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-100 | De Morgan's Laws (Set Theory) | {{:De Morgan's Laws (Set Theory)/Set Difference}} | {{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 (Set Theory)/Set Difference}} | {{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_(Set_Theory) | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_1 | [
"Set Difference",
"Set Union",
"Set Intersection",
"Relative Complement",
"Set Complement",
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Set Theory)/Set Difference/Difference with Intersection"
] |
proofwiki-101 | De Morgan's Laws (Set Theory) | {{:De Morgan's Laws (Set Theory)/Set Difference}} | {{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 Difference}} | {{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_(Set_Theory) | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_2 | [
"Set Difference",
"Set Union",
"Set Intersection",
"Relative Complement",
"Set Complement",
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-102 | De Morgan's Laws (Set Theory) | {{:De Morgan's Laws (Set Theory)/Set Difference}} | {{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 (Set Theory)/Set Difference}} | {{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_(Set_Theory) | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Intersection/Proof_3 | [
"Set Difference",
"Set Union",
"Set Intersection",
"Relative Complement",
"Set Complement",
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Union",
"Complement of Complement",
"Definition:Set Complement",
"Complement of Complement"
] |
proofwiki-103 | De Morgan's Laws (Set Theory) | {{:De Morgan's Laws (Set Theory)/Set Difference}} | {{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 Difference}} | {{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_(Set_Theory) | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Complement/Complement_of_Union/Proof_2 | [
"Set Difference",
"Set Union",
"Set Intersection",
"Relative Complement",
"Set Complement",
"De Morgan's Laws"
] | [] | [
"De Morgan's Laws (Logic)/Conjunction of Negations",
"Definition:Set Equality/Definition 1"
] |
proofwiki-104 | De Morgan's Laws (Set Theory) | {{:De Morgan's Laws (Set Theory)/Set Difference}} | 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 (Set Theory)/Set Difference}} | 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_(Set_Theory) | https://proofwiki.org/wiki/De_Morgan's_Laws_(Set_Theory)/Set_Difference/General_Case/Difference_with_Intersection/Proof | [
"Set Difference",
"Set Union",
"Set Intersection",
"Relative Complement",
"Set Complement",
"De Morgan's Laws"
] | [] | [
"Set Difference is Subset",
"De Morgan's Laws (Predicate Logic)/Denial of Universality"
] |
proofwiki-105 | De Moivre's Formula | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | $\cos x + i \sin x$ is a complex number expressed in polar form $\left\langle{r, \theta}\right\rangle$ whose complex modulus is $1$ and whose argument is $x$.
From De Moivre's Formula: Positive Integer Index:
:$\forall n \in \Z_{>0}: \left({r \left({\cos x + i \sin x}\right)}\right)^n = r^n \left({\cos \left({n x}\righ... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | $\cos x + i \sin x$ is a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|polar form]] $\left\langle{r, \theta}\right\rangle$ whose [[Definition:Complex Modulus|complex modulus]] is $1$ and whose [[Definition:Argument of Complex Number|argument]] is $x$.
From [[De Moi... | De Moivre's Formula/Positive Integer Index/Corollary/Proof 1 | https://proofwiki.org/wiki/De_Moivre's_Formula | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Corollary/Proof_1 | [
"De Moivre's Formula",
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form",
"Definition:Complex Modulus",
"Definition:Argument of Complex Number",
"De Moivre's Formula/Positive Integer Index"
] |
proofwiki-106 | De Moivre's Formula | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | {{begin-eqn}}
{{eqn | l = \paren {\cos \theta + i \sin \theta}^n
| r = \paren {e^{i \theta} }^n
| c = Euler's Formula
}}
{{eqn | r = e^{i n \theta}
| c = Exponential of Product
}}
{{eqn | r = \cos n \theta + i \sin n \theta
| c = Euler's Formula
}}
{{end-eqn}} | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | {{begin-eqn}}
{{eqn | l = \paren {\cos \theta + i \sin \theta}^n
| r = \paren {e^{i \theta} }^n
| c = [[Euler's Formula]]
}}
{{eqn | r = e^{i n \theta}
| c = [[Exponential of Product]]
}}
{{eqn | r = \cos n \theta + i \sin n \theta
| c = [[Euler's Formula]]
}}
{{end-eqn}} | De Moivre's Formula/Positive Integer Index/Corollary/Proof 2 | https://proofwiki.org/wiki/De_Moivre's_Formula | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Corollary/Proof_2 | [
"De Moivre's Formula",
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Euler's Formula",
"Exponential of Product",
"Euler's Formula"
] |
proofwiki-107 | De Moivre's Formula | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | Proof by induction:
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$
$\map P 1$ is the case:
:$\paren {r \paren {\cos x + i \sin x} }^1 = r^1 \paren {\map \cos {1 x} + i \, \map \sin {1 x} }$
which is triviall... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | Proof by [[Principle of Mathematical Induction|induction]]:
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\paren {r \paren {\cos x + i \sin x} }^n = r^n \paren {\map \cos {n x} + i \, \map \sin {n x} }$
$\map P 1$ is the case:
:$\paren {r \paren {\cos x + i \sin x} }^1 = ... | De Moivre's Formula/Positive Integer Index/Proof 1 | https://proofwiki.org/wiki/De_Moivre's_Formula | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Proof_1 | [
"De Moivre's Formula",
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Product of Complex Numbers in Polar Form",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"Principle of Mathematical Induction",
"De Moivre's Formula/Positive Integer Index/Proof 1",
"Product of Complex Nu... |
proofwiki-108 | De Moivre's Formula | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | From Product of Complex Numbers in Polar Form: General Result:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \, \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Setting $z_1 = z_2 = \cdots = z_n = r \paren {\cos x + i \sin x}$ gives the result. | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | From [[Product of Complex Numbers in Polar Form/General Result|Product of Complex Numbers in Polar Form: General Result]]:
:$z_1 z_2 \cdots z_n = r_1 r_2 \cdots r_n \paren {\map \cos {\theta_1 + \theta_2 + \cdots + \theta_n} + i \, \map \sin {\theta_1 + \theta_2 + \cdots + \theta_n} }$
Setting $z_1 = z_2 = \cdots = z_... | De Moivre's Formula/Positive Integer Index/Proof 2 | https://proofwiki.org/wiki/De_Moivre's_Formula | https://proofwiki.org/wiki/De_Moivre's_Formula/Positive_Integer_Index/Proof_2 | [
"De Moivre's Formula",
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Product of Complex Numbers in Polar Form/General Result"
] |
proofwiki-109 | De Moivre's Formula | Let $z \in \C$ be a complex number expressed in complex form:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | {{begin-eqn}}
{{eqn | l = \paren {r \paren {\cos x + i \sin x} }^\omega
| r = \paren {r e^{i x} }^\omega
| c = Euler's Formula
}}
{{eqn | r = r^\omega e^{i \omega x}
| c = Power of Power
}}
{{eqn | r = r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }
| c = Euler's Formula
}}
{{en... | Let $z \in \C$ be a [[Definition:Complex Number|complex number]] expressed in [[Definition:Polar Form of Complex Number|complex form]]:
:$z = r \paren {\cos x + i \sin x}$
Then:
:$\forall \omega \in \C: \paren {r \paren {\cos x + i \sin x} }^\omega = r^\omega \paren {\map \cos {\omega x} + i \, \map \sin {\omega x} }$ | {{begin-eqn}}
{{eqn | l = \paren {r \paren {\cos x + i \sin x} }^\omega
| r = \paren {r e^{i x} }^\omega
| c = [[Euler's Formula]]
}}
{{eqn | r = r^\omega e^{i \omega x}
| c = [[Power of Power]]
}}
{{eqn | r = r^\omega \paren {\map \cos {\omega x} + i \map \sin {\omega x} }
| c = [[Euler's Formu... | De Moivre's Formula/Proof 1 | https://proofwiki.org/wiki/De_Moivre's_Formula | https://proofwiki.org/wiki/De_Moivre's_Formula/Proof_1 | [
"De Moivre's Formula",
"Complex Analysis"
] | [
"Definition:Complex Number",
"Definition:Complex Number/Polar Form"
] | [
"Euler's Formula",
"Exponent Combination Laws/Power of Power",
"Euler's Formula"
] |
proofwiki-110 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | This can be done in sections. | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | This can be done in sections. | Power Rule for Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [] |
proofwiki-111 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = x^{1 / n}$.
From the definition of the power to a rational number, or alternatively from the definition of the root of a number, $\map f x$ is defined when $x \ge 0$.
(However, see the special case where $x = 0$.)
From Continuity of Root Function, $\map f x$ is continuous over... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = x^{1 / n}$.
From the definition of the [[Definition:Rational Power|power to a rational number]], or alternatively from the definition of the [[Definition:Root of Number|root]] of a [[Definition:Number|number]], $\map f x$ is defined when $x \ge 0$.
(However, see the [[Defin... | Power Rule for Derivatives/Fractional Index/Proof 1 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Fractional_Index/Proof_1 | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Power (Algebra)/Rational Number",
"Definition:Root of Number",
"Definition:Number",
"Definition:Power (Algebra)/Power of Zero",
"Continuity of Root Function",
"Definition:Continuous Real Function/Interval",
"Definition:Real Interval/Open",
"Definition:Continuous Real Function/Right-Continu... |
proofwiki-112 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = y = x^{1/n}$.
Thus $\map {f^{-1} } y = x = y^n$ from the definition of root.
So:
{{begin-eqn}}
{{eqn | l = D x^{1/n}
| r = \frac 1 {D y^n}
| c = Derivative of Inverse Function
}}
{{eqn | r = \frac 1 {n y^{n - 1} }
| c = Power Rule for Derivatives: Integer Ind... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $n \in \N_{>0}$.
Thus, let $\map f x = y = x^{1/n}$.
Thus $\map {f^{-1} } y = x = y^n$ from the definition of [[Definition:Root of Number|root]].
So:
{{begin-eqn}}
{{eqn | l = D x^{1/n}
| r = \frac 1 {D y^n}
| c = [[Derivative of Inverse Function]]
}}
{{eqn | r = \frac 1 {n y^{n - 1} }
| c = [... | Power Rule for Derivatives/Fractional Index/Proof 2 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Fractional_Index/Proof_2 | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Root of Number",
"Derivative of Inverse Function",
"Power Rule for Derivatives/Integer Index"
] |
proofwiki-113 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
By the definition of the derivative:
:$\ds \dfrac \d {\d x} \map f x = \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h = \lim_{h \mathop \to 0} \dfrac {\paren {x + h}^n - x^n} h$
Using the Binomial Theorem this simplifies to:
{{begin-eqn}}
{{eqn | o =
| ... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
By the definition of the [[Definition:Derivative|derivative]]:
:$\ds \dfrac \d {\d x} \map f x = \lim_{h \mathop \to 0} \dfrac {\map f {x + h} - \map f x} h = \lim_{h \mathop \to 0} \dfrac {\paren {x + h}^n - x^n} h$
Using the [[Binomial Theorem/Integral Index|Binomial ... | Power Rule for Derivatives/Natural Number Index/Proof by Binomial Theorem | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Binomial_Theorem | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Derivative",
"Binomial Theorem/Integral Index",
"Binomial Coefficient with One"
] |
proofwiki-114 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the derivative:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
{{begin-eqn}}
{{eqn | l = \map {f'} ... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Let $\map f x = x^n$ for $x \in \R, n \in \N$.
Let $a \in \R$.
By definition of the [[Definition:Derivative|derivative]]:
:$\ds \map {f'} a = \lim_{x \mathop \to a} \frac {\map f x - \map f a} {x - a} = \lim_{x \mathop \to a} \frac {x^n - a^n} {x - a}$
=== Case $\text I$ ===
For $n = 0$ it is possible to do:
{{be... | Power Rule for Derivatives/Natural Number Index/Proof by Difference of Two Powers | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Difference_of_Two_Powers | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Derivative",
"Derivative of Identity Function/Real",
"Definition:Commutative Ring",
"Difference of Two Powers",
"Real Polynomial Function is Continuous"
] |
proofwiki-115 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We will use the notation $D \map f x = \map {f'} x$ as it is convenient.
Let $n = 0$.
Then:
:$\forall x \in \R: x^n = 1$
Thus $\map f x$ is the constant function $\map {f_1} x$ on $\R$.
Thus from Derivative of Constant, $D \map f x = \map D {x^0} = 0 x^{-1}$, except where $x = 0$.
So the result holds for $n = 0$.
Let $... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We will use the notation $D \map f x = \map {f'} x$ as it is convenient.
Let $n = 0$.
Then:
:$\forall x \in \R: x^n = 1$
Thus $\map f x$ is the [[Definition:Constant Mapping|constant function]] $\map {f_1} x$ on $\R$.
Thus from [[Derivative of Constant]], $D \map f x = \map D {x^0} = 0 x^{-1}$, except where $x = ... | Power Rule for Derivatives/Natural Number Index/Proof by Induction | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Natural_Number_Index/Proof_by_Induction | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Constant Mapping",
"Derivative of Constant",
"Derivative of Identity Function",
"Product Rule for Derivatives",
"Principle of Mathematical Induction"
] |
proofwiki-116 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We are going to prove that $\map {f'} x = n x^{n - 1}$ holds for all real $n$.
To do this, we compute the limit $\ds \lim_{h \mathop \to 0} \frac {\paren {x + h}^n - x^n} h$:
{{begin-eqn}}
{{eqn | l = \frac {\paren {x + h}^n - x^n} h
| r = \frac {x^n} h \paren {\paren {1 + \frac h x}^n - 1}
| c =
}}
{{eqn ... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | We are going to prove that $\map {f'} x = n x^{n - 1}$ holds for all [[Definition:Real Number|real]] $n$.
To do this, we compute the limit $\ds \lim_{h \mathop \to 0} \frac {\paren {x + h}^n - x^n} h$:
{{begin-eqn}}
{{eqn | l = \frac {\paren {x + h}^n - x^n} h
| r = \frac {x^n} h \paren {\paren {1 + \frac h x}... | Power Rule for Derivatives/Real Number Index/Proof 1 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Real_Number_Index/Proof_1 | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Real Number",
"Derivative of Exponential at Zero",
"Derivative of Logarithm at One"
] |
proofwiki-117 | Power Rule for Derivatives | Let $n \in \R$.
Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Note this proof does not hold for $x = 0$.
Let $y$ = $\map f x$.
Then $y = x^n$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = x^n
}}
{{eqn | ll= \leadsto
| l = \size y
| r = \size {x^n}
| c = taking the absolute value of both sides
}}
{{eqn | r = \size x^n
| c = Absolute Value of Power
}}
{{eqn... | Let $n \in \R$.
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as $\map f x = x^n$.
Then:
:$\map {f'} x = n x^{n - 1}$
everywhere that $\map f x = x^n$ is defined.
When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined. | Note this proof does not hold for $x = 0$.
Let $y$ = $\map f x$.
Then $y = x^n$.
Then:
{{begin-eqn}}
{{eqn | l = y
| r = x^n
}}
{{eqn | ll= \leadsto
| l = \size y
| r = \size {x^n}
| c = taking the [[Definition:Absolute Value|absolute value]] of both sides
}}
{{eqn | r = \size x^n
| c ... | Power Rule for Derivatives/Real Number Index/Proof 2 | https://proofwiki.org/wiki/Power_Rule_for_Derivatives | https://proofwiki.org/wiki/Power_Rule_for_Derivatives/Real_Number_Index/Proof_2 | [
"Power Rule for Derivatives",
"Derivatives"
] | [
"Definition:Real Function"
] | [
"Definition:Absolute Value",
"Absolute Value of Power",
"Definition:Natural Logarithm",
"Logarithm of Power",
"Derivative of Composite Function",
"Derivative of Constant Multiple",
"Exponent Combination Laws/Quotient of Powers"
] |
proofwiki-118 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | By Riemann Zeta Function as a Multiple Integral,
:$\ds \map \zeta 2 = \int_0^1 \int_0^1 \frac 1 {1 - x y} \rd x\rd y$
Let $\tuple {u, v} = \tuple {\dfrac {x + y} 2, \dfrac{y - x} 2}$ so that:
:$\tuple {x, y} = \tuple {u - v, u + v}$
The Jacobian is:
:$\size J = \size {\dfrac {\partial \tuple {x, y} } {\partial \tuple... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | By [[Riemann Zeta Function as a Multiple Integral]],
:$\ds \map \zeta 2 = \int_0^1 \int_0^1 \frac 1 {1 - x y} \rd x\rd y$
Let $\tuple {u, v} = \tuple {\dfrac {x + y} 2, \dfrac{y - x} 2}$ so that:
:$\tuple {x, y} = \tuple {u - v, u + v}$
The [[Definition:Jacobian Determinant|Jacobian]] is:
:$\size J = \size {\dfrac... | Basel Problem/Proof 1 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_1 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Riemann Zeta Function as a Multiple Integral",
"Definition:Jacobian/Determinant",
"Change of Variables Theorem (Multivariable Calculus)",
"Integration by Substitution",
"Definition:Triangle (Geometry)/Right-Angled",
"Definition:Polygon/Side",
"Pythagoras's Theorem",
"Sum of Squares of Sine and Cosine... |
proofwiki-119 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | For $z \ne 0$, from Mittag-Leffler Expansion for Hyperbolic Cotangent Function, we have:
:$\ds \frac 1 {2 z} \paren {\pi \map \coth {\pi z} - \frac 1 z} = \sum_{n \mathop = 1}^\infty \frac 1 {z^2 + n^2}$
Consider the set:
:$A = \set {z \in \C : \size z \le \dfrac 1 2}$.
Then for each $n \in \N$ and $z \in A$, we have... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | For $z \ne 0$, from [[Mittag-Leffler Expansion for Hyperbolic Cotangent Function]], we have:
:$\ds \frac 1 {2 z} \paren {\pi \map \coth {\pi z} - \frac 1 z} = \sum_{n \mathop = 1}^\infty \frac 1 {z^2 + n^2}$
Consider the set:
:$A = \set {z \in \C : \size z \le \dfrac 1 2}$.
Then for each $n \in \N$ and $z \in A$,... | Basel Problem/Proof 10 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_10 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Mittag-Leffler Expansion for Hyperbolic Cotangent Function",
"Reverse Triangle Inequality",
"Fundamental Theorem of Calculus",
"Primitive of Power",
"Cauchy Integral Test",
"Definition:Convergent Series",
"Definition:Convergent Series",
"Weierstrass M-Test",
"Definition:Uniform Convergence/Infinite... |
proofwiki-120 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | Let:
:$\ds P_k = x \prod_{n \mathop = 1}^k \paren {1 - \frac {x^2} {n^2 \pi^2} }$
We note that:
{{begin-eqn}}
{{eqn | l = P_k - P_{k - 1}
| r = \paren {-\frac {x^3} {k^2 \pi^2} } \prod_{n \mathop = 1}^{k - 1} \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = -\frac {x^3} {k^2 \pi ^2} + \map O {x^5... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | Let:
:$\ds P_k = x \prod_{n \mathop = 1}^k \paren {1 - \frac {x^2} {n^2 \pi^2} }$
We note that:
{{begin-eqn}}
{{eqn | l = P_k - P_{k - 1}
| r = \paren {-\frac {x^3} {k^2 \pi^2} } \prod_{n \mathop = 1}^{k - 1} \paren {1 - \frac {x^2} {n^2 \pi^2} }
| c =
}}
{{eqn | r = -\frac {x^3} {k^2 \pi ^2} + \map O {x... | Basel Problem/Proof 2 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_2 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Definition:Telescoping Series",
"Definition:Coefficient",
"Definition:Sine",
"Euler Formula for Sine Function",
"Power Series Expansion for Sine Function",
"Definition:Limit of Sequence/Real Numbers",
"Power Series Expansion for Sine Function"
] |
proofwiki-121 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | Let $x \in \openint 0 {\dfrac \pi 2}$ and let $n$ be a non-negative integer.
{{begin-eqn}}
{{eqn | l = \frac {\map \cos {2 n + 1} x + i \map \sin {2 n + 1} x} {\sin^{2 n + 1} x}
| r = \frac {\paren {\cos x + i \sin x}^{2 n + 1} } {\sin^{2 n + 1} x}
| c = De Moivre's Formula
}}
{{eqn | r = \paren {\cot x + ... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | Let $x \in \openint 0 {\dfrac \pi 2}$ and let $n$ be a [[Definition:Positive Integer|non-negative integer]].
{{begin-eqn}}
{{eqn | l = \frac {\map \cos {2 n + 1} x + i \map \sin {2 n + 1} x} {\sin^{2 n + 1} x}
| r = \frac {\paren {\cos x + i \sin x}^{2 n + 1} } {\sin^{2 n + 1} x}
| c = [[De Moivre's Formu... | Basel Problem/Proof 3 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_3 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Definition:Positive/Integer",
"De Moivre's Formula",
"Binomial Theorem",
"Definition:Complex Number/Imaginary Part",
"Definition:Distinct",
"Definition:Real Interval/Open",
"Shape of Cotangent Function",
"Definition:Positive",
"Definition:Injective",
"Definition:Real Interval/Open",
"Definition... |
proofwiki-122 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | From Sum of Reciprocals of Squares of Odd Integers:
:$\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} = \frac {\pi^2} 8$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {n^2}
| r = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^2} + \sum_{n \mathop = 0}^\infty \frac 1 {\paren ... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | From [[Sum of Reciprocals of Squares of Odd Integers]]:
:$\ds \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1}^2} = \frac {\pi^2} 8$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {n^2}
| r = \sum_{n \mathop = 1}^\infty \frac 1 {\paren {2 n}^2} + \sum_{n \mathop = 0}^\infty \frac 1 ... | Basel Problem/Proof 4 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_4 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Sum of Reciprocals of Squares of Odd Integers"
] |
proofwiki-123 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | Let the function $f_n$ be defined as:
:$(1): \quad \map {f_n} x := \dfrac 1 2 + \cos x + \cos 2 x + \cdots + \cos n x$
By Lagrange's Cosine Identity:
:$(2): \quad \map {f_n} x = \dfrac {\map \sin {\paren {2 n + 1} x / 2} } {2 \map \sin {x / 2} }$
Let $E_n$ be defined as:
{{begin-eqn}}
{{eqn | l = E_n
| r = \int_0... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | Let the [[Definition:Real Function|function]] $f_n$ be defined as:
:$(1): \quad \map {f_n} x := \dfrac 1 2 + \cos x + \cos 2 x + \cdots + \cos n x$
By [[Lagrange's Cosine Identity]]:
:$(2): \quad \map {f_n} x = \dfrac {\map \sin {\paren {2 n + 1} x / 2} } {2 \map \sin {x / 2} }$
Let $E_n$ be defined as:
{{begin-eqn}... | Basel Problem/Proof 5 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_5 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Definition:Real Function",
"Lagrange's Trigonometric Identities/Cosine",
"Primitive of x by Cosine of a x",
"Definition:Even Integer",
"Definition:Real Function",
"Integration by Parts",
"Definition:Increasing/Mapping",
"Definition:Real Interval/Closed",
"Definition:Bounded Mapping",
"Definition:... |
proofwiki-124 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | {{begin-eqn}}
{{eqn | l = \map \zeta 2
| r = \paren {-1}^2 \frac {B_2 2^1 \pi^2} {2!}
| c = Riemann Zeta Function at Even Integers
}}
{{eqn | r = \paren {-1}^2 \paren {\frac 1 6} \frac {2^1 \pi^2} {2!}
| c = {{Defof|Sequence of Bernoulli Numbers}}
}}
{{eqn | r = \paren {\frac 1 6} \paren {\frac 2 2} \... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | {{begin-eqn}}
{{eqn | l = \map \zeta 2
| r = \paren {-1}^2 \frac {B_2 2^1 \pi^2} {2!}
| c = [[Riemann Zeta Function at Even Integers]]
}}
{{eqn | r = \paren {-1}^2 \paren {\frac 1 6} \frac {2^1 \pi^2} {2!}
| c = {{Defof|Sequence of Bernoulli Numbers}}
}}
{{eqn | r = \paren {\frac 1 6} \paren {\frac 2 ... | Basel Problem/Proof 6 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_6 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Riemann Zeta Function at Even Integers"
] |
proofwiki-125 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | By Fourier Series of $x^2$, for $x \in \openint {-\pi} \pi$:
:$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Letting $x \to \pi$ from the left:
{{begin-eqn}}
{{eqn | l = \pi^2
| r = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | By [[Fourier Series of x squared|Fourier Series of $x^2$]], for $x \in \openint {-\pi} \pi$:
:$\ds x^2 = \frac {\pi^2} 3 + \sum_{n \mathop = 1}^\infty \paren {\paren {-1}^n \frac 4 {n^2} \cos n x}$
Letting $x \to \pi$ [[Definition:Limit from Left|from the left]]:
{{begin-eqn}}
{{eqn | l = \pi^2
| r = \frac {\p... | Basel Problem/Proof 7 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_7 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Fourier Series/x squared over Minus Pi to Pi",
"Definition:Limit of Real Function/Left",
"Cosine of Integer Multiple of Pi"
] |
proofwiki-126 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | By Fourier Series of Identity Function over $-\pi$ to $\pi$, for $x \in \openint {-\pi} \pi$:
:$\ds x \sim 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \map \sin {n x}$
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 \pi \int_{-\pi}^\pi x^2 \rd x
| r = \sum_{n \mathop = 1}^\infty \paren {\frac{2 \paren ... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | By [[Fourier Series/Identity Function over Minus Pi to Pi|Fourier Series of Identity Function over $-\pi$ to $\pi$]], for $x \in \openint {-\pi} \pi$:
:$\ds x \sim 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n \map \sin {n x}$
Hence:
{{begin-eqn}}
{{eqn | l = \frac 1 \pi \int_{-\pi}^\pi x^2 \rd x
... | Basel Problem/Proof 8 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_8 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Fourier Series/Identity Function over Minus Pi to Pi",
"Parseval's Theorem",
"Definite Integral of Even Function"
] |
proofwiki-127 | Basel Problem | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the Riemann zeta function. | Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as:
:$\map f x = \begin {cases} \paren {x - \pi}^2 & : 0 < x \le \pi \\ \pi^2 & : \pi < x < 2 \pi \end {cases}$
From Fourier Series: Square of x minus pi, Square of pi, its Fourier series can be expressed as:
:$\map f x \sim \ds \dfrac {2 \pi^2} 3 + \s... | :$\ds \map \zeta 2 = \sum_{n \mathop = 1}^\infty {\frac 1 {n^2} } = \frac {\pi^2} 6$
where $\zeta$ denotes the [[Definition:Riemann Zeta Function|Riemann zeta function]]. | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 {2 \pi}$ as:
:$\map f x = \begin {cases} \paren {x - \pi}^2 & : 0 < x \le \pi \\ \pi^2 & : \pi < x < 2 \pi \end {cases}$
From [[Fourier Series/Square of x minus pi, Square of pi|Fourier Series: Square of x minus pi, Square of pi]]... | Basel Problem/Proof 9 | https://proofwiki.org/wiki/Basel_Problem | https://proofwiki.org/wiki/Basel_Problem/Proof_9 | [
"Basel Problem",
"Riemann Zeta Function at Even Integers",
"Examples of Riemann Zeta Function",
"Formulas for Pi",
"Named Theorems"
] | [
"Definition:Riemann Zeta Function"
] | [
"Definition:Real Function",
"Fourier Series/Square of x minus pi, Square of pi",
"Definition:Fourier Series/Range 2 Pi",
"Sine of Zero is Zero",
"Cosine of Zero is One"
] |
proofwiki-128 | 1+2+...+n+(n-1)+...+1 = n^2 | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | {{begin-eqn}}
{{eqn | o =
| r = 1 + 2 + \cdots + \paren {n - 1} + n + \paren {n - 1} + \cdots + 1
| c =
}}
{{eqn | r = 1 + 2 + \cdots + \paren {n - 1} + \paren {n - 1} + \cdots + 1 + n
| c =
}}
{{eqn | r = 2 \paren {1 + 2 + \cdots + \paren {n - 1} } + n
| c =
}}
{{eqn | r = 2 \paren {\frac {\... | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | {{begin-eqn}}
{{eqn | o =
| r = 1 + 2 + \cdots + \paren {n - 1} + n + \paren {n - 1} + \cdots + 1
| c =
}}
{{eqn | r = 1 + 2 + \cdots + \paren {n - 1} + \paren {n - 1} + \cdots + 1 + n
| c =
}}
{{eqn | r = 2 \paren {1 + 2 + \cdots + \paren {n - 1} } + n
| c =
}}
{{eqn | r = 2 \paren {\frac {\... | 1+2+...+n+(n-1)+...+1 = n^2/Proof 1 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2/Proof_1 | [
"Sums of Sequences",
"Square Numbers",
"1+2+...+n+(n-1)+...+1 = n^2"
] | [] | [
"Closed Form for Triangular Numbers"
] |
proofwiki-129 | 1+2+...+n+(n-1)+...+1 = n^2 | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | {{begin-eqn}}
{{eqn | o =
| r = 1 + 2 + \cdots + \paren {n - 1} + n + \paren {n - 1} + \cdots + 1
| c =
}}
{{eqn | r = \paren {1 + 2 + \cdots + \paren {n - 1} } + \paren {1 + 2 + \cdots + \paren {n - 1} + n}
| c =
}}
{{eqn | r = \frac {\paren {n - 1} n} 2 + \frac {n \paren {n + 1} } 2
| c = C... | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | {{begin-eqn}}
{{eqn | o =
| r = 1 + 2 + \cdots + \paren {n - 1} + n + \paren {n - 1} + \cdots + 1
| c =
}}
{{eqn | r = \paren {1 + 2 + \cdots + \paren {n - 1} } + \paren {1 + 2 + \cdots + \paren {n - 1} + n}
| c =
}}
{{eqn | r = \frac {\paren {n - 1} n} 2 + \frac {n \paren {n + 1} } 2
| c = [... | 1+2+...+n+(n-1)+...+1 = n^2/Proof 2 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2/Proof_2 | [
"Sums of Sequences",
"Square Numbers",
"1+2+...+n+(n-1)+...+1 = n^2"
] | [] | [
"Closed Form for Triangular Numbers"
] |
proofwiki-130 | 1+2+...+n+(n-1)+...+1 = n^2 | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | Proof by induction:
=== Basis for the Induction ===
$n = 1$ holds trivially.
Just to make sure, we try $n = 2$:
:$1 + 2 + 1 = 4$
Likewise $n^2 = 2^2 = 4$.
So shown for basis for the induction.
=== Induction Hypothesis ===
This is the induction hypothesis:
:$1 + 2 + \cdots + k + \paren {k - 1} + \cdots + 1 = k^2$
Now we... | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | Proof by [[Principle of Mathematical Induction|induction]]:
=== Basis for the Induction ===
$n = 1$ holds trivially.
Just to make sure, we try $n = 2$:
:$1 + 2 + 1 = 4$
Likewise $n^2 = 2^2 = 4$.
So shown for [[Definition:Basis for the Induction|basis for the induction]].
=== Induction Hypothesis ===
This is th... | 1+2+...+n+(n-1)+...+1 = n^2/Proof 3 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2/Proof_3 | [
"Sums of Sequences",
"Square Numbers",
"1+2+...+n+(n-1)+...+1 = n^2"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"1+2+...+n+(n-1)+...+1 = n^2/Proof 3",
"Principle of Mathematical Induction"
] |
proofwiki-131 | 1+2+...+n+(n-1)+...+1 = n^2 | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | Let $T_n = 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1$.
We have $T_1 = 1$
and
{{begin-eqn}}
{{eqn | l = T_n - T_{n - 1}
| r = \paren {1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 }
| c = Definition of $T_n$
}}
{{eqn | o =
| ro= -
| r = \paren {1 + 2 + \cdots + \paren {n - 1} + \paren... | :$\forall n \in \N: 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 = n^2$ | Let $T_n = 1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1$.
We have $T_1 = 1$
and
{{begin-eqn}}
{{eqn | l = T_n - T_{n - 1}
| r = \paren {1 + 2 + \cdots + n + \paren {n - 1} + \cdots + 1 }
| c = Definition of $T_n$
}}
{{eqn | o =
| ro= -
| r = \paren {1 + 2 + \cdots + \paren {n - 1} + \pa... | 1+2+...+n+(n-1)+...+1 = n^2/Proof 4 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2 | https://proofwiki.org/wiki/1+2+...+n+(n-1)+...+1_=_n^2/Proof_4 | [
"Sums of Sequences",
"Square Numbers",
"1+2+...+n+(n-1)+...+1 = n^2"
] | [] | [
"Integer Addition is Associative",
"Integer Addition is Commutative",
"Odd Number Theorem"
] |
proofwiki-132 | Necessary Conditions for Existence of Skolem Sequence | A Skolem sequence of order $n$ can only exist if $n \equiv 0, 1 \pmod 4$. | Let $S$ be a Skolem sequence of order $n$.
Let $a_i$ and $b_i$ be the positions of the first and second occurrences respectively of the integer $i$ in $S$, where $1 \le i \le n$.
We can thus conclude that:
:$b_i - a_i = i$
for each $i$ from 1 to $n$.
Summing both sides of this equation we obtain:
:$\ds \sum_i^n b_i - \... | A [[Definition:Skolem Sequence|Skolem sequence]] of order $n$ can only exist if $n \equiv 0, 1 \pmod 4$. | Let $S$ be a [[Definition:Skolem Sequence|Skolem sequence]] of order $n$.
Let $a_i$ and $b_i$ be the positions of the first and second occurrences respectively of the integer $i$ in $S$, where $1 \le i \le n$.
We can thus conclude that:
:$b_i - a_i = i$
for each $i$ from 1 to $n$.
Summing both sides of this equatio... | Necessary Conditions for Existence of Skolem Sequence | https://proofwiki.org/wiki/Necessary_Conditions_for_Existence_of_Skolem_Sequence | https://proofwiki.org/wiki/Necessary_Conditions_for_Existence_of_Skolem_Sequence | [
"Combinatorics"
] | [
"Definition:Skolem Sequence"
] | [
"Definition:Skolem Sequence",
"Definition:Integer",
"Definition:Integer"
] |
proofwiki-133 | Divisibility by 9 | A number expressed in decimal notation is divisible by $9$ {{iff}} the sum of its digits is divisible by $9$.
That is:
:$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $9$
{{iff}}:
:$a_0 + a_1 + \cdots + a_n$ is divisible by $9$. | Let $N$ be divisible by $9$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 9
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n
| o = \equiv
| r = 0 \pmod 9
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + a_1 1 + a_2 1^2 + \cdots + a_n 1^n
... | A number expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $9$ {{iff}} the sum of its [[Definition:Digit|digits]] is [[Definition:Divisor of Integer|divisible]] by $9$.
That is:
:$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots ... | Let $N$ be [[Definition:Divisor of Integer|divisible]] by $9$.
Then:
{{begin-eqn}}
{{eqn | l = N
| o = \equiv
| r = 0 \pmod 9
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n
| o = \equiv
| r = 0 \pmod 9
}}
{{eqn | ll= \leadstoandfrom
| l = a_0 + a_1 ... | Divisibility by 9/Proof 1 | https://proofwiki.org/wiki/Divisibility_by_9 | https://proofwiki.org/wiki/Divisibility_by_9/Proof_1 | [
"Divisibility by 9",
"Divisibility Tests",
"9"
] | [
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Divisor (Algebra)/Integer"
] |
proofwiki-134 | Divisibility by 9 | A number expressed in decimal notation is divisible by $9$ {{iff}} the sum of its digits is divisible by $9$.
That is:
:$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ is divisible by $9$
{{iff}}:
:$a_0 + a_1 + \cdots + a_n$ is divisible by $9$. | This is a special case of Congruence of Sum of Digits to Base Less 1.
{{qed}} | A number expressed in [[Definition:Decimal Notation|decimal notation]] is [[Definition:Divisor of Integer|divisible]] by $9$ {{iff}} the sum of its [[Definition:Digit|digits]] is [[Definition:Divisor of Integer|divisible]] by $9$.
That is:
:$N = \sqbrk {a_0 a_1 a_2 \ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \cdots ... | This is a special case of [[Congruence of Sum of Digits to Base Less 1]].
{{qed}} | Divisibility by 9/Proof 2 | https://proofwiki.org/wiki/Divisibility_by_9 | https://proofwiki.org/wiki/Divisibility_by_9/Proof_2 | [
"Divisibility by 9",
"Divisibility Tests",
"9"
] | [
"Definition:Decimal Notation",
"Definition:Divisor (Algebra)/Integer",
"Definition:Digit",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Congruence of Sum of Digits to Base Less 1"
] |
proofwiki-135 | Existence of Rational Powers of Irrational Numbers | There exist irrational numbers $a$ and $b$ such that $a^b$ is rational. | We have that:
: $\sqrt 2$ is irrational.
: $2$ is trivially rational, as $2 = \dfrac 2 1$.
Consider the number $q = \sqrt 2^{\sqrt 2}$, which is irrational by the Gelfond-Schneider Theorem.
Thus:
: $q^{\sqrt 2} = \left({\sqrt 2^{\sqrt 2}}\right)^{\sqrt 2} = \sqrt 2 ^{\left({\sqrt 2}\right) \left({\sqrt 2}\right)} = \sq... | There exist [[Definition:Irrational Number|irrational numbers]] $a$ and $b$ such that $a^b$ is [[Definition:Rational Number|rational]]. | We have that:
: [[Square Root of 2 is Irrational|$\sqrt 2$ is irrational]].
: $2$ is trivially rational, as $2 = \dfrac 2 1$.
Consider the number $q = \sqrt 2^{\sqrt 2}$, which is [[Definition:Irrational Number|irrational]] by the [[Gelfond-Schneider Theorem]].
Thus:
: $q^{\sqrt 2} = \left({\sqrt 2^{\sqrt 2}}\right)^... | Existence of Rational Powers of Irrational Numbers/Proof 1 | https://proofwiki.org/wiki/Existence_of_Rational_Powers_of_Irrational_Numbers | https://proofwiki.org/wiki/Existence_of_Rational_Powers_of_Irrational_Numbers/Proof_1 | [
"Number Theory",
"Roots of Numbers",
"Existence of Rational Powers of Irrational Numbers"
] | [
"Definition:Irrational Number",
"Definition:Rational Number"
] | [
"Square Root of 2 is Irrational",
"Definition:Irrational Number",
"Gelfond-Schneider Theorem",
"Definition:Rational Number",
"Definition:Irrational Number"
] |
proofwiki-136 | Existence of Rational Powers of Irrational Numbers | There exist irrational numbers $a$ and $b$ such that $a^b$ is rational. | Given that $2$ is rational and $\sqrt 2$ is irrational, consider the number $q = \sqrt 2^{\sqrt 2}$.
We consider the two cases.
:$(1): \quad$ If $q$ is rational then $a = \sqrt 2$ and $b = \sqrt 2$ are the desired irrational numbers.
:$(2): \quad$ If $q$ is irrational then $q^{\sqrt 2} = \paren {\sqrt 2^{\sqrt 2} }^{\s... | There exist [[Definition:Irrational Number|irrational numbers]] $a$ and $b$ such that $a^b$ is [[Definition:Rational Number|rational]]. | Given that $2$ is [[Definition:Rational Number|rational]] and [[Square Root of 2 is Irrational|$\sqrt 2$ is irrational]], consider the number $q = \sqrt 2^{\sqrt 2}$.
We consider the two cases.
:$(1): \quad$ If $q$ is [[Definition:Rational Number|rational]] then $a = \sqrt 2$ and $b = \sqrt 2$ are the desired [[Defin... | Existence of Rational Powers of Irrational Numbers/Proof 2 | https://proofwiki.org/wiki/Existence_of_Rational_Powers_of_Irrational_Numbers | https://proofwiki.org/wiki/Existence_of_Rational_Powers_of_Irrational_Numbers/Proof_2 | [
"Number Theory",
"Roots of Numbers",
"Existence of Rational Powers of Irrational Numbers"
] | [
"Definition:Irrational Number",
"Definition:Rational Number"
] | [
"Definition:Rational Number",
"Square Root of 2 is Irrational",
"Definition:Rational Number",
"Definition:Irrational Number",
"Definition:Irrational Number",
"Definition:Rational Number",
"Definition:Irrational Number"
] |
proofwiki-137 | Existence of Rational Powers of Irrational Numbers | There exist irrational numbers $a$ and $b$ such that $a^b$ is rational. | Consider the equation:
:$\paren {\sqrt 2}^{\log_{\sqrt 2} 3} = 3$
We have that $\sqrt 2$ is irrational and $3$ is (trivially) rational.
It remains to be proved $\log_{\sqrt 2} 3$ is irrational.
We have:
{{begin-eqn}}
{{eqn | l = \log_{\sqrt 2} 3
| r = \frac {\log_2 3} {\log_2 \sqrt 2}
| c = Change of Base o... | There exist [[Definition:Irrational Number|irrational numbers]] $a$ and $b$ such that $a^b$ is [[Definition:Rational Number|rational]]. | Consider the equation:
:$\paren {\sqrt 2}^{\log_{\sqrt 2} 3} = 3$
We have that [[Square Root of 2 is Irrational|$\sqrt 2$ is irrational]] and $3$ is (trivially) [[Definition:Rational Number|rational]].
It remains to be proved $\log_{\sqrt 2} 3$ is [[Definition:Irrational Number|irrational]].
We have:
{{begin-eqn}... | Existence of Rational Powers of Irrational Numbers/Proof 3 | https://proofwiki.org/wiki/Existence_of_Rational_Powers_of_Irrational_Numbers | https://proofwiki.org/wiki/Existence_of_Rational_Powers_of_Irrational_Numbers/Proof_3 | [
"Number Theory",
"Roots of Numbers",
"Existence of Rational Powers of Irrational Numbers"
] | [
"Definition:Irrational Number",
"Definition:Rational Number"
] | [
"Square Root of 2 is Irrational",
"Definition:Rational Number",
"Definition:Irrational Number",
"Change of Base of Logarithm",
"Logarithm of Power",
"Irrationality of Logarithm",
"Definition:Irrational Number",
"Definition:Irrational Number",
"Definition:Irrational Number",
"Definition:Rational Nu... |
proofwiki-138 | Intersection is Associative | Set intersection is associative:
:$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$ | {{begin-eqn}}
{{eqn | o =
| r = x \in A \cap \paren {B \cap C}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land \paren {x \in B \land x \in C}
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {x \in A \land x \in B} \land x \in C
| c = Rule of Association: Co... | [[Definition:Set Intersection|Set intersection]] is [[Definition:Associative Operation|associative]]:
:$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$ | {{begin-eqn}}
{{eqn | o =
| r = x \in A \cap \paren {B \cap C}
}}
{{eqn | o = \leadstoandfrom
| r = x \in A \land \paren {x \in B \land x \in C}
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \leadstoandfrom
| r = \paren {x \in A \land x \in B} \land x \in C
| c = [[Rule of Association/C... | Intersection is Associative | https://proofwiki.org/wiki/Intersection_is_Associative | https://proofwiki.org/wiki/Intersection_is_Associative | [
"Intersection is Associative",
"Set Intersection",
"Associative Laws of Set Theory",
"Examples of Associative Operations",
"Direct Proofs"
] | [
"Definition:Set Intersection",
"Definition:Associative Operation"
] | [
"Rule of Association/Conjunction"
] |
proofwiki-139 | Intersection is Associative | Set intersection is associative:
:$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$ | For every $\lambda \in \Lambda$, let $\ds T_\lambda = \bigcap_{i \mathop \in I_\lambda} S_i$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcap_{i \mathop \in I} S_i
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall i \in I
| l = x
| o = \in
| r = S_i
| c = {{De... | [[Definition:Set Intersection|Set intersection]] is [[Definition:Associative Operation|associative]]:
:$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$ | For every $\lambda \in \Lambda$, let $\ds T_\lambda = \bigcap_{i \mathop \in I_\lambda} S_i$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \bigcap_{i \mathop \in I} S_i
| c =
}}
{{eqn | ll= \leadstoandfrom
| q = \forall i \in I
| l = x
| o = \in
| r = S_i
| c = {... | Intersection is Associative/Family of Sets/Proof 1 | https://proofwiki.org/wiki/Intersection_is_Associative | https://proofwiki.org/wiki/Intersection_is_Associative/Family_of_Sets/Proof_1 | [
"Intersection is Associative",
"Set Intersection",
"Associative Laws of Set Theory",
"Examples of Associative Operations",
"Direct Proofs"
] | [
"Definition:Set Intersection",
"Definition:Associative Operation"
] | [] |
proofwiki-140 | Intersection is Associative | Set intersection is associative:
:$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$ | {{begin-eqn}}
{{eqn | l = \bigcap_{i \mathop \in I} S_i
| r = \map \complement {\map \complement {\bigcap_{i \mathop \in I} S_i} }
| c = Complement of Complement
}}
{{eqn | r = \map \complement {\bigcup_{i \mathop \in I} \map \complement {S_i} }
| c = De Morgan's Laws (Set Theory)/Set Complement/Famil... | [[Definition:Set Intersection|Set intersection]] is [[Definition:Associative Operation|associative]]:
:$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$ | {{begin-eqn}}
{{eqn | l = \bigcap_{i \mathop \in I} S_i
| r = \map \complement {\map \complement {\bigcap_{i \mathop \in I} S_i} }
| c = [[Complement of Complement]]
}}
{{eqn | r = \map \complement {\bigcup_{i \mathop \in I} \map \complement {S_i} }
| c = [[De Morgan's Laws (Set Theory)/Set Complement... | Intersection is Associative/Family of Sets/Proof 2 | https://proofwiki.org/wiki/Intersection_is_Associative | https://proofwiki.org/wiki/Intersection_is_Associative/Family_of_Sets/Proof_2 | [
"Intersection is Associative",
"Set Intersection",
"Associative Laws of Set Theory",
"Examples of Associative Operations",
"Direct Proofs"
] | [
"Definition:Set Intersection",
"Definition:Associative Operation"
] | [
"Complement of Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection",
"Union is Associative/Family of Sets",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/... |
proofwiki-141 | Center of Symmetric Group is Trivial | Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group of order $n$.
Let $n \ge 3$.
Then the center $\map Z {S_n}$ of $S_n$ is trivial. | From its definition, the identity (here denoted by $e$) of a group $G$ commutes with all elements of $G$.
So by definition of center:
:$e \in \map Z {S_n}$
By definition of center:
:$\map Z {S_n} = \set {\tau \in S_n: \forall \sigma \in S_n: \tau \sigma = \sigma \tau}$
Let $\pi, \rho \in S_n$ be permutations of $\N_n$.... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ denote the [[Definition:Symmetric Group|symmetric group]] of [[Definition:Order of Structure|order]] $n$.
Let $n \ge 3$.
Then the [[Definition:Center of Group|center]] $\map Z {S_n}$ of $S_n$ is [[Definition:Trivial Group|trivial]]. | From its definition, the [[Definition:Identity Element|identity]] (here denoted by $e$) of a [[Definition:Group|group]] $G$ [[Definition:Commute|commutes]] with all [[Definition:Element|elements]] of $G$.
So by definition of [[Definition:Center of Group|center]]:
:$e \in \map Z {S_n}$
By definition of [[Definition:C... | Center of Symmetric Group is Trivial/Proof 1 | https://proofwiki.org/wiki/Center_of_Symmetric_Group_is_Trivial | https://proofwiki.org/wiki/Center_of_Symmetric_Group_is_Trivial/Proof_1 | [
"Symmetric Groups",
"Center of Symmetric Group is Trivial",
"Centers of Groups"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group",
"Definition:Order of Structure",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Trivial Group"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Group",
"Definition:Commutative/Elements",
"Definition:Element",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Permutation",
"Definition:Fixed Element under Permutation",
... |
proofwiki-142 | Center of Symmetric Group is Trivial | Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group of order $n$.
Let $n \ge 3$.
Then the center $\map Z {S_n}$ of $S_n$ is trivial. | Let $\pi \in S_n$ such that $\pi \ne e$ be arbitrary.
Let $i, j \in \set {1, 2, \ldots, n}$ such that $\map \pi i = j \ne i$.
Let $m = \map \pi j$.
Then $m \ne j$.
Since $n \ge 3$, there exists $k \in \N$ such that $k \ne j, k \ne m$.
Let $\rho \in S_n$ interchange $j, k$ and fix everything else.
Then $\rho$ fixes $m$.... | Let $n \in \N$ be a [[Definition:Natural Number|natural number]].
Let $S_n$ denote the [[Definition:Symmetric Group|symmetric group]] of [[Definition:Order of Structure|order]] $n$.
Let $n \ge 3$.
Then the [[Definition:Center of Group|center]] $\map Z {S_n}$ of $S_n$ is [[Definition:Trivial Group|trivial]]. | Let $\pi \in S_n$ such that $\pi \ne e$ be arbitrary.
Let $i, j \in \set {1, 2, \ldots, n}$ such that $\map \pi i = j \ne i$.
Let $m = \map \pi j$.
Then $m \ne j$.
Since $n \ge 3$, there exists $k \in \N$ such that $k \ne j, k \ne m$.
Let $\rho \in S_n$ interchange $j, k$ and [[Definition:Fixed Element under Permu... | Center of Symmetric Group is Trivial/Proof 2 | https://proofwiki.org/wiki/Center_of_Symmetric_Group_is_Trivial | https://proofwiki.org/wiki/Center_of_Symmetric_Group_is_Trivial/Proof_2 | [
"Symmetric Groups",
"Center of Symmetric Group is Trivial",
"Centers of Groups"
] | [
"Definition:Natural Numbers",
"Definition:Symmetric Group",
"Definition:Order of Structure",
"Definition:Center (Abstract Algebra)/Group",
"Definition:Trivial Group"
] | [
"Definition:Fixed Element under Permutation",
"Definition:Fixed Element under Permutation",
"Definition:Permutation",
"Definition:Injection",
"Definition:Commutative/Elements",
"Definition:Trivial Group"
] |
proofwiki-143 | Square Root of 2 is Irrational | :$\sqrt 2$ is irrational. | First we note that, from Parity of Integer equals Parity of its Square, if an integer is even, its square root, if an integer, is also even.
Thus it follows that:
:$(1): \quad 2 \divides p^2 \implies 2 \divides p$
where $2 \divides p$ indicates that $2$ is a divisor of $p$.
{{AimForCont}} that $\sqrt 2$ is rational.
So... | :$\sqrt 2$ is [[Definition:Irrational Number|irrational]]. | First we note that, from [[Parity of Integer equals Parity of its Square]], if an [[Definition:Integer|integer]] is [[Definition:Even Integer|even]], its [[Definition:Square Root|square root]], if an [[Definition:Integer|integer]], is also [[Definition:Even Integer|even]].
Thus it follows that:
:$(1): \quad 2 \divides... | Square Root of 2 is Irrational/Classic Proof | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational/Classic_Proof | [
"Square Root of 2 is Irrational",
"Square Root of 2",
"Irrationality Proofs"
] | [
"Definition:Irrational Number"
] | [
"Parity of Integer equals Parity of its Square",
"Definition:Integer",
"Definition:Even Integer",
"Definition:Square Root",
"Definition:Integer",
"Definition:Even Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Rational Number",
"Definition:Greatest Common Divisor/Integers",
"Definit... |
proofwiki-144 | Square Root of 2 is Irrational | :$\sqrt 2$ is irrational. | Special case of Square Root of Prime is Irrational.
{{qed}} | :$\sqrt 2$ is [[Definition:Irrational Number|irrational]]. | Special case of [[Square Root of Prime is Irrational]].
{{qed}} | Square Root of 2 is Irrational/Proof 2 | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational/Proof_2 | [
"Square Root of 2 is Irrational",
"Square Root of 2",
"Irrationality Proofs"
] | [
"Definition:Irrational Number"
] | [
"Square Root of Prime is Irrational"
] |
proofwiki-145 | Square Root of 2 is Irrational | :$\sqrt 2$ is irrational. | {{AimForCont}} that $\sqrt 2$ is rational.
Then $\sqrt 2 = \dfrac p q$ for some $p, q \in \Z_{>0}$
Consider the quantity $\paren {\sqrt 2 - 1}$:
{{begin-eqn}}
{{eqn | ll= 1
| l = <
| o = \sqrt 2
| r = <
| rr = 2
| c = Ordering of Squares in Reals
}}
{{eqn | lll = \leadsto
| ll= 0
... | :$\sqrt 2$ is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} that $\sqrt 2$ is [[Definition:Rational Number|rational]].
Then $\sqrt 2 = \dfrac p q$ for some $p, q \in \Z_{>0}$
Consider the quantity $\paren {\sqrt 2 - 1}$:
{{begin-eqn}}
{{eqn | ll= 1
| l = <
| o = \sqrt 2
| r = <
| rr = 2
| c = [[Ordering of Squares in Reals]]
}}
{{... | Square Root of 2 is Irrational/Proof 3 | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational/Proof_3 | [
"Square Root of 2 is Irrational",
"Square Root of 2",
"Irrationality Proofs"
] | [
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Ordering of Squares in Reals",
"Binomial Theorem",
"Definition:Integer",
"Definition:Fraction/Numerator",
"Sequence of Powers of Number less than One",
"Definition:Limit of Sequence/Real Numbers",
"Lower and Upper Bounds for Sequences",
"Proof by Contradiction"
] |
proofwiki-146 | Square Root of 2 is Irrational | :$\sqrt 2$ is irrational. | {{AimForCont}} that $\sqrt 2$ is rational.
Let $n$ be the smallest positive integer such that:
:$\sqrt 2 = \dfrac m n$
for some $m \in \Z_{>0}$
Then:
:$m = n \sqrt2 > n$
so:
:$(1): \quad m - n > 0$
We also have:
:$m = n \sqrt2 < 2 n$
so:
:$m < 2 n$
and therefore:
:$(2): \quad m - n < n$
Finally, we have
{{begin-eqn}}
{... | :$\sqrt 2$ is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} that $\sqrt 2$ is [[Definition:Rational Number|rational]].
Let $n$ be the smallest [[Definition:Positive Integer|positive integer]] such that:
:$\sqrt 2 = \dfrac m n$
for some $m \in \Z_{>0}$
Then:
:$m = n \sqrt2 > n$
so:
:$(1): \quad m - n > 0$
We also have:
:$m = n \sqrt2 < 2 n$
so:
:$m < 2 n$
and ... | Square Root of 2 is Irrational/Proof 4 | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational/Proof_4 | [
"Square Root of 2 is Irrational",
"Square Root of 2",
"Irrationality Proofs"
] | [
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Positive/Integer",
"Definition:Fraction/Denominator",
"Definition:Positive/Real Number",
"Definition:Fraction/Denominator",
"Definition:Fraction",
"Definition:Fraction/Denominator",
"Definition:Contradiction",
"Definition:Fraction/Denominator",
"Proof by C... |
proofwiki-147 | Square Root of 2 is Irrational | :$\sqrt 2$ is irrational. | By the Rational Root Theorem, every rational root of the polynomial $n^2 - 2$ would have:
:a numerator that divides $2$
and:
:a denominator that divides $1$.
That is, the only possible rational roots are:
:$n = \pm 1, \pm 2$
But:
:$\paren {\pm 1}^2 - 2 \ne 0 \ne \paren {\pm 2}^2 - 2$
So its root $n = \sqrt 2$ is not a ... | :$\sqrt 2$ is [[Definition:Irrational Number|irrational]]. | By the [[Rational Root Theorem]], every [[Definition:Rational Number|rational]] [[Definition:Root of Polynomial|root]] of the [[Definition:Polynomial|polynomial]] $n^2 - 2$ would have:
:a [[Definition:Numerator|numerator]] that [[Definition:Divisor of Integer|divides]] $2$
and:
:a [[Definition:Denominator|denominator]]... | Square Root of 2 is Irrational/Proof 5 | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational | https://proofwiki.org/wiki/Square_Root_of_2_is_Irrational/Proof_5 | [
"Square Root of 2 is Irrational",
"Square Root of 2",
"Irrationality Proofs"
] | [
"Definition:Irrational Number"
] | [
"Rational Root Theorem",
"Definition:Rational Number",
"Definition:Root of Polynomial",
"Definition:Polynomial",
"Definition:Fraction/Numerator",
"Definition:Divisor (Algebra)/Integer",
"Definition:Fraction/Denominator",
"Definition:Divisor (Algebra)/Integer",
"Definition:Rational Number",
"Defini... |
proofwiki-148 | Empty Set is Unique | The empty set is unique. | Let $\O$ and $\O'$ both be empty sets.
From Empty Set is Subset of All Sets, $\O \subseteq \O'$, because $\O$ is empty.
Likewise, we have $\O' \subseteq \O$, since $\O'$ is empty.
Together, by the definition of set equality, this implies that $\O = \O'$.
Thus there is only one empty set.
{{Qed}} | The [[Definition:Empty Set|empty set]] is [[Definition:Unique|unique]]. | Let $\O$ and $\O'$ both be [[Definition:Empty Set|empty sets]].
From [[Empty Set is Subset of All Sets]], $\O \subseteq \O'$, because $\O$ is [[Definition:Empty Set|empty]].
Likewise, we have $\O' \subseteq \O$, since $\O'$ is [[Definition:Empty Set|empty]].
Together, by the definition of [[Definition:Set Equality/D... | Empty Set is Unique/Proof 1 | https://proofwiki.org/wiki/Empty_Set_is_Unique | https://proofwiki.org/wiki/Empty_Set_is_Unique/Proof_1 | [
"Empty Set",
"Empty Set is Unique"
] | [
"Definition:Empty Set",
"Definition:Unique"
] | [
"Definition:Empty Set",
"Empty Set is Subset of All Sets",
"Definition:Empty Set",
"Definition:Empty Set",
"Definition:Set Equality/Definition 2",
"Definition:Empty Set"
] |
proofwiki-149 | Empty Set is Unique | The empty set is unique. | Let $A$ and $B$ both be empty sets.
Thus:
:$\forall x: \neg \paren {x \in A}$
and:
:$\forall x: \neg \paren {x \in B}$
Hence:
:$x \notin A \iff x \notin B$
and so:
:$x \in A \iff x \in B$
vacuously.
From the Axiom of Extension:
:$\forall x: \paren {x \in A \iff x \in B} \iff A = B$
Hence the result.
{{Qed}} | The [[Definition:Empty Set|empty set]] is [[Definition:Unique|unique]]. | Let $A$ and $B$ both be [[Definition:Empty Set|empty sets]].
Thus:
:$\forall x: \neg \paren {x \in A}$
and:
:$\forall x: \neg \paren {x \in B}$
Hence:
:$x \notin A \iff x \notin B$
and so:
:$x \in A \iff x \in B$
[[Definition:Vacuous Truth|vacuously]].
From the [[Axiom:Axiom of Extension (Sets)|Axiom of Extension... | Empty Set is Unique/Proof 2 | https://proofwiki.org/wiki/Empty_Set_is_Unique | https://proofwiki.org/wiki/Empty_Set_is_Unique/Proof_2 | [
"Empty Set",
"Empty Set is Unique"
] | [
"Definition:Empty Set",
"Definition:Unique"
] | [
"Definition:Empty Set",
"Definition:Vacuous Truth",
"Axiom:Axiom of Extension/Set Theory"
] |
proofwiki-150 | Empty Set is Unique | The empty set is unique. | From {{axiom-link|the Empty Set|Class Theory}} in the context of class theory, the empty class is a set.
The result follows from Empty Class Exists and is Unique.
{{Qed}} | The [[Definition:Empty Set|empty set]] is [[Definition:Unique|unique]]. | From {{axiom-link|the Empty Set|Class Theory}} in the context of [[Definition:Class Theory|class theory]], the [[Definition:Empty Class (Class Theory)|empty class]] is a [[Definition:Set|set]].
The result follows from [[Empty Class Exists and is Unique]].
{{Qed}} | Empty Set is Unique/Proof 3 | https://proofwiki.org/wiki/Empty_Set_is_Unique | https://proofwiki.org/wiki/Empty_Set_is_Unique/Proof_3 | [
"Empty Set",
"Empty Set is Unique"
] | [
"Definition:Empty Set",
"Definition:Unique"
] | [
"Definition:Class Theory",
"Definition:Empty Class (Class Theory)",
"Definition:Set",
"Empty Class Exists and is Unique"
] |
proofwiki-151 | Principle of Non-Contradiction | The '''Principle of Non-Contradiction''' is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.
This includes classical propositional logic and predicate logic, and in particular natural deduction.
=== Proof Rule ===
{{:Principle of Non-Contradiction/Proof Rule}}
=== Sequent Form =... | {{BeginTableau|p, \neg p \vdash \bot}}
{{Premise|1|p}}
{{Premise|2|\neg p}}
{{NonContradiction|3|1, 2|1|2}}
{{EndTableau}}
{{Qed}} | The '''[[Principle of Non-Contradiction]]''' is a [[Definition:Valid Argument|valid argument]] in types of [[Definition:Logic|logic]] dealing with [[Definition:Logical Not|negation]] $\neg$ and [[Definition:Contradiction|contradiction]] $\bot$.
This includes [[Definition:Classical Propositional Logic|classical proposi... | {{BeginTableau|p, \neg p \vdash \bot}}
{{Premise|1|p}}
{{Premise|2|\neg p}}
{{NonContradiction|3|1, 2|1|2}}
{{EndTableau}}
{{Qed}} | Principle of Non-Contradiction/Sequent Form/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Principle_of_Non-Contradiction | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_1/Proof_1 | [
"Principle of Non-Contradiction",
"Rules of Inference",
"Contradiction"
] | [
"Principle of Non-Contradiction",
"Definition:Valid Argument",
"Definition:Logic",
"Definition:Logical Not",
"Definition:Contradiction",
"Definition:Classical Propositional Logic",
"Definition:Predicate Logic",
"Definition:Natural Deduction",
"Principle of Non-Contradiction/Proof Rule",
"Principle... | [] |
proofwiki-152 | Principle of Non-Contradiction | The '''Principle of Non-Contradiction''' is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.
This includes classical propositional logic and predicate logic, and in particular natural deduction.
=== Proof Rule ===
{{:Principle of Non-Contradiction/Proof Rule}}
=== Sequent Form =... | We apply the Method of Truth Tables.
:<nowiki>$\begin {array} {|cccc||c|} \hline
p & \land & \neg & p & \bot \\
\hline
\F & \F & \T & \F & \F \\
\T & \F & \F & \T & \F \\
\hline
\end {array}$</nowiki>
As can be seen by inspection, the truth value of the main connective, that is $\land$, is $F$ for each boolean interpre... | The '''[[Principle of Non-Contradiction]]''' is a [[Definition:Valid Argument|valid argument]] in types of [[Definition:Logic|logic]] dealing with [[Definition:Logical Not|negation]] $\neg$ and [[Definition:Contradiction|contradiction]] $\bot$.
This includes [[Definition:Classical Propositional Logic|classical proposi... | We apply the [[Method of Truth Tables]].
:<nowiki>$\begin {array} {|cccc||c|} \hline
p & \land & \neg & p & \bot \\
\hline
\F & \F & \T & \F & \F \\
\T & \F & \F & \T & \F \\
\hline
\end {array}$</nowiki>
As can be seen by inspection, the [[Definition:Truth Value|truth value]] of the [[Definition:Main Connective (Pro... | Principle of Non-Contradiction/Sequent Form/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Principle_of_Non-Contradiction | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_1/Proof_by_Truth_Table | [
"Principle of Non-Contradiction",
"Rules of Inference",
"Contradiction"
] | [
"Principle of Non-Contradiction",
"Definition:Valid Argument",
"Definition:Logic",
"Definition:Logical Not",
"Definition:Contradiction",
"Definition:Classical Propositional Logic",
"Definition:Predicate Logic",
"Definition:Natural Deduction",
"Principle of Non-Contradiction/Proof Rule",
"Principle... | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-153 | Principle of Non-Contradiction | The '''Principle of Non-Contradiction''' is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.
This includes classical propositional logic and predicate logic, and in particular natural deduction.
=== Proof Rule ===
{{:Principle of Non-Contradiction/Proof Rule}}
=== Sequent Form =... | {{BeginTableau|\vdash \neg \left({p \land \neg p}\right)}}
{{Assumption|1|p \land \neg p}}
{{Simplification|2|1|p|1|1}}
{{Simplification|3|1|\neg p|1|2}}
{{NonContradiction|4|1|2|3}}
{{Contradiction|5||\neg \left({p \land \neg p}\right)|1|4}}
{{EndTableau|qed}} | The '''[[Principle of Non-Contradiction]]''' is a [[Definition:Valid Argument|valid argument]] in types of [[Definition:Logic|logic]] dealing with [[Definition:Logical Not|negation]] $\neg$ and [[Definition:Contradiction|contradiction]] $\bot$.
This includes [[Definition:Classical Propositional Logic|classical proposi... | {{BeginTableau|\vdash \neg \left({p \land \neg p}\right)}}
{{Assumption|1|p \land \neg p}}
{{Simplification|2|1|p|1|1}}
{{Simplification|3|1|\neg p|1|2}}
{{NonContradiction|4|1|2|3}}
{{Contradiction|5||\neg \left({p \land \neg p}\right)|1|4}}
{{EndTableau|qed}} | Principle of Non-Contradiction/Sequent Form/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Principle_of_Non-Contradiction | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_2/Proof_1 | [
"Principle of Non-Contradiction",
"Rules of Inference",
"Contradiction"
] | [
"Principle of Non-Contradiction",
"Definition:Valid Argument",
"Definition:Logic",
"Definition:Logical Not",
"Definition:Contradiction",
"Definition:Classical Propositional Logic",
"Definition:Predicate Logic",
"Definition:Natural Deduction",
"Principle of Non-Contradiction/Proof Rule",
"Principle... | [] |
proofwiki-154 | Principle of Non-Contradiction | The '''Principle of Non-Contradiction''' is a valid argument in types of logic dealing with negation $\neg$ and contradiction $\bot$.
This includes classical propositional logic and predicate logic, and in particular natural deduction.
=== Proof Rule ===
{{:Principle of Non-Contradiction/Proof Rule}}
=== Sequent Form =... | We apply the Method of Truth Tables to the proposition $\neg \paren {p \land \neg p}$.
As can be seen by inspection, the truth value of the main connective, that is $\neg$, is $T$ for each boolean interpretation for $p$.
:<nowiki>$\begin {array} {|ccccc|} \hline
\neg & (p & \land & \neg & p)\\
\hline
\T & \F & \F & \T ... | The '''[[Principle of Non-Contradiction]]''' is a [[Definition:Valid Argument|valid argument]] in types of [[Definition:Logic|logic]] dealing with [[Definition:Logical Not|negation]] $\neg$ and [[Definition:Contradiction|contradiction]] $\bot$.
This includes [[Definition:Classical Propositional Logic|classical proposi... | We apply the [[Method of Truth Tables]] to the proposition $\neg \paren {p \land \neg p}$.
As can be seen by inspection, the [[Definition:Truth Value|truth value]] of the [[Definition:Main Connective (Propositional Logic)|main connective]], that is $\neg$, is $T$ for each [[Definition:Boolean Interpretation|boolean in... | Principle of Non-Contradiction/Sequent Form/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/Principle_of_Non-Contradiction | https://proofwiki.org/wiki/Principle_of_Non-Contradiction/Sequent_Form/Formulation_2/Proof_by_Truth_Table | [
"Principle of Non-Contradiction",
"Rules of Inference",
"Contradiction"
] | [
"Principle of Non-Contradiction",
"Definition:Valid Argument",
"Definition:Logic",
"Definition:Logical Not",
"Definition:Contradiction",
"Definition:Classical Propositional Logic",
"Definition:Predicate Logic",
"Definition:Natural Deduction",
"Principle of Non-Contradiction/Proof Rule",
"Principle... | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-155 | Rule of Substitution | Let $S$ be a sequent of propositional logic that has been proved.
Then we may infer any sequent $S'$ resulting from $S$ by substitutions for letters. | This is apparent from inspection of the proof rules themselves.
The rules concern only the broad structure of the propositional formulas involved, and this structure is unaffected by substitution of letters.
By performing the substitutions systematically throughout the given sequent, all applications of proof rules rem... | Let $S$ be a [[Definition:Sequent|sequent]] of [[Definition:Propositional Logic|propositional logic]] that has been [[Definition:Formal Proof|proved]].
Then we may infer any [[Definition:Sequent|sequent]] $S'$ resulting from $S$ by [[Definition:Substitution for Letter|substitutions for letters]]. | This is apparent from inspection of the [[Definition:Proof Rule|proof rules]] themselves.
The rules concern only the broad structure of the [[Definition:Propositional Formula|propositional formulas]] involved, and this structure is unaffected by substitution of letters.
By performing the substitutions systematically ... | Rule of Substitution | https://proofwiki.org/wiki/Rule_of_Substitution | https://proofwiki.org/wiki/Rule_of_Substitution | [
"Rule of Substitution",
"Natural Deduction",
"Propositional Logic"
] | [
"Definition:Sequent",
"Definition:Propositional Logic",
"Definition:Proof System/Formal Proof",
"Definition:Sequent",
"Definition:Substitution (Formal Systems)/Letter"
] | [
"Definition:Rule of Inference",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Sequent",
"Definition:Rule of Inference",
"Definition:Sequent"
] |
proofwiki-156 | Rule of Sequent Introduction | Let the statements $P_1, P_2, \ldots, P_n$ be conclusions in a proof, on various assumptions.
Let $P_1, P_2, \ldots, P_n \vdash Q$ be a sequent for which we already have a proof.
Then we may infer, at any stage of a proof (citing '''SI'''), the conclusion $Q$ of the sequent already proved.
This conclusion depends upon ... | By hypothesis we have a proof of:
:$P_1, P_2, \ldots, P_n \vdash Q$
Therefore we can include this proof in our current proof and arrive at $Q$ with the pool of assumptions upon which $P_1, P_2, \ldots, P_n$ rest.
{{qed}} | Let the statements $P_1, P_2, \ldots, P_n$ be [[Definition:Conclusion|conclusions]] in a [[Definition:Proof|proof]], on various [[Definition:Assumption|assumptions]].
Let $P_1, P_2, \ldots, P_n \vdash Q$ be a [[Definition:Sequent|sequent]] for which we already have a proof.
Then we may infer, at any stage of a proof ... | [[Definition:By Hypothesis|By hypothesis]] we have a [[Definition:Formal Proof|proof]] of:
:$P_1, P_2, \ldots, P_n \vdash Q$
Therefore we can include this [[Definition:Formal Proof|proof]] in our current [[Definition:Proof|proof]] and arrive at $Q$ with the [[Definition:Pool of Assumptions|pool of assumptions]] upon w... | Rule of Sequent Introduction | https://proofwiki.org/wiki/Rule_of_Sequent_Introduction | https://proofwiki.org/wiki/Rule_of_Sequent_Introduction | [
"Rule of Sequent Introduction",
"Rules of Inference",
"Logic"
] | [
"Definition:Conclusion",
"Definition:Proof",
"Definition:Assumption",
"Definition:Sequent",
"Definition:Conclusion",
"Definition:Sequent",
"Definition:Pool of Assumptions",
"Rule of Sequent Introduction"
] | [
"Definition:By Hypothesis",
"Definition:Proof System/Formal Proof",
"Definition:Proof System/Formal Proof",
"Definition:Proof",
"Definition:Pool of Assumptions"
] |
proofwiki-157 | Law of Identity | Every proposition entails itself:
=== Formulation 1 ===
{{:Law of Identity/Formulation 1}}
=== Formulation 2 ===
{{:Law of Identity/Formulation 2}} | {{BeginTableau|p \vdash p}}
{{Premise|1|p}}
{{EndTableau|qed}}
This is the shortest tableau proof possible. | Every [[Definition:Proposition|proposition]] entails itself:
=== [[Law of Identity/Formulation 1|Formulation 1]] ===
{{:Law of Identity/Formulation 1}}
=== [[Law of Identity/Formulation 2|Formulation 2]] ===
{{:Law of Identity/Formulation 2}} | {{BeginTableau|p \vdash p}}
{{Premise|1|p}}
{{EndTableau|qed}}
This is the shortest [[Definition:Tableau Proof (Formal Systems)|tableau proof]] possible. | Law of Identity/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Law_of_Identity | https://proofwiki.org/wiki/Law_of_Identity/Formulation_1/Proof_1 | [
"Law of Identity",
"Conditional",
"Propositional Logic"
] | [
"Definition:Proposition",
"Law of Identity/Formulation 1",
"Law of Identity/Formulation 2"
] | [
"Definition:Tableau Proof (Natural Deduction)"
] |
proofwiki-158 | Law of Identity | Every proposition entails itself:
=== Formulation 1 ===
{{:Law of Identity/Formulation 1}}
=== Formulation 2 ===
{{:Law of Identity/Formulation 2}} | We apply the Method of Truth Tables (trivially) to the proposition.
:<nowiki>$\begin{array}{|c|c|} \hline
p & p \\
\hline
\F & \F \\
\T & \T \\
\hline
\end{array}$</nowiki>
{{qed}} | Every [[Definition:Proposition|proposition]] entails itself:
=== [[Law of Identity/Formulation 1|Formulation 1]] ===
{{:Law of Identity/Formulation 1}}
=== [[Law of Identity/Formulation 2|Formulation 2]] ===
{{:Law of Identity/Formulation 2}} | We apply the [[Method of Truth Tables]] (trivially) to the proposition.
:<nowiki>$\begin{array}{|c|c|} \hline
p & p \\
\hline
\F & \F \\
\T & \T \\
\hline
\end{array}$</nowiki>
{{qed}} | Law of Identity/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Law_of_Identity | https://proofwiki.org/wiki/Law_of_Identity/Formulation_1/Proof_by_Truth_Table | [
"Law of Identity",
"Conditional",
"Propositional Logic"
] | [
"Definition:Proposition",
"Law of Identity/Formulation 1",
"Law of Identity/Formulation 2"
] | [
"Method of Truth Tables"
] |
proofwiki-159 | Law of Identity | Every proposition entails itself:
=== Formulation 1 ===
{{:Law of Identity/Formulation 1}}
=== Formulation 2 ===
{{:Law of Identity/Formulation 2}} | {{BeginTableau|\vdash p \implies p}}
{{Premise|1|p}}
{{Implication|2||p \implies p|1|1}}
{{EndTableau|qed}} | Every [[Definition:Proposition|proposition]] entails itself:
=== [[Law of Identity/Formulation 1|Formulation 1]] ===
{{:Law of Identity/Formulation 1}}
=== [[Law of Identity/Formulation 2|Formulation 2]] ===
{{:Law of Identity/Formulation 2}} | {{BeginTableau|\vdash p \implies p}}
{{Premise|1|p}}
{{Implication|2||p \implies p|1|1}}
{{EndTableau|qed}} | Law of Identity/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Law_of_Identity | https://proofwiki.org/wiki/Law_of_Identity/Formulation_2/Proof_1 | [
"Law of Identity",
"Conditional",
"Propositional Logic"
] | [
"Definition:Proposition",
"Law of Identity/Formulation 1",
"Law of Identity/Formulation 2"
] | [] |
proofwiki-160 | Law of Identity | Every proposition entails itself:
=== Formulation 1 ===
{{:Law of Identity/Formulation 1}}
=== Formulation 2 ===
{{:Law of Identity/Formulation 2}} | Using a tableau proof for instance 1 of a Hilbert proof system:
{{BeginTableau|p \implies p|nohead = 1}}
{{TableauLine|n = 1
| f = \paren {p \implies \paren {\paren {p \implies p} \implies p} } \implies \paren {\paren {p \implies \paren {p \implies p} } \implies \paren {p \implies p} }
| rtxt = Axiom 2
| c = $\ma... | Every [[Definition:Proposition|proposition]] entails itself:
=== [[Law of Identity/Formulation 1|Formulation 1]] ===
{{:Law of Identity/Formulation 1}}
=== [[Law of Identity/Formulation 2|Formulation 2]] ===
{{:Law of Identity/Formulation 2}} | Using a [[Definition:Tableau Proof (Formal Systems)|tableau proof]] for [[Definition:Hilbert Proof System/Instance 1|instance 1 of a Hilbert proof system]]:
{{BeginTableau|p \implies p|nohead = 1}}
{{TableauLine|n = 1
| f = \paren {p \implies \paren {\paren {p \implies p} \implies p} } \implies \paren {\paren {p \imp... | Law of Identity/Formulation 2/Proof 2 | https://proofwiki.org/wiki/Law_of_Identity | https://proofwiki.org/wiki/Law_of_Identity/Formulation_2/Proof_2 | [
"Law of Identity",
"Conditional",
"Propositional Logic"
] | [
"Definition:Proposition",
"Law of Identity/Formulation 1",
"Law of Identity/Formulation 2"
] | [
"Definition:Tableau Proof (Natural Deduction)",
"Definition:Hilbert Proof System/Instance 1"
] |
proofwiki-161 | Law of Identity | Every proposition entails itself:
=== Formulation 1 ===
{{:Law of Identity/Formulation 1}}
=== Formulation 2 ===
{{:Law of Identity/Formulation 2}} | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth value under the main connective is $\T$ throughout.
:<nowiki>$\begin{array}{|ccc|} \hline
p & \implies & p \\
\hline
\F & \T & \F \\
\T & \T & \T \\
\hline
\end{array}$</nowiki>
{{qed}} | Every [[Definition:Proposition|proposition]] entails itself:
=== [[Law of Identity/Formulation 1|Formulation 1]] ===
{{:Law of Identity/Formulation 1}}
=== [[Law of Identity/Formulation 2|Formulation 2]] ===
{{:Law of Identity/Formulation 2}} | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, the [[Definition:Truth Value|truth value]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is $\T$ throughout.
:<nowiki>$\begin{array}{|ccc|} \hline
p & \implies & p \\
\hline
\F & \T & \F \\
\T & ... | Law of Identity/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/Law_of_Identity | https://proofwiki.org/wiki/Law_of_Identity/Formulation_2/Proof_by_Truth_Table | [
"Law of Identity",
"Conditional",
"Propositional Logic"
] | [
"Definition:Proposition",
"Law of Identity/Formulation 1",
"Law of Identity/Formulation 2"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic"
] |
proofwiki-162 | Rule of Idempotence | The '''rule of idempotence''' is two-fold:
=== Conjunction ===
{{:Rule of Idempotence/Conjunction}}
=== Disjunction ===
{{:Rule of Idempotence/Disjunction}} | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for each boolean interpretations.
$\begin{array} {|c||ccc|} \hline
p & p & \land & p \\
\hline
\T & \T & \T & \T \\
\F & \F & \F & \F \\
\hline
\end{array}$
{{qed}} | The '''[[Rule of Idempotence|rule of idempotence]]''' is two-fold:
=== [[Rule of Idempotence/Conjunction|Conjunction]] ===
{{:Rule of Idempotence/Conjunction}}
=== [[Rule of Idempotence/Disjunction|Disjunction]] ===
{{:Rule of Idempotence/Disjunction}} | 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 each [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array} {|c||ccc|} \hline
p & p & \lan... | Rule of Idempotence/Conjunction/Formulation 1/Proof | https://proofwiki.org/wiki/Rule_of_Idempotence | https://proofwiki.org/wiki/Rule_of_Idempotence/Conjunction/Formulation_1/Proof | [
"Rule of Idempotence",
"Conjunction",
"Disjunction",
"Examples of Idempotence"
] | [
"Rule of Idempotence",
"Rule of Idempotence/Conjunction",
"Rule of Idempotence/Disjunction"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-163 | Rule of Idempotence | The '''rule of idempotence''' is two-fold:
=== Conjunction ===
{{:Rule of Idempotence/Conjunction}}
=== Disjunction ===
{{:Rule of Idempotence/Disjunction}} | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for each boolean interpretations.
$\begin{array}{|c||ccc|} \hline
p & p & \lor & p \\
\hline
\T & \T & \T & \T \\
\F & \F & \F & \F \\
\hline
\end{array}$
{{qed}} | The '''[[Rule of Idempotence|rule of idempotence]]''' is two-fold:
=== [[Rule of Idempotence/Conjunction|Conjunction]] ===
{{:Rule of Idempotence/Conjunction}}
=== [[Rule of Idempotence/Disjunction|Disjunction]] ===
{{:Rule of Idempotence/Disjunction}} | 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 each [[Definition:Boolean Interpretation|boolean interpretations]].
$\begin{array}{|c||ccc|} \hline
p & p & \lor ... | Rule of Idempotence/Disjunction/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Rule_of_Idempotence | https://proofwiki.org/wiki/Rule_of_Idempotence/Disjunction/Formulation_1/Proof_by_Truth_Table | [
"Rule of Idempotence",
"Conjunction",
"Disjunction",
"Examples of Idempotence"
] | [
"Rule of Idempotence",
"Rule of Idempotence/Conjunction",
"Rule of Idempotence/Disjunction"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-164 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau|\vdash p \implies q, q \implies r \vdash p \implies r}}
{{Premise|1|p \implies q}}
{{Premise|2|q \implies r}}
{{Assumption|3|p}}
{{ModusPonens|4|1, 3|q|1|3}}
{{ModusPonens|5|1, 2, 3|r|2|4}}
{{Implication|6|1, 2|p \implies r|3|5}}
{{EndTableau}}
{{qed}} | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau|\vdash p \implies q, q \implies r \vdash p \implies r}}
{{Premise|1|p \implies q}}
{{Premise|2|q \implies r}}
{{Assumption|3|p}}
{{ModusPonens|4|1, 3|q|1|3}}
{{ModusPonens|5|1, 2, 3|r|2|4}}
{{Implication|6|1, 2|p \implies r|3|5}}
{{EndTableau}}
{{qed}} | Hypothetical Syllogism/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_1/Proof_1 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [] |
proofwiki-165 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | This proof uses $\mathscr H_2$, Instance 2 of the Hilbert proof systems.
Recall the sequent form of the Hypothetical Syllogism:
:$\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }$
Applying the Rule of Detachment $\text {RST} 3$ twice, we obtain:
:$\vdash \paren {p \im... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | This proof uses $\mathscr H_2$, [[Definition:Hilbert Proof System/Instance 2|Instance 2]] of the [[Definition:Hilbert Proof System|Hilbert proof systems]].
Recall the [[Hypothetical Syllogism/Formulation 5/Proof 2|sequent form of the Hypothetical Syllogism]]:
:$\vdash \paren {q \implies r} \implies \paren {\paren {p ... | Hypothetical Syllogism/Formulation 1/Proof 2 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_1/Proof_2 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Definition:Hilbert Proof System/Instance 2",
"Definition:Hilbert Proof System",
"Hypothetical Syllogism/Formulation 5/Proof 2",
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-166 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | We apply the Method of Truth Tables to the propositions in turn.
As can be seen for all boolean interpretations by inspection, where the truth values under the main connective on the {{LHS}} is $\T$, that under the one on the {{RHS}} is also $\T$:
:<nowiki>$\begin{array}{|ccccccc||ccc|} \hline
(p & \implies & q) & \lan... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | We apply the [[Method of Truth Tables]] to the propositions in turn.
As can be seen for all [[Definition:Boolean Interpretation|boolean interpretations]] by inspection, where the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] on the {{LHS}} is $\T... | Hypothetical Syllogism/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_1/Proof_by_Truth_Table | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Method of Truth Tables",
"Definition:Boolean Interpretation",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic"
] |
proofwiki-167 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau|p \implies q, q \implies r, p \vdash r}}
{{Premise|1|p \implies q}}
{{Premise|2|q \implies r}}
{{Premise|3|p}}
{{SequentIntro|4|1, 2|p \implies r|1, 2|Hypothetical Syllogism: Formulation 1}}
{{ModusPonens|5|1, 2, 3|r|4|3}}
{{EndTableau}}
{{qed}} | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau|p \implies q, q \implies r, p \vdash r}}
{{Premise|1|p \implies q}}
{{Premise|2|q \implies r}}
{{Premise|3|p}}
{{SequentIntro|4|1, 2|p \implies r|1, 2|[[Hypothetical Syllogism/Formulation 1|Hypothetical Syllogism: Formulation 1]]}}
{{ModusPonens|5|1, 2, 3|r|4|3}}
{{EndTableau}}
{{qed}} | Hypothetical Syllogism/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_2/Proof_1 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Hypothetical Syllogism/Formulation 1"
] |
proofwiki-168 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau|p \implies q, q \implies r, p \vdash r}}
{{Premise|1|p \implies q}}
{{Premise|2|q \implies r}}
{{Premise|3|p}}
{{ModusPonens|4|1, 3|q|1|3}}
{{ModusPonens|5|1, 2, 3|r|2|4}}
{{EndTableau}}
{{qed}} | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau|p \implies q, q \implies r, p \vdash r}}
{{Premise|1|p \implies q}}
{{Premise|2|q \implies r}}
{{Premise|3|p}}
{{ModusPonens|4|1, 3|q|1|3}}
{{ModusPonens|5|1, 2, 3|r|2|4}}
{{EndTableau}}
{{qed}} | Hypothetical Syllogism/Formulation 2/Proof 2 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_2/Proof_2 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [] |
proofwiki-169 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | We apply the Method of Truth Tables to the propositions in turn.
As can be seen for all boolean interpretations by inspection, where the truth values under the main connective on the {{LHS}} (that is, the rightmost $\land$) is $\T$, that under the instance of $r$ on the {{RHS}} is also $\T$:
:<nowiki>$\begin{array}{|cc... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | We apply the [[Method of Truth Tables]] to the propositions in turn.
As can be seen for all [[Definition:Boolean Interpretation|boolean interpretations]] by inspection, where the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] on the {{LHS}} (that ... | Hypothetical Syllogism/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_2/Proof_by_Truth_Table | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Method of Truth Tables",
"Definition:Boolean Interpretation",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic"
] |
proofwiki-170 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | Let us use the following abbreviations:
{{begin-eqn}}
{{eqn | l = \phi
| o = \text{ for }
| r = p \implies q
| c =
}}
{{eqn | l = \psi
| o = \text{ for }
| r = q \implies r
| c =
}}
{{eqn | l = \chi
| o = \text{ for }
| r = p \implies r
| c =
}}
{{end-eqn}}
{{Beg... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | Let us use the following abbreviations:
{{begin-eqn}}
{{eqn | l = \phi
| o = \text{ for }
| r = p \implies q
| c =
}}
{{eqn | l = \psi
| o = \text{ for }
| r = q \implies r
| c =
}}
{{eqn | l = \chi
| o = \text{ for }
| r = p \implies r
| c =
}}
{{end-eqn}}
{{... | Hypothetical Syllogism/Formulation 3/Proof 1 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_3/Proof_1 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [] |
proofwiki-171 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | 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}{|ccccccc|c|ccc|} \hline
((p & \implies & q) & \land & (q & \implies & r)) & \implies & (p & \implies & r) \\
\hline
\F & \T & \F & \T & \F & \T &... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | 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}{|... | Hypothetical Syllogism/Formulation 3/Proof by Truth Table | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_3/Proof_by_Truth_Table | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-172 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | Let us use the following abbreviations
{{begin-eqn}}
{{eqn | l = \phi
| o = \text{ for }
| r = p \implies q
}}
{{eqn | l = \psi
| o = \text{ for }
| r = q \implies r
}}
{{eqn | l = \chi
| o = \text{ for }
| r = p \implies r
}}
{{end-eqn}}
{{BeginTableau|\paren {p \implies q} \implies... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | Let us use the following abbreviations
{{begin-eqn}}
{{eqn | l = \phi
| o = \text{ for }
| r = p \implies q
}}
{{eqn | l = \psi
| o = \text{ for }
| r = q \implies r
}}
{{eqn | l = \chi
| o = \text{ for }
| r = p \implies r
}}
{{end-eqn}}
{{BeginTableau|\paren {p \implies q} \impl... | Hypothetical Syllogism/Formulation 4/Proof 1 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_4/Proof_1 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Hypothetical Syllogism/Formulation 3",
"Rule of Exportation"
] |
proofwiki-173 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau|\paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} } }}
{{Assumption|1|p \implies q}}
{{Assumption|2|p}}
{{Assumption|3|q \implies r}}
{{ModusPonens|4|1,2|q|1|2}}
{{ModusPonens|5|1,2,3|r|3|4}}
{{Implication|6|1,3|p \implies r|2|5}}
{{Implication|7|1|\paren {q \imp... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau|\paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} } }}
{{Assumption|1|p \implies q}}
{{Assumption|2|p}}
{{Assumption|3|q \implies r}}
{{ModusPonens|4|1,2|q|1|2}}
{{ModusPonens|5|1,2,3|r|3|4}}
{{Implication|6|1,3|p \implies r|2|5}}
{{Implication|7|1|\paren {q \imp... | Hypothetical Syllogism/Formulation 4/Proof 2 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_4/Proof_2 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [] |
proofwiki-174 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau|\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }|instance 1 of a Hilbert proof system}}
{{Assumption|1|p}}
{{Assumption|2|p \implies q}}
{{ModusPonens|3|1, 2|q|1|2}}
{{Assumption|4|q \implies r}}
{{ModusPonens|5|1, 2, 4|r|3|4}}
{{TableauLine|n = 6
| p... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau|\vdash \paren {p \implies q} \implies \paren {\paren {q \implies r} \implies \paren {p \implies r} }|[[Definition:Hilbert Proof System/Instance 1|instance 1 of a Hilbert proof system]]}}
{{Assumption|1|p}}
{{Assumption|2|p \implies q}}
{{ModusPonens|3|1, 2|q|1|2}}
{{Assumption|4|q \implies r}}
{{ModusPon... | Hypothetical Syllogism/Formulation 4/Proof 3 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_4/Proof_3 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Definition:Hilbert Proof System/Instance 1",
"Definition:Assumption",
"Definition:Discharged Assumption",
"Definition:Assumption",
"Definition:Discharged Assumption",
"Definition:Assumption",
"Definition:Discharged Assumption"
] |
proofwiki-175 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | Let us use the following abbreviations
{{begin-eqn}}
{{eqn | l = \phi
| o = \text{ for }
| r = p \implies q
| c =
}}
{{eqn | l = \psi
| o = \text{ for }
| r = q \implies r
| c =
}}
{{eqn | l = \chi
| o = \text{ for }
| r = p \implies r
| c =
}}
{{end-eqn}}
From H... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | Let us use the following abbreviations
{{begin-eqn}}
{{eqn | l = \phi
| o = \text{ for }
| r = p \implies q
| c =
}}
{{eqn | l = \psi
| o = \text{ for }
| r = q \implies r
| c =
}}
{{eqn | l = \chi
| o = \text{ for }
| r = p \implies r
| c =
}}
{{end-eqn}}
Fro... | Hypothetical Syllogism/Formulation 5/Proof 1 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_5/Proof_1 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Hypothetical Syllogism/Formulation 3",
"Hypothetical Syllogism/Formulation 3",
"Rule of Exportation"
] |
proofwiki-176 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau |\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }|Instance 2 of the Hilbert-style systems}}
{{TableauLine
| n = 1
| f = \paren {q \implies r} \implies \paren {\paren {p \lor q} \implies \paren {p \lor r} }
| rlnk = Definition:Hilbert Proof System/Ins... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau |\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} }|[[Definition:Hilbert Proof System/Instance 2|Instance 2 of the Hilbert-style systems]]}}
{{TableauLine
| n = 1
| f = \paren {q \implies r} \implies \paren {\paren {p \lor q} \implies \paren {p \lor r} ... | Hypothetical Syllogism/Formulation 5/Proof 2 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_5/Proof_2 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [
"Definition:Hilbert Proof System/Instance 2"
] |
proofwiki-177 | Hypothetical Syllogism | The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{item||that $q$ implies $r$}}
{{end-itemize}} }}
{{item||then we may infer that $p$ implies $r$.}}
{{end-itemize}} | {{BeginTableau|\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption|1|q \implies r}}
{{Assumption|2|p \implies q}}
{{Assumption|3|p}}
{{ModusPonens|4|2,3|q|2|3}}
{{ModusPonens|5|1,2,3|r|1|4}}
{{Implication|6|1,2|p \implies r|3|5}}
{{Implication|7|1|\paren ... | The '''(rule of the) [[Hypothetical Syllogism|hypothetical syllogism]]''' is a [[Definition:Valid Argument|valid]] deduction [[Definition:Sequent|sequent]] in [[Definition:Propositional Logic|propositional logic]]:
{{begin-itemize}}
{{item||If we can conclude both:
{{begin-itemize}}
{{item||that $p$ implies $q$}}
{{it... | {{BeginTableau|\vdash \paren {q \implies r} \implies \paren {\paren {p \implies q} \implies \paren {p \implies r} } }}
{{Assumption|1|q \implies r}}
{{Assumption|2|p \implies q}}
{{Assumption|3|p}}
{{ModusPonens|4|2,3|q|2|3}}
{{ModusPonens|5|1,2,3|r|1|4}}
{{Implication|6|1,2|p \implies r|3|5}}
{{Implication|7|1|\paren ... | Hypothetical Syllogism/Formulation 5/Proof 3 | https://proofwiki.org/wiki/Hypothetical_Syllogism | https://proofwiki.org/wiki/Hypothetical_Syllogism/Formulation_5/Proof_3 | [
"Hypothetical Syllogism",
"Conditional"
] | [
"Hypothetical Syllogism",
"Definition:Valid Argument",
"Definition:Sequent",
"Definition:Propositional Logic"
] | [] |
proofwiki-178 | Extended Rule of Implication | Any sequent can be expressed as a theorem.
That is:
:$P_1, P_2, P_3, \ldots, P_n \vdash Q$
means the same thing as:
:$\vdash P_1 \implies \paren {P_2 \implies \paren {P_3 \implies \paren {\ldots \implies \paren {P_n \implies Q} \ldots} } }$
The latter expression is known as the corresponding conditional of the former.
... | From the Rule of Conjunction, we note the following.
Any sequent:
:$P_1, P_2 \vdash Q$
can be expressed as:
:$P_1 \land p_2 \vdash Q$
Also, from the Rule of Simplification, any sequent:
:$P_1 \land P_2 \vdash Q$
can be expressed as:
:$P_1, P_2 \vdash Q$
Consider the expression:
:$P_1, P_2, P_3, \ldots, P_{n - 1}, P_n \... | Any [[Definition:Sequent|sequent]] can be expressed as a [[Definition:Theorem of Logic|theorem]].
That is:
:$P_1, P_2, P_3, \ldots, P_n \vdash Q$
means the same thing as:
:$\vdash P_1 \implies \paren {P_2 \implies \paren {P_3 \implies \paren {\ldots \implies \paren {P_n \implies Q} \ldots} } }$
The latter expressi... | From the [[Rule of Conjunction/Proof Rule|Rule of Conjunction]], we note the following.
Any sequent:
:$P_1, P_2 \vdash Q$
can be expressed as:
:$P_1 \land p_2 \vdash Q$
Also, from the [[Rule of Simplification/Proof Rule|Rule of Simplification]], any [[Definition:Sequent|sequent]]:
:$P_1 \land P_2 \vdash Q$
can ... | Extended Rule of Implication | https://proofwiki.org/wiki/Extended_Rule_of_Implication | https://proofwiki.org/wiki/Extended_Rule_of_Implication | [
"Conditional",
"Conjunction"
] | [
"Definition:Sequent",
"Definition:Theorem/Logic",
"Definition:Corresponding Conditional",
"Definition:Sequent",
"Definition:Symbol",
"Definition:Theorem/Logic"
] | [
"Rule of Conjunction/Proof Rule",
"Rule of Simplification/Proof Rule",
"Definition:Sequent",
"Rule of Substitution",
"Rule of Implication",
"Rule of Exportation",
"Rule of Substitution",
"Rule of Exportation"
] |
proofwiki-179 | Equivalences are Interderivable | If two propositional formulas are interderivable, they are equivalent:
:$\paren {p \dashv \vdash q} \dashv \vdash \paren {p \iff q}$ | Let $v$ be an arbitrary interpretation.
Then by definition of interderivable:
:$\map v {p \iff q}$ {{iff}} $\map v p = \map v q$
Since $v$ is arbitrary, $\map v p = \map v q$ holds in all interpretations.
That is:
:$p \dashv \vdash q$
{{qed}} | If two [[Definition:Propositional Formula|propositional formulas]] are [[Definition:Interderivable|interderivable]], they are [[Definition:Biconditional|equivalent]]:
:$\paren {p \dashv \vdash q} \dashv \vdash \paren {p \iff q}$ | Let $v$ be an arbitrary [[Definition:Boolean Interpretation|interpretation]].
Then by definition of [[Definition:Interderivable|interderivable]]:
:$\map v {p \iff q}$ {{iff}} $\map v p = \map v q$
Since $v$ is arbitrary, $\map v p = \map v q$ holds in all [[Definition:Boolean Interpretation|interpretations]].
That i... | Equivalences are Interderivable/Proof 2 | https://proofwiki.org/wiki/Equivalences_are_Interderivable | https://proofwiki.org/wiki/Equivalences_are_Interderivable/Proof_2 | [
"Equivalences are Interderivable",
"Biconditional"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Logical Equivalence",
"Definition:Biconditional"
] | [
"Definition:Boolean Interpretation",
"Definition:Logical Equivalence",
"Definition:Boolean Interpretation"
] |
proofwiki-180 | Equivalences are Interderivable | If two propositional formulas are interderivable, they are equivalent:
:$\paren {p \dashv \vdash q} \dashv \vdash \paren {p \iff q}$ | The result follows directly from the truth table for the biconditional:
:<nowiki>$\begin {array} {|cc||ccc|} \hline
p & q & p & \iff & q \\
\hline
\F & \F & \F & \T & \F \\
\F & \T & \F & \F & \T \\
\T & \F & \T & \F & \F \\
\T & \T & \T & \T & \T \\
\hline
\end {array}$</nowiki>
By inspection, it is seen that $\map \M... | If two [[Definition:Propositional Formula|propositional formulas]] are [[Definition:Interderivable|interderivable]], they are [[Definition:Biconditional|equivalent]]:
:$\paren {p \dashv \vdash q} \dashv \vdash \paren {p \iff q}$ | The result follows directly from the [[Definition:Truth Table|truth table]] for the [[Definition:Biconditional|biconditional]]:
:<nowiki>$\begin {array} {|cc||ccc|} \hline
p & q & p & \iff & q \\
\hline
\F & \F & \F & \T & \F \\
\F & \T & \F & \F & \T \\
\T & \F & \T & \F & \F \\
\T & \T & \T & \T & \T \\
\hline
\end ... | Equivalences are Interderivable/Proof by Truth Table | https://proofwiki.org/wiki/Equivalences_are_Interderivable | https://proofwiki.org/wiki/Equivalences_are_Interderivable/Proof_by_Truth_Table | [
"Equivalences are Interderivable",
"Biconditional"
] | [
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Definition:Logical Equivalence",
"Definition:Biconditional"
] | [
"Definition:Truth Table",
"Definition:Biconditional"
] |
proofwiki-181 | Union is Commutative | Set union is commutative:
:$S \cup T = T \cup S$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \paren {S \cup T}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x \in S
| o = \lor
| r = x \in T
| c = {{Defof|Set Union}}
}}
{{eqn | ll= \leadstoandfrom
| l = x \in T
| o = \lor
| r = x \in S
| c = Disjunction i... | [[Definition:Set Union|Set union]] is [[Definition:Commutative Operation|commutative]]:
:$S \cup T = T \cup S$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \paren {S \cup T}
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = x \in S
| o = \lor
| r = x \in T
| c = {{Defof|Set Union}}
}}
{{eqn | ll= \leadstoandfrom
| l = x \in T
| o = \lor
| r = x \in S
| c = [[Disjunction... | Union is Commutative | https://proofwiki.org/wiki/Union_is_Commutative | https://proofwiki.org/wiki/Union_is_Commutative | [
"Union is Commutative",
"Set Union",
"Commutative Laws of Set Theory",
"Examples of Commutative Operations"
] | [
"Definition:Set Union",
"Definition:Commutative/Operation"
] | [
"Rule of Commutation/Disjunction"
] |
proofwiki-182 | Intersection is Commutative | Set intersection is commutative:
:$S \cap T = T \cap S$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \paren {S \cap T}
}}
{{eqn | ll= \leadstoandfrom
| l = x \in S
| o = \land
| r = x \in T
| c = {{Defof|Set Intersection}}
}}
{{eqn | ll= \leadstoandfrom
| l = x \in T
| o = \land
| r = x \in S
| c = Conjunction is Co... | [[Definition:Set Intersection|Set intersection]] is [[Definition:Commutative Operation|commutative]]:
:$S \cap T = T \cap S$ | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = \paren {S \cap T}
}}
{{eqn | ll= \leadstoandfrom
| l = x \in S
| o = \land
| r = x \in T
| c = {{Defof|Set Intersection}}
}}
{{eqn | ll= \leadstoandfrom
| l = x \in T
| o = \land
| r = x \in S
| c = [[Conjunction is ... | Intersection is Commutative | https://proofwiki.org/wiki/Intersection_is_Commutative | https://proofwiki.org/wiki/Intersection_is_Commutative | [
"Intersection is Commutative",
"Set Intersection",
"Commutative Laws of Set Theory",
"Examples of Commutative Operations"
] | [
"Definition:Set Intersection",
"Definition:Commutative/Operation"
] | [
"Rule of Commutation/Conjunction"
] |
proofwiki-183 | Intersection is Commutative | Set intersection is commutative:
:$S \cap T = T \cap S$ | We have that both $\ds \bigcap_{j \mathop \in J} S_j$ and $\ds \bigcap_{k \mathop \in \relcomp I J} S_k$ are sets.
Hence by Intersection is Commutative we have:
:$\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j... | [[Definition:Set Intersection|Set intersection]] is [[Definition:Commutative Operation|commutative]]:
:$S \cap T = T \cap S$ | We have that both $\ds \bigcap_{j \mathop \in J} S_j$ and $\ds \bigcap_{k \mathop \in \relcomp I J} S_k$ are [[Definition:Set|sets]].
Hence by [[Intersection is Commutative]] we have:
:$\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigc... | Intersection is Commutative/Family of Sets/Proof 1 | https://proofwiki.org/wiki/Intersection_is_Commutative | https://proofwiki.org/wiki/Intersection_is_Commutative/Family_of_Sets/Proof_1 | [
"Intersection is Commutative",
"Set Intersection",
"Commutative Laws of Set Theory",
"Examples of Commutative Operations"
] | [
"Definition:Set Intersection",
"Definition:Commutative/Operation"
] | [
"Definition:Set",
"Intersection is Commutative",
"Definition:Set Equality"
] |
proofwiki-184 | Intersection is Commutative | Set intersection is commutative:
:$S \cap T = T \cap S$ | {{begin-eqn}}
{{eqn | l = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k
| r = \map \complement {\map \complement {\bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k} }
| c = Complement of Complement
}}
{{eqn | r = \map \complement {\map \complement {\bi... | [[Definition:Set Intersection|Set intersection]] is [[Definition:Commutative Operation|commutative]]:
:$S \cap T = T \cap S$ | {{begin-eqn}}
{{eqn | l = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k
| r = \map \complement {\map \complement {\bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k} }
| c = [[Complement of Complement]]
}}
{{eqn | r = \map \complement {\map \complement ... | Intersection is Commutative/Family of Sets/Proof 2 | https://proofwiki.org/wiki/Intersection_is_Commutative | https://proofwiki.org/wiki/Intersection_is_Commutative/Family_of_Sets/Proof_2 | [
"Intersection is Commutative",
"Set Intersection",
"Commutative Laws of Set Theory",
"Examples of Commutative Operations"
] | [
"Definition:Set Intersection",
"Definition:Commutative/Operation"
] | [
"Complement of Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection",
"Union is Commutative/Family of Sets",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of I... |
proofwiki-185 | Russell's Paradox | The {{axiom-link|Abstraction}} leads to a contradiction. | {{AimForCont}} there does exist such an $x$.
Let $\RR$ be such that $\map \RR {x, x}$.
Then $\neg \map \RR {x, x}$.
Hence it cannot be the case that $\map \RR {x, x}$.
Now suppose that $\neg \map \RR {x, x}$.
Then by definition of $x$ it follows that $\map \RR {x, x}$.
In both cases a contradiction results.
Hence there... | The {{axiom-link|Abstraction}} leads to a [[Definition:Contradiction|contradiction]]. | {{AimForCont}} there does exist such an $x$.
Let $\RR$ be such that $\map \RR {x, x}$.
Then $\neg \map \RR {x, x}$.
Hence it cannot be the case that $\map \RR {x, x}$.
Now suppose that $\neg \map \RR {x, x}$.
Then by definition of $x$ it follows that $\map \RR {x, x}$.
In both cases a [[Definition:Contradiction|c... | Russell's Paradox/Corollary/Proof 1 | https://proofwiki.org/wiki/Russell's_Paradox | https://proofwiki.org/wiki/Russell's_Paradox/Corollary/Proof_1 | [
"Russell's Paradox",
"Axiom of Abstraction",
"Naive Set Theory",
"Subsets",
"Antinomies",
"Logical Paradoxes"
] | [
"Definition:Contradiction"
] | [
"Definition:Contradiction"
] |
proofwiki-186 | Russell's Paradox | The {{axiom-link|Abstraction}} leads to a contradiction. | {{AimForCont}}:
:$\exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$
By Existential Instantiation:
:$\forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$
By Universal Instantiation:
:$\map \RR {x, x} \iff \neg \map \RR {x, x} $
But this contradicts Biconditional of Proposition and its... | The {{axiom-link|Abstraction}} leads to a [[Definition:Contradiction|contradiction]]. | {{AimForCont}}:
:$\exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$
By [[Existential Instantiation]]:
:$\forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$
By [[Universal Instantiation]]:
:$\map \RR {x, x} \iff \neg \map \RR {x, x} $
But this [[Definition:Contradiction|contradict... | Russell's Paradox/Corollary/Proof 2 | https://proofwiki.org/wiki/Russell's_Paradox | https://proofwiki.org/wiki/Russell's_Paradox/Corollary/Proof_2 | [
"Russell's Paradox",
"Axiom of Abstraction",
"Naive Set Theory",
"Subsets",
"Antinomies",
"Logical Paradoxes"
] | [
"Definition:Contradiction"
] | [
"Existential Instantiation",
"Universal Instantiation",
"Definition:Contradiction",
"Biconditional of Proposition and its Negation"
] |
proofwiki-187 | Russell's Paradox | The {{axiom-link|Abstraction}} leads to a contradiction. | Sets have elements.
Some of those elements may themselves be sets.
So, given two sets $S$ and $T$, we can ask the question:
:''Is $S$ an element of $T$?''
The answer will either be ''yes'' or ''no''.
In particular, given any set $S$, we can ask the question:
:''Is $S$ an element of $S$?''
Again, the answer will either ... | The {{axiom-link|Abstraction}} leads to a [[Definition:Contradiction|contradiction]]. | [[Definition:Set|Sets]] have [[Definition:Element|elements]].
Some of those [[Definition:Element|elements]] may themselves be [[Definition:Set|sets]].
So, given two [[Definition:Set|sets]] $S$ and $T$, we can ask the question:
:''Is $S$ an [[Definition:Element|element]] of $T$?''
The answer will either be ''yes'' or ... | Russell's Paradox/Proof 1 | https://proofwiki.org/wiki/Russell's_Paradox | https://proofwiki.org/wiki/Russell's_Paradox/Proof_1 | [
"Russell's Paradox",
"Axiom of Abstraction",
"Naive Set Theory",
"Subsets",
"Antinomies",
"Logical Paradoxes"
] | [
"Definition:Contradiction"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Element",
"Definition:Set",
"Definition:Set",
"Definition:Element",
"Definition:Set",
"Definition:Element",
"Definition:Set",
"Definition:Ordinary Set",
"Definition:Element",
"Definition:Extraordinary Set",
"Definition:Element",
"Definiti... |
proofwiki-188 | Russell's Paradox | The {{axiom-link|Abstraction}} leads to a contradiction. | {{AimForCont}} the {{axiom-link|Abstraction}}, that for all predicates $P$ where $S$ is not free:
:$\exists S : \forall x : \paren {x \in S \iff \map P x}$
Since $x \notin x$ is a predicate where $S$ is not free, it follows that:
:$\exists S : \forall x : \paren {x \in S \iff x \notin x}$
is an instance of the {{axiom-... | The {{axiom-link|Abstraction}} leads to a [[Definition:Contradiction|contradiction]]. | {{AimForCont}} the {{axiom-link|Abstraction}}, that for all [[Definition:Predicate|predicates]] $P$ where $S$ is not [[Definition:Free Variable|free]]:
:$\exists S : \forall x : \paren {x \in S \iff \map P x}$
Since $x \notin x$ is a [[Definition:Predicate|predicate]] where $S$ is not [[Definition:Free Variable|free]]... | Russell's Paradox/Proof 2 | https://proofwiki.org/wiki/Russell's_Paradox | https://proofwiki.org/wiki/Russell's_Paradox/Proof_2 | [
"Russell's Paradox",
"Axiom of Abstraction",
"Naive Set Theory",
"Subsets",
"Antinomies",
"Logical Paradoxes"
] | [
"Definition:Contradiction"
] | [
"Definition:Predicate",
"Definition:Free Variable",
"Definition:Predicate",
"Definition:Free Variable",
"Existential Instantiation",
"Universal Instantiation",
"Definition:Contradiction",
"Biconditional of Proposition and its Negation",
"Definition:Contradiction"
] |
proofwiki-189 | Solution to Quadratic Equation | The quadratic equation of the form $a x^2 + b x + c = 0$ has solutions:
:$x = \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}$ | {{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = 0
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = \dfrac {\paren {2 a x + b}^2 - b^2 + 4 a c} {4 a}
| r = 0
| c = Completing the Square
}}
{{eqn | ll= \leadsto
| l = \paren {2 a x + b}^2
| r = b^2 - 4 a c
| c = simplifica... | The [[Definition:Quadratic Equation|quadratic equation]] of the form $a x^2 + b x + c = 0$ has [[Definition:Solution to Equation|solutions]]:
:$x = \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}$ | {{begin-eqn}}
{{eqn | l = a x^2 + b x + c
| r = 0
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = \dfrac {\paren {2 a x + b}^2 - b^2 + 4 a c} {4 a}
| r = 0
| c = [[Completing the Square]]
}}
{{eqn | ll= \leadsto
| l = \paren {2 a x + b}^2
| r = b^2 - 4 a c
| c = simpli... | Solution to Quadratic Equation | https://proofwiki.org/wiki/Solution_to_Quadratic_Equation | https://proofwiki.org/wiki/Solution_to_Quadratic_Equation | [
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Fiber of Truth/Solution"
] | [
"Completing the Square"
] |
proofwiki-190 | Normal Subgroup Test | Let $G$ be a group and $H \le G$.
Then $H$ is a normal subgroup (by definition 1) of $G$ {{iff}}:
:$\forall x \in G: x H x^{-1} \subseteq H$. | Let $H$ be a subgroup of $G$.
Suppose $H$ is normal in $G$.
Then $\forall x \in G, a \in H: \exists b \in H: x a = b x$.
Thus, $x a x^{-1} = b \in H$ implying $x H x^{-1} \subseteq H$.
Conversely, suppose $\forall x \in G: x H x^{-1} \subseteq H$.
Then for $g \in G$, we have $g H g^{-1} \subseteq H$, which implies $g H... | Let $G$ be a [[Definition:Group|group]] and $H \le G$.
Then $H$ is a [[Definition:Normal Subgroup/Definition 1|normal subgroup (by definition 1)]] of $G$ {{iff}}:
:$\forall x \in G: x H x^{-1} \subseteq H$. | Let $H$ be a [[Definition:Subgroup|subgroup]] of $G$.
Suppose $H$ is [[Definition:Normal Subgroup|normal]] in $G$.
Then $\forall x \in G, a \in H: \exists b \in H: x a = b x$.
Thus, $x a x^{-1} = b \in H$ implying $x H x^{-1} \subseteq H$.
Conversely, suppose $\forall x \in G: x H x^{-1} \subseteq H$.
Then for $... | Normal Subgroup Test | https://proofwiki.org/wiki/Normal_Subgroup_Test | https://proofwiki.org/wiki/Normal_Subgroup_Test | [
"Normal Subgroups"
] | [
"Definition:Group",
"Definition:Normal Subgroup/Definition 1"
] | [
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Category:Normal Subgroups"
] |
proofwiki-191 | Equality of Ordered Pairs | Two ordered pairs are equal {{iff}} corresponding coordinates are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | === Necessary Condition ===
{{:Equality of Ordered Pairs/Necessary Condition}}{{qed|lemma}} | Two [[Definition:Ordered Pair|ordered pairs]] are [[Definition:Equality|equal]] {{iff}} corresponding [[Definition:Coordinate of Ordered Pair|coordinates]] are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | === [[Equality of Ordered Pairs/Necessary Condition|Necessary Condition]] ===
{{:Equality of Ordered Pairs/Necessary Condition}}{{qed|lemma}} | Equality of Ordered Pairs | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs | [
"Equality of Ordered Pairs",
"Cartesian Product",
"Axiomatic Set Theory",
"Equality of Ordered Tuples",
"Equality"
] | [
"Definition:Ordered Pair",
"Definition:Equals",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair"
] | [
"Equality of Ordered Pairs/Necessary Condition"
] |
proofwiki-192 | Equality of Ordered Pairs | Two ordered pairs are equal {{iff}} corresponding coordinates are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | First a lemma:
{{:Equality of Ordered Pairs/Lemma}}{{qed|lemma}}
Let $\tuple {a, b} = \tuple {c, d}$.
From the empty set formalization:
:$\set {\set {\O, a}, \set {\set \O, b} } = \set {\set {\O, c}, \set {\set \O, d} }$
First we note the special case where $a = \set \O$ and $b = \O$.
Then we have:
{{begin-eqn}}
{{eqn ... | Two [[Definition:Ordered Pair|ordered pairs]] are [[Definition:Equality|equal]] {{iff}} corresponding [[Definition:Coordinate of Ordered Pair|coordinates]] are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | First a [[Equality of Ordered Pairs/Lemma|lemma]]:
{{:Equality of Ordered Pairs/Lemma}}{{qed|lemma}}
Let $\tuple {a, b} = \tuple {c, d}$.
From the [[Definition:Empty Set Formalization of Ordered Pair|empty set formalization]]:
:$\set {\set {\O, a}, \set {\set \O, b} } = \set {\set {\O, c}, \set {\set \O, d} }$
Fir... | Equality of Ordered Pairs/Necessary Condition/Proof from Empty Set Formalization | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs/Necessary_Condition/Proof_from_Empty_Set_Formalization | [
"Equality of Ordered Pairs",
"Cartesian Product",
"Axiomatic Set Theory",
"Equality of Ordered Tuples",
"Equality"
] | [
"Definition:Ordered Pair",
"Definition:Equals",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair"
] | [
"Equality of Ordered Pairs/Lemma",
"Definition:Ordered Pair/Empty Set Formalization",
"Doubleton Class of Equal Sets is Singleton Class",
"Proof by Contradiction",
"Equality of Ordered Pairs/Lemma",
"Proof by Contradiction",
"Equality of Ordered Pairs/Lemma"
] |
proofwiki-193 | Equality of Ordered Pairs | Two ordered pairs are equal {{iff}} corresponding coordinates are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | First a lemma:
{{:Equality of Ordered Pairs/Lemma}}{{qed|lemma}}
Let $\tuple {a, b} = \tuple {c, d}$.
From the Kuratowski formalization:
:$\set {\set a, \set {a, b} } = \set {\set c, \set {c, d} }$
There are two cases: either $a = b$, or $a \ne b$.
==== Case 1 ====
Suppose $a = b$.
Then:
:$\set {\set a, \set {a, b} } ... | Two [[Definition:Ordered Pair|ordered pairs]] are [[Definition:Equality|equal]] {{iff}} corresponding [[Definition:Coordinate of Ordered Pair|coordinates]] are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | First a [[Equality of Ordered Pairs/Lemma|lemma]]:
{{:Equality of Ordered Pairs/Lemma}}{{qed|lemma}}
Let $\tuple {a, b} = \tuple {c, d}$.
From the [[Definition:Kuratowski Formalization of Ordered Pair|Kuratowski formalization]]:
:$\set {\set a, \set {a, b} } = \set {\set c, \set {c, d} }$
There are two cases: eith... | Equality of Ordered Pairs/Necessary Condition/Proof from Kuratowski Formalization | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs/Necessary_Condition/Proof_from_Kuratowski_Formalization | [
"Equality of Ordered Pairs",
"Cartesian Product",
"Axiomatic Set Theory",
"Equality of Ordered Tuples",
"Equality"
] | [
"Definition:Ordered Pair",
"Definition:Equals",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair"
] | [
"Equality of Ordered Pairs/Lemma",
"Definition:Ordered Pair/Kuratowski Formalization",
"Definition:Element",
"Definition:Distinct/Plural",
"Equality of Ordered Pairs/Lemma"
] |
proofwiki-194 | Equality of Ordered Pairs | Two ordered pairs are equal {{iff}} corresponding coordinates are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | First a lemma:
{{:Equality of Ordered Pairs/Lemma}}{{qed|lemma}}
Let $\tuple {a, b} = \tuple {c, d}$.
From the Wiener formalization:
:$\set {\set {\O, \set a}, \set {\set b} } = \set {\set {\O, \set c}, \set {\set d} }$
Let $x \in \set {\set {\O, \set a}, \set {\set b} }$.
Then either:
:$x = \set {\O, \set a}$
or:
:$x ... | Two [[Definition:Ordered Pair|ordered pairs]] are [[Definition:Equality|equal]] {{iff}} corresponding [[Definition:Coordinate of Ordered Pair|coordinates]] are equal:
:$\tuple {a, b} = \tuple {c, d} \iff a = c \land b = d$ | First a [[Equality of Ordered Pairs/Lemma|lemma]]:
{{:Equality of Ordered Pairs/Lemma}}{{qed|lemma}}
Let $\tuple {a, b} = \tuple {c, d}$.
From the [[Definition:Wiener Formalization of Ordered Pair|Wiener formalization]]:
:$\set {\set {\O, \set a}, \set {\set b} } = \set {\set {\O, \set c}, \set {\set d} }$
Let $x ... | Equality of Ordered Pairs/Necessary Condition/Proof from Wiener Formalization | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs | https://proofwiki.org/wiki/Equality_of_Ordered_Pairs/Necessary_Condition/Proof_from_Wiener_Formalization | [
"Equality of Ordered Pairs",
"Cartesian Product",
"Axiomatic Set Theory",
"Equality of Ordered Tuples",
"Equality"
] | [
"Definition:Ordered Pair",
"Definition:Equals",
"Definition:Coordinate System/Coordinate/Element of Ordered Pair"
] | [
"Equality of Ordered Pairs/Lemma",
"Definition:Ordered Pair/Wiener Formalization",
"Definition:Contradiction",
"Definition:Empty Set",
"Proof by Contradiction",
"Equality of Ordered Pairs/Lemma",
"Singleton Equality",
"Proof by Contradiction",
"Singleton Equality",
"Singleton Equality"
] |
proofwiki-195 | Identity is Unique | Let $\struct {S, \circ}$ be an algebraic structure that has an identity element $e \in S$.
Then $e$ is unique. | Suppose $e_1$ and $e_2$ are both identity elements of $\struct {S, \circ}$.
Then by the definition of identity element:
:$\forall s \in S: s \circ e_1 = s = e_2 \circ s$
Then:
:$e_1 = e_2 \circ e_1 = e_2$
So:
:$e_1 = e_2$
and there is only one identity element after all.
{{qed}} | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] that has an [[Definition:Identity Element|identity element]] $e \in S$.
Then $e$ is [[Definition:Unique|unique]]. | Suppose $e_1$ and $e_2$ are both [[Definition:Identity Element|identity elements]] of $\struct {S, \circ}$.
Then by the definition of [[Definition:Identity Element|identity element]]:
:$\forall s \in S: s \circ e_1 = s = e_2 \circ s$
Then:
:$e_1 = e_2 \circ e_1 = e_2$
So:
:$e_1 = e_2$
and there is only one [[Definit... | Identity is Unique/Proof 1 | https://proofwiki.org/wiki/Identity_is_Unique | https://proofwiki.org/wiki/Identity_is_Unique/Proof_1 | [
"Identity Elements",
"Identity is Unique",
"Algebraic Structures"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unique"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-196 | Identity is Unique | Let $\struct {S, \circ}$ be an algebraic structure that has an identity element $e \in S$.
Then $e$ is unique. | Let $e_S$ be an identity of $\struct {S, \circ}$.
Then by definition, $e_S$ is both a left identity and a right identity.
By More than one Left Identity then no Right Identity, if there is more than one of either, there cannot be one of the other.
So there can be only one of each.
By Left and Right Identity are the Sam... | Let $\struct {S, \circ}$ be an [[Definition:Algebraic Structure with One Operation|algebraic structure]] that has an [[Definition:Identity Element|identity element]] $e \in S$.
Then $e$ is [[Definition:Unique|unique]]. | Let $e_S$ be an identity of $\struct {S, \circ}$.
Then by [[Definition:Identity Element|definition]], $e_S$ is both a [[Definition:Left Identity|left identity]] and a [[Definition:Right Identity|right identity]].
By [[More than one Left Identity then no Right Identity]], if there is more than one of either, there can... | Identity is Unique/Proof 2 | https://proofwiki.org/wiki/Identity_is_Unique | https://proofwiki.org/wiki/Identity_is_Unique/Proof_2 | [
"Identity Elements",
"Identity is Unique",
"Algebraic Structures"
] | [
"Definition:Algebraic Structure/One Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Unique"
] | [
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Left Identity",
"Definition:Identity (Abstract Algebra)/Right Identity",
"More than one Left Identity then no Right Identity",
"Left and Right Identity are the Same"
] |
proofwiki-197 | Cancellation Laws | Let $G$ be a group.
Let $a, b, c \in G$.
Then the following hold:
;Right cancellation law
:$b a = c a \implies b = c$
;Left cancellation law
:$a b = a c \implies b = c$ | {{begin-eqn}}
{{eqn | l = g h
| r = g
| c =
}}
{{eqn | ll= \leadsto
| l = g h
| r = g e
| c = {{Group-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = h
| r = e
| c = Left Cancellation Law
}}
{{end-eqn}}
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $a, b, c \in G$.
Then the following hold:
;Right cancellation law
:$b a = c a \implies b = c$
;Left cancellation law
:$a b = a c \implies b = c$ | {{begin-eqn}}
{{eqn | l = g h
| r = g
| c =
}}
{{eqn | ll= \leadsto
| l = g h
| r = g e
| c = {{Group-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = h
| r = e
| c = [[Left Cancellation Law]]
}}
{{end-eqn}}
{{qed}} | Cancellation Laws/Corollary 1/Proof 1 | https://proofwiki.org/wiki/Cancellation_Laws | https://proofwiki.org/wiki/Cancellation_Laws/Corollary_1/Proof_1 | [
"Cancellation Laws",
"Group Theory",
"Cancellability",
"Named Theorems"
] | [
"Definition:Group"
] | [
"Cancellation Laws"
] |
proofwiki-198 | Cancellation Laws | Let $G$ be a group.
Let $a, b, c \in G$.
Then the following hold:
;Right cancellation law
:$b a = c a \implies b = c$
;Left cancellation law
:$a b = a c \implies b = c$ | {{begin-eqn}}
{{eqn | l = g h
| r = g
| c =
}}
{{eqn | ll= \leadsto
| l = g^{-1} \paren {g h}
| r = g^{-1} g
| c = {{Group-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = \paren {g^{-1} g} h
| r = g^{-1} g
| c = {{Group-axiom|1}}
}}
{{eqn | ll= \leadsto
| l = e h
| r... | Let $G$ be a [[Definition:Group|group]].
Let $a, b, c \in G$.
Then the following hold:
;Right cancellation law
:$b a = c a \implies b = c$
;Left cancellation law
:$a b = a c \implies b = c$ | {{begin-eqn}}
{{eqn | l = g h
| r = g
| c =
}}
{{eqn | ll= \leadsto
| l = g^{-1} \paren {g h}
| r = g^{-1} g
| c = {{Group-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = \paren {g^{-1} g} h
| r = g^{-1} g
| c = {{Group-axiom|1}}
}}
{{eqn | ll= \leadsto
| l = e h
| r... | Cancellation Laws/Corollary 1/Proof 2 | https://proofwiki.org/wiki/Cancellation_Laws | https://proofwiki.org/wiki/Cancellation_Laws/Corollary_1/Proof_2 | [
"Cancellation Laws",
"Group Theory",
"Cancellability",
"Named Theorems"
] | [
"Definition:Group"
] | [] |
proofwiki-199 | Cancellation Laws | Let $G$ be a group.
Let $a, b, c \in G$.
Then the following hold:
;Right cancellation law
:$b a = c a \implies b = c$
;Left cancellation law
:$a b = a c \implies b = c$ | {{begin-eqn}}
{{eqn | l = h g
| r = g
| c =
}}
{{eqn | ll= \leadsto
| l = h g
| r = e g
| c = {{Group-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = h
| r = e
| c = Right Cancellation Law
}}
{{end-eqn}}
{{qed}} | Let $G$ be a [[Definition:Group|group]].
Let $a, b, c \in G$.
Then the following hold:
;Right cancellation law
:$b a = c a \implies b = c$
;Left cancellation law
:$a b = a c \implies b = c$ | {{begin-eqn}}
{{eqn | l = h g
| r = g
| c =
}}
{{eqn | ll= \leadsto
| l = h g
| r = e g
| c = {{Group-axiom|2}}
}}
{{eqn | ll= \leadsto
| l = h
| r = e
| c = [[Right Cancellation Law]]
}}
{{end-eqn}}
{{qed}} | Cancellation Laws/Corollary 2/Proof 1 | https://proofwiki.org/wiki/Cancellation_Laws | https://proofwiki.org/wiki/Cancellation_Laws/Corollary_2/Proof_1 | [
"Cancellation Laws",
"Group Theory",
"Cancellability",
"Named Theorems"
] | [
"Definition:Group"
] | [
"Cancellation Laws"
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.