complement {\map \complement A} \cap \map \complement {\map \complement B}
| c = [[De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|De Morgan's Laws: Complement of Union]]
}}
{{eqn | r = A \ca... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection/Proof 3 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_3 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Union",
"Complement of Complement",
"Definition:Set Complement",
"Complement of Complement"
] |
proofwiki-2277 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cup T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eqn | o = \leadstoan... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | {{begin-eqn}}
{{eqn | o =
| r = x \in \overline {T_1 \cup T_2}
}}
{{eqn | o = \leadstoandfrom
| r = x \notin \paren {T_1 \cup T_2}
| c = {{Defof|Set Complement}}
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {x \in T_1 \lor x \in T_2}
| c = {{Defof|Set Union}}
}}
{{eqn | o = \leadstoan... | De Morgan's Laws (Set Theory)/Set Complement/Complement of Union/Proof 2 | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Union/Proof_2 | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"De Morgan's Laws (Logic)/Conjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-2278 | De Morgan's Laws | '''De Morgan's Laws''' are a suite of theorems in logic, which are also applied in set theory, as follows:
=== Propositional Logic ===
{{:De Morgan's Laws (Logic)}}
=== Predicate Logic ===
{{:De Morgan's Laws (Predicate Logic)}}
=== Set Theory ===
{{:De Morgan's Laws (Set Theory)}}
=== Boolean Algebras ===
{{:De Morgan... | Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by Set Difference is Subset we have that $x \in S$ (we need this later).
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S \setminus \bigcap \mathbb T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \notin
| r = \big... | '''[[De Morgan's Laws]]''' are a suite of [[Definition:Theorem|theorems]] in [[Definition:Logic|logic]], which are also applied in [[Definition:Set Theory|set theory]], as follows:
=== [[De Morgan's Laws (Logic)|Propositional Logic]] ===
{{:De Morgan's Laws (Logic)}}
=== [[De Morgan's Laws (Predicate Logic)|Predicat... | Suppose:
:$\ds x \in S \setminus \bigcap \mathbb T$
Note that by [[Set Difference is Subset]] we have that $x \in S$ (we need this later).
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S \setminus \bigcap \mathbb T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \notin
| r ... | De Morgan's Laws (Set Theory)/Set Difference/General Case/Difference with Intersection/Proof | https://proofwiki.org/wiki/De_Morgan's_Laws | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Difference/General_Case/Difference_with_Intersection/Proof | [
"De Morgan's Laws",
"Logical Negation"
] | [
"De Morgan's Laws",
"Definition:Theorem",
"Definition:Logic",
"Definition:Set Theory",
"De Morgan's Laws (Logic)",
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Set Theory)",
"De Morgan's Laws (Boolean Algebras)"
] | [
"Set Difference is Subset",
"De Morgan's Laws (Predicate Logic)/Denial of Universality"
] |
proofwiki-2279 | Four Color Theorem | Let $G$ be a planar graph.
Then:
:$\map \chi G \le 4$
where the $\map \chi G$ denotes the chromatic number of $G$.
That is, $G$ can be assigned a proper vertex $k$-coloring such that $k \le 4$. | It can be shown that it is necessary to consider only planar graphs which are simple.
Hence any discussion below about a graph will carry the assumption that it is both simple and planar.
{{finish|The thesis follows.}}
{{ProofWanted}} | Let $G$ be a [[Definition:Planar Graph|planar graph]].
Then:
:$\map \chi G \le 4$
where the $\map \chi G$ denotes the [[Definition:Chromatic Number|chromatic number]] of $G$.
That is, $G$ can be assigned a [[Definition:Proper Vertex Coloring|proper vertex $k$-coloring]] such that $k \le 4$. | It can be shown that it is necessary to consider only [[Definition:Planar Graph|planar graphs]] which are [[Definition:Simple Graph|simple]].
Hence any discussion below about a [[Definition:Graph (Graph Theory)|graph]] will carry the assumption that it is both [[Definition:Simple Graph|simple]] and [[Definition:Planar... | Four Color Theorem | https://proofwiki.org/wiki/Four_Color_Theorem | https://proofwiki.org/wiki/Four_Color_Theorem | [
"Four Color Theorem",
"Chromatic Numbers",
"Graph Colorings",
"4",
"Named Theorems"
] | [
"Definition:Planar Graph",
"Definition:Chromatic Number",
"Definition:Proper Coloring/Vertex Coloring"
] | [
"Definition:Planar Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)",
"Definition:Simple Graph",
"Definition:Planar Graph"
] |
proofwiki-2280 | Cycle does not Contain Subcycles | Let $G$ be a cycle graph.
Then the only cycle graph that is a subgraph of $G$ is $G$ itself. | {{AimForCont}} $G$ contains a subgraph $C$ such that:
:$C$ is a cycle graph
:$C \ne G$ is non-empty.
Then there exists some vertex $v$ that is not in $C$.
Let $u$ be any vertex of $C$.
Since $G$ is a cycle graph, it is connected.
Therefore there is a walk from $u$ to $v$ in $G$.
There must be some vertex $x$ that is th... | Let $G$ be a [[Definition:Cycle Graph|cycle graph]].
Then the only [[Definition:Cycle Graph|cycle graph]] that is a [[Definition:Subgraph|subgraph]] of $G$ is $G$ itself. | {{AimForCont}} $G$ contains a [[Definition:Subgraph|subgraph]] $C$ such that:
:$C$ is a [[Definition:Cycle Graph|cycle graph]]
:$C \ne G$ is non-[[Definition:Empty Set|empty]].
Then there exists some [[Definition:Vertex of Graph|vertex]] $v$ that is not in $C$.
Let $u$ be any [[Definition:Vertex of Graph|vertex]] of ... | Cycle does not Contain Subcycles | https://proofwiki.org/wiki/Cycle_does_not_Contain_Subcycles | https://proofwiki.org/wiki/Cycle_does_not_Contain_Subcycles | [
"Cycles (Graph Theory)"
] | [
"Definition:Cycle Graph",
"Definition:Cycle Graph",
"Definition:Subgraph"
] | [
"Definition:Subgraph",
"Definition:Cycle Graph",
"Definition:Empty Set",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Cycle Graph",
"Definition:Connected (Graph Theory)/Graph",
"Definition:Walk (Graph Theory)",
"Definition:Graph (Graph Theory)/Verte... |
proofwiki-2281 | Dirac's Theorem | Let $G$ be a connected simple graph with $n$ vertices such that $n > 3$.
Let the degree of each vertex be at least $\dfrac n 2$.
Then $G$ is Hamiltonian. | Let $P = p_1 p_2 \ldots p_k$ be the longest path in $G$.
If $p_1$ is adjacent to some vertex $v$ not in $P$, then the path $v p_1 p_2 \ldots p_k$ would be longer than $P$, contradicting the choice of $P$.
The same argument can be made for $p_k$.
So both $p_1$ and $p_k$ are adjacent only to vertices in $P$.
Since $\map... | Let $G$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] with $n$ [[Definition:Vertex of Graph|vertices]] such that $n > 3$.
Let the [[Definition:Degree of Vertex|degree]] of each [[Definition:Vertex of Graph|vertex]] be at least $\dfrac n 2$.
Then $G$ is [[Definition:Hamiltonian... | Let $P = p_1 p_2 \ldots p_k$ be the longest [[Definition:Path (Graph Theory)|path]] in $G$.
If $p_1$ is [[Definition:Adjacent Vertices (Undirected Graph)|adjacent]] to some [[Definition:Vertex of Graph|vertex]] $v$ not in $P$, then the [[Definition:Path (Graph Theory)|path]] $v p_1 p_2 \ldots p_k$ would be longer tha... | Dirac's Theorem/Proof 1 | https://proofwiki.org/wiki/Dirac's_Theorem | https://proofwiki.org/wiki/Dirac's_Theorem/Proof_1 | [
"Dirac's Theorem",
"Hamiltonian Graphs"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Hamiltonian Graph"
] | [
"Definition:Path (Graph Theory)",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Path (Graph Theory)",
"Definition:Adjacent (Graph Theory)/Vertices/Undirected Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Adjacent (Gr... |
proofwiki-2282 | Dirac's Theorem | Let $G$ be a connected simple graph with $n$ vertices such that $n > 3$.
Let the degree of each vertex be at least $\dfrac n 2$.
Then $G$ is Hamiltonian. | Take any two non-adjacent vertices $u, v \in G$.
Then:
:$\deg u + \deg v \ge \dfrac n 2 + \dfrac n 2 = n$
The result follows by a direct application of Ore's Theorem.
{{qed}} | Let $G$ be a [[Definition:Connected Graph|connected]] [[Definition:Simple Graph|simple graph]] with $n$ [[Definition:Vertex of Graph|vertices]] such that $n > 3$.
Let the [[Definition:Degree of Vertex|degree]] of each [[Definition:Vertex of Graph|vertex]] be at least $\dfrac n 2$.
Then $G$ is [[Definition:Hamiltonian... | Take any two [[Definition:Non-Adjacent Vertices (Graph Theory)|non-adjacent vertices]] $u, v \in G$.
Then:
:$\deg u + \deg v \ge \dfrac n 2 + \dfrac n 2 = n$
The result follows by a direct application of [[Ore's Theorem]].
{{qed}} | Dirac's Theorem/Proof 2 | https://proofwiki.org/wiki/Dirac's_Theorem | https://proofwiki.org/wiki/Dirac's_Theorem/Proof_2 | [
"Dirac's Theorem",
"Hamiltonian Graphs"
] | [
"Definition:Connected (Graph Theory)/Graph",
"Definition:Simple Graph",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Degree of Vertex",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Hamiltonian Graph"
] | [
"Definition:Adjacent (Graph Theory)/Vertices/Non-Adjacent",
"Ore's Theorem"
] |
proofwiki-2283 | Three-Way Exclusive Or and Equivalence | Let $p \iff q$ be the biconditional operator, and $p \oplus q$ be the exclusive or operator.
Then:
: $p \iff q \iff r \dashv \vdash p \oplus q \oplus r$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, in each case, the truth values under the main connectives match for all boolean interpretations.
$\begin{array}{|ccccc||ccccc|} \hline
(p & \iff & q) & \iff & r & (p & \oplus & q) & \oplus & r \\
\hline
F & T & F & F & F & F & F & F &... | Let $p \iff q$ be the [[Definition:Biconditional|biconditional operator]], and $p \oplus q$ be the [[Definition:Exclusive Or|exclusive or operator]].
Then:
: $p \iff q \iff r \dashv \vdash p \oplus q \oplus r$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, in each case, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{arra... | Three-Way Exclusive Or and Equivalence | https://proofwiki.org/wiki/Three-Way_Exclusive_Or_and_Equivalence | https://proofwiki.org/wiki/Three-Way_Exclusive_Or_and_Equivalence | [
"Biconditional",
"Exclusive Or"
] | [
"Definition:Biconditional",
"Definition:Exclusive Or"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation",
"Biconditional is Associative",
"Exclusive Or is Associative",
"Definition:Associative Operation",
"Definition:Parenthesis"
] |
proofwiki-2284 | NAND and NOR are Functionally Complete | The NAND and NOR operators are each functionally complete.
That is, NAND and NOR are Sheffer operators. | In NAND is Functionally Complete it is demonstrated that NAND is functionally complete.
In NOR is Functionally Complete it is demonstrated that NOR is functionally complete. | The [[Definition:Logical NAND|NAND]] and [[Definition:Logical NOR|NOR]] operators are each [[Definition:Functionally Complete|functionally complete]].
That is, [[Definition:Logical NAND|NAND]] and [[Definition:Logical NOR|NOR]] are [[Definition:Sheffer Operator|Sheffer operators]]. | In [[NAND is Functionally Complete]] it is demonstrated that [[Definition:Logical NAND|NAND]] is [[Definition:Functionally Complete|functionally complete]].
In [[NOR is Functionally Complete]] it is demonstrated that [[Definition:Logical NOR|NOR]] is [[Definition:Functionally Complete|functionally complete]]. | NAND and NOR are Functionally Complete | https://proofwiki.org/wiki/NAND_and_NOR_are_Functionally_Complete | https://proofwiki.org/wiki/NAND_and_NOR_are_Functionally_Complete | [
"Logical NAND",
"Logical NOR",
"Functional Completeness"
] | [
"Definition:Logical NAND",
"Definition:Logical NOR",
"Definition:Functionally Complete",
"Definition:Logical NAND",
"Definition:Logical NOR",
"Definition:Sheffer Operator"
] | [
"Functionally Complete Logical Connectives/NAND",
"Definition:Logical NAND",
"Definition:Functionally Complete",
"Functionally Complete Logical Connectives/NOR",
"Definition:Logical NOR",
"Definition:Functionally Complete"
] |
proofwiki-2285 | Proof of Theorem by Truth Table | Let $\phi$ be a propositional formula whose atoms are $p_1, p_2, \ldots, p_n$.
Let $l$ be the line number of any row in the truth table of $\phi$.
Let $\hat {p_i}$ be defined as:
:<nowiki>$\hat {p_i} = \begin{cases}
p_i & : \text {the entry in line } l \text { of } p_i \text { is } \T \\
\neg p_i & : \text {the entry i... | :$(1): \quad$ Suppose $\phi$ is an atom $p$.
Then we need to show that $p \vdash p$ and $\neg p \vdash \neg p$.
These are proved in one line in the proof of the Law of Identity.
:$(2): \quad$ Suppose $\phi$ is of the form $\neg \phi_1$.
There are two cases to consider:
Suppose $\phi$ evaluates to $\T$.
{{finish|More ha... | Let $\phi$ be a [[Definition:Propositional Formula|propositional formula]] whose [[Definition:Simple Statement|atoms]] are $p_1, p_2, \ldots, p_n$.
Let $l$ be the line number of any row in the [[Definition:Truth Table|truth table]] of $\phi$.
Let $\hat {p_i}$ be defined as:
:<nowiki>$\hat {p_i} = \begin{cases}
p_i &... | :$(1): \quad$ Suppose $\phi$ is an [[Definition:Simple Statement|atom]] $p$.
Then we need to show that $p \vdash p$ and $\neg p \vdash \neg p$.
These are proved in one line in the proof of the [[Law of Identity]].
:$(2): \quad$ Suppose $\phi$ is of the form $\neg \phi_1$.
There are two cases to consider:
Suppose ... | Proof of Theorem by Truth Table | https://proofwiki.org/wiki/Proof_of_Theorem_by_Truth_Table | https://proofwiki.org/wiki/Proof_of_Theorem_by_Truth_Table | [
"Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Simple Statement",
"Definition:Truth Table",
"Definition:Provable Consequence",
"Definition:Provable Consequence"
] | [
"Definition:Simple Statement",
"Law of Identity",
"Category:Propositional Logic"
] |
proofwiki-2286 | WFFs of PropLog of Length 1 | The only WFFs of propositional logic of length $1$ are:
* The letters of the formal grammar of propositional logic $\LL_0$
* The tautology symbol $\top$
* The contradiction symbol $\bot$. | We refer to the rules of formation.
From $\mathbf W: \T \F$, $\top$ and $\bot$ (both of length 1) are WFFs.
From $\mathbf W: \PP_0$, all elements of $\PP_0$ (all of length 1) are WFFs.
Every other rule of formation of the formal grammar of propositional logic consists of an existing WFF in addition to at least one othe... | The only [[Definition:WFF of Propositional Logic|WFFs of propositional logic]] of [[Definition:Length of String|length]] $1$ are:
* The [[Definition:Letter of Formal Language|letters]] of the [[Definition:Formal Grammar of Propositional Logic|formal grammar of propositional logic]] $\LL_0$
* The [[Definition:Top (Logic... | We refer to the [[Definition:Bottom-Up Specification of Propositional Logic|rules of formation]].
From $\mathbf W: \T \F$, $\top$ and $\bot$ (both of [[Definition:Length of String|length]] 1) are WFFs.
From $\mathbf W: \PP_0$, all elements of $\PP_0$ (all of [[Definition:Length of String|length]] 1) are WFFs.
Every ... | WFFs of PropLog of Length 1 | https://proofwiki.org/wiki/WFFs_of_PropLog_of_Length_1 | https://proofwiki.org/wiki/WFFs_of_PropLog_of_Length_1 | [
"Language of Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Length of String",
"Definition:Formal Language/Alphabet/Letter",
"Definition:Language of Propositional Logic/Formal Grammar",
"Definition:Top (Logic)",
"Definition:Bottom (Logic)"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/Bottom-Up Specification",
"Definition:Length of String",
"Definition:Length of String",
"Definition:Language of Propositional Logic/Formal Grammar"
] |
proofwiki-2287 | WFF of PropLog is Balanced | Let $\mathbf A$ be a WFF of propositional logic.
Then $\mathbf A$ is a balanced string. | We will prove by strong induction on $n$ that:
:All WFFs of length $n$ are balanced.
Let $\map l {\mathbf A}$ denote the number of left brackets in a string $\mathbf A$.
Let $\map r {\mathbf A}$ denote the number of right brackets in a string $\mathbf A$. | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Then $\mathbf A$ is a [[Definition:Balanced String|balanced string]]. | We will prove by [[Second Principle of Mathematical Induction|strong induction]] on $n$ that:
:All [[Definition:WFF of Propositional Logic|WFFs]] of [[Definition:Length of String|length]] $n$ are [[Definition:Balanced String|balanced]].
Let $\map l {\mathbf A}$ denote the number of left brackets in a [[Definition:St... | WFF of PropLog is Balanced | https://proofwiki.org/wiki/WFF_of_PropLog_is_Balanced | https://proofwiki.org/wiki/WFF_of_PropLog_is_Balanced | [
"Language of Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Balanced String"
] | [
"Second Principle of Mathematical Induction",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Length of String",
"Definition:Balanced String",
"Definition:String",
"Definition:String",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Length of S... |
proofwiki-2288 | Prefix of WFF of PropLog is not WFF | Let $\mathbf A$ be a WFF of propositional logic.
Let $\mathbf S$ be a prefix of $\mathbf A$.
Then $\mathbf S$ is not a WFF of propositional logic. | The proof proceeds by strong induction on the length of a WFF of propositional logic.
Let $\map l {\mathbf Q}$ denote the length of a string $\mathbf Q$.
For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
:A prefix of $\mathbf A$ such that $\map l {\mathbf A} = n$ is not a WFF of propositional logic.
By defin... | Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Let $\mathbf S$ be a [[Definition:Prefix|prefix]] of $\mathbf A$.
Then $\mathbf S$ is not a [[Definition:WFF of Propositional Logic|WFF of propositional logic]]. | The proof proceeds by [[Second Principle of Mathematical Induction|strong induction]] on the [[Definition:Length of String|length]] of a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Let $\map l {\mathbf Q}$ denote the [[Definition:Length of String|length]] of a [[Definition:String|string]] $\... | Prefix of WFF of PropLog is not WFF | https://proofwiki.org/wiki/Prefix_of_WFF_of_PropLog_is_not_WFF | https://proofwiki.org/wiki/Prefix_of_WFF_of_PropLog_is_not_WFF | [
"Language of Propositional Logic"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Prefix",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [
"Second Principle of Mathematical Induction",
"Definition:Length of String",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Length of String",
"Definition:String",
"Definition:Proposition",
"Definition:Prefix",
"Definition:Language of Propositional Logic/Formal Grammar/WF... |
proofwiki-2289 | Construction of Parallel Line | Given a straight line, and a given point not on that straight line, it is possible to draw a parallel to the given straight line.
{{:Euclid:Proposition/I/31}} | The transversal $AD$ cuts the lines $BC$ and $AE$ and makes $\angle DAE = \angle ADC$.
From Equal Alternate Angles implies Parallel Lines it follows that $EA \parallel BC$.
{{qed}}
{{Euclid Note|31|I}} | Given a [[Definition:Straight Line|straight line]], and a given [[Definition:Point|point]] not on that straight line, it is possible to draw a [[Definition:Parallel Lines|parallel]] to the given [[Definition:Straight Line|straight line]].
{{:Euclid:Proposition/I/31}} | The [[Definition:Transversal (Geometry)|transversal]] $AD$ cuts the lines $BC$ and $AE$ and makes $\angle DAE = \angle ADC$.
From [[Equal Alternate Angles implies Parallel Lines]] it follows that $EA \parallel BC$.
{{qed}}
{{Euclid Note|31|I}} | Construction of Parallel Line | https://proofwiki.org/wiki/Construction_of_Parallel_Line | https://proofwiki.org/wiki/Construction_of_Parallel_Line | [
"Parallel Lines"
] | [
"Definition:Line/Straight Line",
"Definition:Point",
"Definition:Parallel (Geometry)/Lines",
"Definition:Line/Straight Line"
] | [
"Definition:Transversal (Geometry)",
"Equal Alternate Angles implies Parallel Lines"
] |
proofwiki-2290 | Sum of Angles of Triangle equals Two Right Angles | In a triangle, the sum of the three interior angles equals two right angles.
{{:Euclid:Proposition/I/32}} | :300px
Let $\triangle ABC$ be a triangle.
Let $BC$ be extended to a point $D$.
From External Angle of Triangle equals Sum of other Internal Angles:
: $\angle ACD = \angle ABC + \angle BAC$
Bby by Euclid's Second Common Notion:
: $\angle ACB + \angle ACD = \angle ABC + \angle BAC + \angle ACB$
But from Two Angles on Str... | In a [[Definition:Triangle (Geometry)|triangle]], the sum of the three [[Definition:Internal Angle|interior angles]] equals two [[Definition:Right Angle|right angles]].
{{:Euclid:Proposition/I/32}} | :[[File:Euclid-I-32.png|300px]]
Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $BC$ be extended to a point $D$.
From [[External Angle of Triangle equals Sum of other Internal Angles]]:
: $\angle ACD = \angle ABC + \angle BAC$
Bby [[Axiom:Euclid's Common Notions|by Euclid's Second Common N... | Sum of Angles of Triangle equals Two Right Angles/Proof 1 | https://proofwiki.org/wiki/Sum_of_Angles_of_Triangle_equals_Two_Right_Angles | https://proofwiki.org/wiki/Sum_of_Angles_of_Triangle_equals_Two_Right_Angles/Proof_1 | [
"Sum of Angles of Triangle equals Two Right Angles",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Internal Angle",
"Definition:Right Angle"
] | [
"File:Euclid-I-32.png",
"Definition:Triangle (Geometry)",
"External Angle of Triangle equals Sum of other Internal Angles",
"Axiom:Euclid's Common Notions",
"Two Angles on Straight Line make Two Right Angles",
"Definition:Right Angle",
"Axiom:Euclid's Common Notions",
"Definition:Right Angle"
] |
proofwiki-2291 | Sum of Angles of Triangle equals Two Right Angles | In a triangle, the sum of the three interior angles equals two right angles.
{{:Euclid:Proposition/I/32}} | :480px
Let $\Delta ABC$ be a triangle.
Let $DAE$ be a line such that $DE \parallel BC$.
By Parallelism implies Equal Alternate Angles:
:$\angle DAB = \angle ABC$
and:
:$\angle EAC = \angle ACB$
Therefore, the sum of the three angles is:
:$\angle ABC + \angle BCA + \angle CAB = \angle DAB + \angle BAC + \angle CAE = 180... | In a [[Definition:Triangle (Geometry)|triangle]], the sum of the three [[Definition:Internal Angle|interior angles]] equals two [[Definition:Right Angle|right angles]].
{{:Euclid:Proposition/I/32}} | :[[File:TriangleWithLine.png|480px]]
Let $\Delta ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $DAE$ be a [[Definition:Line|line]] such that $DE \parallel BC$.
By [[Parallelism implies Equal Alternate Angles]]:
:$\angle DAB = \angle ABC$
and:
:$\angle EAC = \angle ACB$
Therefore, the sum of the three [... | Sum of Angles of Triangle equals Two Right Angles/Proof 2 | https://proofwiki.org/wiki/Sum_of_Angles_of_Triangle_equals_Two_Right_Angles | https://proofwiki.org/wiki/Sum_of_Angles_of_Triangle_equals_Two_Right_Angles/Proof_2 | [
"Sum of Angles of Triangle equals Two Right Angles",
"Triangles"
] | [
"Definition:Triangle (Geometry)",
"Definition:Polygon/Internal Angle",
"Definition:Right Angle"
] | [
"File:TriangleWithLine.png",
"Definition:Triangle (Geometry)",
"Definition:Line",
"Parallelism implies Equal Alternate Angles",
"Definition:Angle"
] |
proofwiki-2292 | Equivalence of Logical Implication and Conditional | :$\paren {p \implies q} \dashv \vdash \paren {p \vdash q}$
That is, the conditional is logically equivalent to logical implication. | This directly follows from:
* The Modus Ponendo Ponens: $p \implies q, p \vdash q$
* The Rule of Implication: $\left({p \vdash q}\right) \vdash p \implies q$.
{{qed}} | :$\paren {p \implies q} \dashv \vdash \paren {p \vdash q}$
That is, the [[Definition:Conditional|conditional]] is [[Definition:Logical Equivalence|logically equivalent]] to [[Definition:Logical Implication|logical implication]]. | This directly follows from:
* The [[Modus Ponendo Ponens]]: $p \implies q, p \vdash q$
* The [[Rule of Implication]]: $\left({p \vdash q}\right) \vdash p \implies q$.
{{qed}} | Equivalence of Logical Implication and Conditional | https://proofwiki.org/wiki/Equivalence_of_Logical_Implication_and_Conditional | https://proofwiki.org/wiki/Equivalence_of_Logical_Implication_and_Conditional | [
"Propositional Logic",
"Conditional"
] | [
"Definition:Conditional",
"Definition:Logical Equivalence",
"Definition:Logical Implication"
] | [
"Modus Ponendo Ponens",
"Rule of Implication"
] |
proofwiki-2293 | Soundness Theorem for Propositional Tableaux and Boolean Interpretations | Tableau proofs (in terms of propositional tableaux) are a sound proof system for boolean interpretations.
That is, for every WFF $\mathbf A$:
:$\vdash_{\mathrm{PT} } \mathbf A$ implies $\models_{\mathrm{BI} } \mathbf A$ | This is a corollary of the Extended Soundness Theorem for Propositional Tableaux and Boolean Interpretations:
Let $\mathbf H$ be a countable set of propositional formulas.
Let $\mathbf A$ be a propositional formula.
If $\mathbf H \vdash \mathbf A$, then $\mathbf H \models \mathbf A$.
In this case, we have $\mathbf H = ... | [[Definition:Tableau Proof (Propositional Tableaux)|Tableau proofs]] (in terms of [[Definition:Propositional Tableau|propositional tableaux]]) are a [[Definition:Sound Proof System|sound proof system]] for [[Definition:Boolean Interpretation|boolean interpretations]].
That is, for every [[Definition:WFF of Proposition... | This is a corollary of the [[Extended Soundness Theorem for Propositional Tableaux and Boolean Interpretations]]:
Let $\mathbf H$ be a [[Definition:Countable|countable]] set of [[Definition:Propositional Formula|propositional formulas]].
Let $\mathbf A$ be a [[Definition:Propositional Formula|propositional formula]].... | Soundness Theorem for Propositional Tableaux and Boolean Interpretations | https://proofwiki.org/wiki/Soundness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | https://proofwiki.org/wiki/Soundness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | [
"Propositional Tableaux",
"Named Theorems"
] | [
"Definition:Tableau Proof (Propositional Tableaux)",
"Definition:Propositional Tableau",
"Definition:Sound Proof System",
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [
"Extended Soundness Theorem for Propositional Tableaux and Boolean Interpretations",
"Definition:Countable Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] |
proofwiki-2294 | No Boolean Interpretation Models a WFF and its Negation | Let $v$ be a boolean interpretation.
Let $\mathbf A$ be a WFF of propositional logic.
Then $v$ can not model both $\mathbf A$ and $\neg \mathbf A$. | Suppose that $v$ models $\mathbf A$:
:$v \models \mathbf A$
Then $v \left({\mathbf A}\right) = T$ by definition of models.
By definition of boolean interpretation, $v \left({\neg \mathbf A}\right) = F$.
In particular, $v (\neg \mathbf A) \ne T$, so that:
:$v \not\models \neg \mathbf A$
Hence the result.
{{qed}}
Categor... | Let $v$ be a [[Definition:Boolean Interpretation|boolean interpretation]].
Let $\mathbf A$ be a [[Definition:WFF of Propositional Logic|WFF of propositional logic]].
Then $v$ can not [[Definition:Model (Boolean Interpretations)|model]] both $\mathbf A$ and $\neg \mathbf A$. | Suppose that $v$ [[Definition:Model (Boolean Interpretations)|models]] $\mathbf A$:
:$v \models \mathbf A$
Then $v \left({\mathbf A}\right) = T$ by definition of [[Definition:Model (Boolean Interpretations)|models]].
By definition of [[Definition:Boolean Interpretation|boolean interpretation]], $v \left({\neg \math... | No Boolean Interpretation Models a WFF and its Negation | https://proofwiki.org/wiki/No_Boolean_Interpretation_Models_a_WFF_and_its_Negation | https://proofwiki.org/wiki/No_Boolean_Interpretation_Models_a_WFF_and_its_Negation | [
"Propositional Logic"
] | [
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Model (Boolean Interpretations)"
] | [
"Definition:Model (Boolean Interpretations)",
"Definition:Model (Boolean Interpretations)",
"Definition:Boolean Interpretation",
"Category:Propositional Logic"
] |
proofwiki-2295 | Extended Soundness Theorem for Propositional Tableaux and Boolean Interpretations | Tableau proofs (in terms of propositional tableaux) are a strongly sound proof system for boolean interpretations.
That is, for every collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$:
:$\mathbf H \vdash_{\mathrm{PT}} \mathbf A$ implies $\mathbf H \models_{\mathrm{BI}} \mathbf A$ | By definition of tableau proof, $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$ means:
:There exists a tableau confutation of $\mathbf H \cup \set {\neg\mathbf A}$.
By Tableau Confutation implies Unsatisfiable, it follows that $\mathbf H \cup \set {\neg\mathbf A}$ is unsatisfiable for boolean interpretations.
Therefore, if ... | [[Definition:Tableau Proof (Propositional Tableaux)|Tableau proofs]] (in terms of [[Definition:Propositional Tableau|propositional tableaux]]) are a [[Definition:Strongly Sound Proof System|strongly sound proof system]] for [[Definition:Boolean Interpretation|boolean interpretations]].
That is, for every collection $\... | By definition of [[Definition:Tableau Proof (Propositional Tableaux)|tableau proof]], $\mathbf H \vdash_{\mathrm{PT}} \mathbf A$ means:
:There exists a [[Definition:Tableau Confutation|tableau confutation]] of $\mathbf H \cup \set {\neg\mathbf A}$.
By [[Tableau Confutation implies Unsatisfiable]], it follows that $\m... | Extended Soundness Theorem for Propositional Tableaux and Boolean Interpretations | https://proofwiki.org/wiki/Extended_Soundness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | https://proofwiki.org/wiki/Extended_Soundness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | [
"Propositional Tableaux",
"Named Theorems"
] | [
"Definition:Tableau Proof (Propositional Tableaux)",
"Definition:Propositional Tableau",
"Definition:Sound Proof System/Strongly Sound",
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [
"Definition:Tableau Proof (Propositional Tableaux)",
"Definition:Tableau Confutation",
"Tableau Confutation implies Unsatisfiable",
"Definition:Unsatisfiable",
"Definition:Boolean Interpretation",
"Definition:Boolean Interpretation",
"Definition:Model (Boolean Interpretations)",
"Definition:Unsatisfia... |
proofwiki-2296 | Extended Completeness Theorem for Propositional Tableaux and Boolean Interpretations | Tableau proofs (in terms of propositional tableaux) are a strongly complete proof system for boolean interpretations.
More precisely, for every countable collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$:
:$\mathbf H \models_{\mathrm{BI} } \mathbf A$ implies $\mathbf H \vdash_{\mathrm{PT} ... | Let $\mathbf A$ be a semantic consequence of $\mathbf H$ for boolean interpretations.
That is, if $v \models_{\mathrm{BI} } \mathbf H$, also $v \models_{\mathrm{BI} } \mathbf A$.
By the truth function for $\neg$, it follows that for such $v$:
:$v \not\models_{\mathrm{BI}} \neg \mathbf A$
Therefore, $\mathbf H' := \math... | [[Definition:Tableau Proof (Propositional Tableaux)|Tableau proofs]] (in terms of [[Definition:Propositional Tableau|propositional tableaux]]) are a [[Definition:Strongly Complete Proof System|strongly complete proof system]] for [[Definition:Boolean Interpretation|boolean interpretations]].
More precisely, for every ... | Let $\mathbf A$ be a [[Definition:Semantic Consequence|semantic consequence]] of $\mathbf H$ for [[Definition:Boolean Interpretation|boolean interpretations]].
That is, if $v \models_{\mathrm{BI} } \mathbf H$, also $v \models_{\mathrm{BI} } \mathbf A$.
By the [[Definition:Logical Not/Truth Function|truth function for... | Extended Completeness Theorem for Propositional Tableaux and Boolean Interpretations | https://proofwiki.org/wiki/Extended_Completeness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | https://proofwiki.org/wiki/Extended_Completeness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | [
"Completeness Theorem",
"Boolean Interpretations",
"Propositional Tableaux",
"Named Theorems"
] | [
"Definition:Tableau Proof (Propositional Tableaux)",
"Definition:Propositional Tableau",
"Definition:Complete Proof System/Strongly Complete",
"Definition:Boolean Interpretation",
"Definition:Countable Set",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Language of Proposi... | [
"Definition:Semantic Consequence",
"Definition:Boolean Interpretation",
"Definition:Logical Not/Truth Function",
"Definition:Unsatisfiable",
"Definition:Boolean Interpretation",
"Definition:Countable Set",
"Compactness Theorem for Boolean Interpretations",
"Definition:Finite Set",
"Definition:Unsati... |
proofwiki-2297 | Completeness Theorem for Propositional Tableaux and Boolean Interpretations | Tableau proofs (in terms of propositional tableaux) are a complete proof system for boolean interpretations.
That is, for every WFF $\mathbf A$:
:$\models_{\mathrm{BI} } \mathbf A$ implies $\vdash_{\mathrm{PT} } \mathbf A$ | This is a corollary of the Extended Completeness Theorem for Propositional Tableaux and Boolean Interpretations.
Namely, it is the special case $\mathbf H = \O$.
Hence the result.
{{qed}} | [[Definition:Tableau Proof (Propositional Tableaux)|Tableau proofs]] (in terms of [[Definition:Propositional Tableau|propositional tableaux]]) are a [[Definition:Complete Proof System|complete proof system]] for [[Definition:Boolean Interpretation|boolean interpretations]].
That is, for every [[Definition:WFF of Propo... | This is a corollary of the [[Extended Completeness Theorem for Propositional Tableaux and Boolean Interpretations]].
Namely, it is the special case $\mathbf H = \O$.
Hence the result.
{{qed}} | Completeness Theorem for Propositional Tableaux and Boolean Interpretations | https://proofwiki.org/wiki/Completeness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | https://proofwiki.org/wiki/Completeness_Theorem_for_Propositional_Tableaux_and_Boolean_Interpretations | [
"Completeness Theorem",
"Propositional Tableaux",
"Named Theorems"
] | [
"Definition:Tableau Proof (Propositional Tableaux)",
"Definition:Propositional Tableau",
"Definition:Complete Proof System",
"Definition:Boolean Interpretation",
"Definition:Language of Propositional Logic/Formal Grammar/WFF"
] | [
"Extended Completeness Theorem for Propositional Tableaux and Boolean Interpretations"
] |
proofwiki-2298 | König's Tree Lemma | Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children.
Then $T$ has a branch of infinite length. | We will show that we can choose an infinite sequence of nodes $t_0, t_1, t_2, \ldots$ of $T$ such that:
* $t_0$ is the root node;
* $t_{n + 1}$ is a child of $t_n$;
* Each $t_n$ has infinitely many descendants.
Then the sequence $t_0, t_1, t_2, \ldots$ is such a branch of infinite length.
Take the root node $t_0$.
By d... | Let $T$ be a [[Definition:Rooted Tree|rooted tree]] with an [[Definition:Infinite Set|infinite number]] of [[Definition:Node of Tree|nodes]], each with a [[Definition:Finite Set|finite number]] of [[Definition:Child Node|children]].
Then $T$ has a [[Definition:Branch (Graph Theory)|branch]] of [[Definition:Infinite Br... | We will show that we can choose an [[Definition:Infinite Sequence|infinite sequence]] of [[Definition:Node of Tree|nodes]] $t_0, t_1, t_2, \ldots$ of $T$ such that:
* $t_0$ is the [[Definition:Root Node|root node]];
* $t_{n + 1}$ is a [[Definition:Child Node|child]] of $t_n$;
* Each $t_n$ has [[Definition:Infinite Se... | König's Tree Lemma/Proof 1 | https://proofwiki.org/wiki/König's_Tree_Lemma | https://proofwiki.org/wiki/König's_Tree_Lemma/Proof_1 | [
"Tree Theory",
"König's Tree Lemma"
] | [
"Definition:Rooted Tree",
"Definition:Infinite Set",
"Definition:Tree (Graph Theory)/Node",
"Definition:Finite Set",
"Definition:Rooted Tree/Child Node",
"Definition:Rooted Tree/Branch",
"Definition:Rooted Tree/Branch/Infinite",
"Definition:Rooted Tree/Branch/Length"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Tree (Graph Theory)/Node",
"Definition:Rooted Tree/Root Node",
"Definition:Rooted Tree/Child Node",
"Definition:Infinite Set",
"Definition:Rooted Tree/Descendant",
"Definition:Rooted Tree/Branch",
"Definition:Rooted Tree/Branch/Infinite",
"Definit... |
proofwiki-2299 | König's Tree Lemma | Let $T$ be a rooted tree with an infinite number of nodes, each with a finite number of children.
Then $T$ has a branch of infinite length. | We will show that we can choose an infinite sequence of nodes $t_0, t_1, t_2, \ldots$ of $T$ such that:
:$t_0$ is the root node
:$t_{n + 1}$ is a child of $t_n$
Then the sequence $t_0, t_1, t_2, \ldots$ is such a branch of infinite length.
Let $I$ be the set of all nodes in $T$ that have infinitely many descendants.
De... | Let $T$ be a [[Definition:Rooted Tree|rooted tree]] with an [[Definition:Infinite Set|infinite number]] of [[Definition:Node of Tree|nodes]], each with a [[Definition:Finite Set|finite number]] of [[Definition:Child Node|children]].
Then $T$ has a [[Definition:Branch (Graph Theory)|branch]] of [[Definition:Infinite Br... | We will show that we can choose an [[Definition:Infinite Sequence|infinite sequence]] of [[Definition:Node of Tree|nodes]] $t_0, t_1, t_2, \ldots$ of $T$ such that:
:$t_0$ is the [[Definition:Root Node|root node]]
:$t_{n + 1}$ is a [[Definition:Child Node|child]] of $t_n$
Then the sequence $t_0, t_1, t_2, \ldots$ is ... | König's Tree Lemma/Proof 2 | https://proofwiki.org/wiki/König's_Tree_Lemma | https://proofwiki.org/wiki/König's_Tree_Lemma/Proof_2 | [
"Tree Theory",
"König's Tree Lemma"
] | [
"Definition:Rooted Tree",
"Definition:Infinite Set",
"Definition:Tree (Graph Theory)/Node",
"Definition:Finite Set",
"Definition:Rooted Tree/Child Node",
"Definition:Rooted Tree/Branch",
"Definition:Rooted Tree/Branch/Infinite",
"Definition:Rooted Tree/Branch/Length"
] | [
"Definition:Sequence/Infinite Sequence",
"Definition:Tree (Graph Theory)/Node",
"Definition:Rooted Tree/Root Node",
"Definition:Rooted Tree/Child Node",
"Definition:Rooted Tree/Branch",
"Definition:Rooted Tree/Branch/Infinite",
"Definition:Rooted Tree/Branch/Length",
"Definition:set",
"Definition:In... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.