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-2700 | Derivatives of PGF of Shifted Geometric Distribution | Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$.
Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are:
:$\map {\dfrac {\d^n} {\d s^n} } {\map {\Pi_X} s} = \dfrac {p q^{n - 1} \paren {n - 1}!} {\paren {1 - q s}^{n + 1} }$
where $q = 1 - p$. | The Probability Generating Function of Shifted Geometric Distribution is:
:$\map {\Pi_X} s = \dfrac {p s} {1 - q s}$
where $q = 1 - p$.
First we need to obtain the first derivative:
{{begin-eqn}}
{{eqn | l = \map {\Pi'_X} s
| r = \map {\frac \d {\d s} } {\frac {p s} {1 - q s} }
| c =
}}
{{eqn | r = p s \ma... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Shifted Geometric Distribution|shifted geometric distribution with parameter $p$]].
Then the [[Definition:Higher Derivative|derivatives]] of the [[Definition:Probability Generating Function|PGF]] of $X$ {{WRT|Different... | The [[Probability Generating Function of Shifted Geometric Distribution]] is:
:$\map {\Pi_X} s = \dfrac {p s} {1 - q s}$
where $q = 1 - p$.
First we need to obtain the first derivative:
{{begin-eqn}}
{{eqn | l = \map {\Pi'_X} s
| r = \map {\frac \d {\d s} } {\frac {p s} {1 - q s} }
| c =
}}
{{eqn | r = p... | Derivatives of PGF of Shifted Geometric Distribution | https://proofwiki.org/wiki/Derivatives_of_PGF_of_Shifted_Geometric_Distribution | https://proofwiki.org/wiki/Derivatives_of_PGF_of_Shifted_Geometric_Distribution | [
"Geometric Distribution",
"Derivatives of PGFs"
] | [
"Definition:Random Variable/Discrete",
"Definition:Geometric Distribution/Shifted",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Probability Generating Function"
] | [
"Probability Generating Function of Shifted Geometric Distribution",
"Sum Rule for Derivatives",
"Derivatives of PGF of Geometric Distribution",
"Derivatives of Function of a x + b",
"Nth Derivative of Reciprocal of Mth Power",
"Definition:Rising Factorial",
"Category:Geometric Distribution",
"Categor... |
proofwiki-2701 | Derivatives of PGF of Discrete Uniform Distribution | Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$.
Then the derivatives of the PGF of $X$ {{WRT|Differentiation}} $s$ are:
$\quad\dfrac {\d^m} {\d s^m} \map {\Pi_X} s = \begin{cases}
\ds \dfrac 1 n \sum_{k \mathop = m}^n k^{\underline m} s^{k - m} & : m \le n \\
0 & : k > n... | The Probability Generating Function of Discrete Uniform Distribution is:
:$\map {\Pi_X} s = \dfrac {s \paren {1 - s^n} } {n \paren {1 - s} } = \dfrac 1 n \ds \sum_{k \mathop = 1}^n s^k$
From Nth Derivative of Mth Power:
$\quad\dfrac {\d^k} {\d s^k} s^n = \begin {cases}
n^{\underline k} s^{n - k} & : k \le n \\
0 & : k ... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Discrete Uniform Distribution|discrete uniform distribution with parameter $n$]].
Then the [[Definition:Higher Derivatives|derivatives]] of the [[Definition:Probability Generating Function|PGF]] of $X$ {{WRT|Differentia... | The [[Probability Generating Function of Discrete Uniform Distribution]] is:
:$\map {\Pi_X} s = \dfrac {s \paren {1 - s^n} } {n \paren {1 - s} } = \dfrac 1 n \ds \sum_{k \mathop = 1}^n s^k$
From [[Nth Derivative of Mth Power]]:
$\quad\dfrac {\d^k} {\d s^k} s^n = \begin {cases}
n^{\underline k} s^{n - k} & : k \le n ... | Derivatives of PGF of Discrete Uniform Distribution | https://proofwiki.org/wiki/Derivatives_of_PGF_of_Discrete_Uniform_Distribution | https://proofwiki.org/wiki/Derivatives_of_PGF_of_Discrete_Uniform_Distribution | [
"Discrete Uniform Distribution",
"Derivatives of PGFs"
] | [
"Definition:Random Variable/Discrete",
"Definition:Uniform Distribution/Discrete",
"Definition:Derivative/Higher Derivatives",
"Definition:Probability Generating Function",
"Definition:Falling Factorial"
] | [
"Probability Generating Function of Discrete Uniform Distribution",
"Nth Derivative of Mth Power",
"Category:Discrete Uniform Distribution",
"Category:Derivatives of PGFs"
] |
proofwiki-2702 | Total Expectation Theorem | Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete random variable on $\EE$.
Let $\set {B_1 \mid B_2 \mid \cdots}$ be a partition of $\Omega$ such that $\map \Pr {B_i} > 0$ for each $i$.
Then:
:$\ds \expect X = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}$
whenever this sum converg... | {{begin-eqn}}
{{eqn | l = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}
| r = \sum_i \sum_x x \, \map \Pr {\set {X = x} \cap B_i}
| c = {{Defof|Conditional Expectation}}
}}
{{eqn | r = \sum_x x \map \Pr {\set {X \in x} \cap \paren {\bigcup_i B_i} }
| c =
}}
{{eqn | r = \sum_x x \map \Pr {X = x}
... | Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] on $\EE$.
Let $\set {B_1 \mid B_2 \mid \cdots}$ be a [[Definition:Partition (Probability Theory)|partition]] of $\Omega$ such that $\map \Pr ... | {{begin-eqn}}
{{eqn | l = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}
| r = \sum_i \sum_x x \, \map \Pr {\set {X = x} \cap B_i}
| c = {{Defof|Conditional Expectation}}
}}
{{eqn | r = \sum_x x \map \Pr {\set {X \in x} \cap \paren {\bigcup_i B_i} }
| c =
}}
{{eqn | r = \sum_x x \map \Pr {X = x}
... | Total Expectation Theorem | https://proofwiki.org/wiki/Total_Expectation_Theorem | https://proofwiki.org/wiki/Total_Expectation_Theorem | [
"Total Expectation Theorem",
"Expectation",
"Named Theorems"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Partition (Probability Theory)",
"Definition:Absolutely Convergent Series",
"Definition:Expectation",
"Definition:Conditional Expectation"
] | [] |
proofwiki-2703 | Function of Two Discrete Random Variables | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $g: \R^2 \to \R$ be a real-valued function.
Then $Z = \map g {X, Y}$, defined as:
:$\forall \omega \in \Omega: \map Z \omega = \map g {\map X \omega, \map Y \omega}$
is also a... | As $\Img X$ and $\Img Y$ are countable, then so is $\Img {\map g {X, Y} }$.
Now consider $\map {g^{-1} } Z$.
We have that:
* $\forall x \in \R: \map {X^{-1} } x \in \Sigma$.
* $\forall y \in \R: \map {Y^{-1} } x \in \Sigma$.
We also have that:
: $\ds \forall z \in \R: \map {g^{-1} } z = \bigcup_{\tuple {x, y}: \map g {... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Discrete Random Variable|discrete random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $g: \R^2 \to \R$ be a [[Definition:Real-Valued Function|real-valued function]].
Then $Z = \map g {X... | As $\Img X$ and $\Img Y$ are [[Definition:Countable|countable]], then so is $\Img {\map g {X, Y} }$.
Now consider $\map {g^{-1} } Z$.
We have that:
* $\forall x \in \R: \map {X^{-1} } x \in \Sigma$.
* $\forall y \in \R: \map {Y^{-1} } x \in \Sigma$.
We also have that:
: $\ds \forall z \in \R: \map {g^{-1} } z = \bi... | Function of Two Discrete Random Variables | https://proofwiki.org/wiki/Function_of_Two_Discrete_Random_Variables | https://proofwiki.org/wiki/Function_of_Two_Discrete_Random_Variables | [
"Probability Theory"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Real-Valued Function",
"Definition:Random Variable/Discrete",
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Random Variable/Discrete"
] | [
"Definition:Countable Set",
"Definition:Sigma-Algebra",
"Definition:Sigma-Algebra"
] |
proofwiki-2704 | Expectation of Function of Joint Probability Mass Distribution | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ be the expectation of $X$.
Let $g: \R^2 \to \R$ be a real-valued function
Let $p_{X, Y}$ be the joint probability mass function of $X$ and $Y$.
:$\ds \expect {\map... | Let $\Omega_X = \Img X = I_X$ and $\Omega_Y = \Img Y = I_Y$.
Let $Z = \map g {X, Y}$.
Thus $\Omega_Z = \Img Z = g \sqbrk {I_X, I_Y}$.
So:
{{begin-eqn}}
{{eqn | l = \expect Z
| r = \sum_{z \mathop \in g \sqbrk {I_X, I_Y} } z \map \Pr {Z = z}
| c =
}}
{{eqn | r = \sum_{z \mathop \in g \sqbrk {I_X, I_Y} } z \... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Discrete Random Variable|discrete random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ be the [[Definition:Expectation|expectation]] of $X$.
Let $g: \R^2 \to \R$ be a [[Defin... | Let $\Omega_X = \Img X = I_X$ and $\Omega_Y = \Img Y = I_Y$.
Let $Z = \map g {X, Y}$.
Thus $\Omega_Z = \Img Z = g \sqbrk {I_X, I_Y}$.
So:
{{begin-eqn}}
{{eqn | l = \expect Z
| r = \sum_{z \mathop \in g \sqbrk {I_X, I_Y} } z \map \Pr {Z = z}
| c =
}}
{{eqn | r = \sum_{z \mathop \in g \sqbrk {I_X, I_Y} } ... | Expectation of Function of Joint Probability Mass Distribution | https://proofwiki.org/wiki/Expectation_of_Function_of_Joint_Probability_Mass_Distribution | https://proofwiki.org/wiki/Expectation_of_Function_of_Joint_Probability_Mass_Distribution | [
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Expectation",
"Definition:Real-Valued Function",
"Definition:Probability Mass Function/Joint",
"Definition:Absolutely Convergent Series"
] | [
"Probability Mass Function of Function of Discrete Random Variable",
"Definition:Expectation",
"Definition:Absolutely Convergent Series"
] |
proofwiki-2705 | Expectation is Linear | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be integrable random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ denote the expectation of $X$.
Then:
:$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \, \expect X + \beta \, \expect Y$ | === Discrete Random Variable ===
{{:Expectation is Linear/Discrete}} | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Integrable Random Variable|integrable random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ denote the [[Definition:Expectation|expectation]] of $X$.
Then:
:$\forall \alpha, ... | === [[Expectation is Linear/Discrete|Discrete Random Variable]] ===
{{:Expectation is Linear/Discrete}} | Expectation is Linear | https://proofwiki.org/wiki/Expectation_is_Linear | https://proofwiki.org/wiki/Expectation_is_Linear | [
"Expectation is Linear",
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Random Variable",
"Definition:Expectation"
] | [
"Expectation is Linear/Discrete"
] |
proofwiki-2706 | Condition for Independence of Discrete Random Variables | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ and $Y$ are independent {{iff}} there exist functions $f, g: \R \to \R$ such that the joint mass function of $X$ and $Y$ satisfies:
:$\forall x, y \in \R: \map {p_{X, Y} ... | We have by definition of joint mass function that:
:$x \notin \Omega_X \implies \map {p_{X, Y} } {x, y} = 0$
:$y \notin \Omega_Y \implies \map {p_{X, Y} } {x, y} = 0$
Hence we only need to worry about values of $x$ and $y$ in their appropriate $\Omega$ spaces. | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Discrete Random Variable|discrete random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ and $Y$ are [[Definition:Independent Random Variables|independent]] {{iff}} there exist [[Defi... | We have by definition of [[Definition:Joint Probability Mass Function|joint mass function]] that:
:$x \notin \Omega_X \implies \map {p_{X, Y} } {x, y} = 0$
:$y \notin \Omega_Y \implies \map {p_{X, Y} } {x, y} = 0$
Hence we only need to worry about values of $x$ and $y$ in their appropriate $\Omega$ spaces. | Condition for Independence of Discrete Random Variables | https://proofwiki.org/wiki/Condition_for_Independence_of_Discrete_Random_Variables | https://proofwiki.org/wiki/Condition_for_Independence_of_Discrete_Random_Variables | [
"Independent Random Variables",
"Discrete Random Variables"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Independent Random Variables",
"Definition:Real Function",
"Definition:Probability Mass Function/Joint"
] | [
"Definition:Probability Mass Function/Joint"
] |
proofwiki-2707 | Condition for Independence from Product of Expectations | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ denote the expectation of $X$.
Then $X$ and $Y$ are independent {{iff}}:
:$\expect {\map g x \map h y} = \expect {\map g x} \expect {\map h y}$
for all functions $... | === Sufficient Condition ===
{{AimForCont}} $X$ and $Y$ are ''not'' independent.
That is:
:$\map \Pr {X = a, Y = b} \ne \map \Pr {X = a} \map \Pr {Y = b}$
for some $a, b \in \R$.
Now, suppose that:
:$\expect {\map g x \map h y} = \expect {\map g x} \expect {\map h y}$
for all functions $g, h: \R \to \R$ for which the l... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Discrete Random Variable|discrete random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\expect X$ denote the [[Definition:Expectation|expectation]] of $X$.
Then $X$ and $Y$ are [[Defin... | === Sufficient Condition ===
{{AimForCont}} $X$ and $Y$ are ''not'' [[Definition:Independent Random Variables|independent]].
That is:
:$\map \Pr {X = a, Y = b} \ne \map \Pr {X = a} \map \Pr {Y = b}$
for some $a, b \in \R$.
Now, suppose that:
:$\expect {\map g x \map h y} = \expect {\map g x} \expect {\map h y}$
for... | Condition for Independence from Product of Expectations | https://proofwiki.org/wiki/Condition_for_Independence_from_Product_of_Expectations | https://proofwiki.org/wiki/Condition_for_Independence_from_Product_of_Expectations | [
"Condition for Independence from Product of Expectations",
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Expectation",
"Definition:Independent Random Variables",
"Definition:Real Function"
] | [
"Definition:Independent Random Variables",
"Definition:Real Function",
"Definition:Expectation",
"Definition:Real Number",
"Definition:Real Function",
"Definition:Independent Random Variables",
"Definition:Independent Random Variables",
"Definition:Real Function"
] |
proofwiki-2708 | PGF of Sum of Independent Discrete Random Variables | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be independent discrete random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $Z$ be a discrete random variable such that $Z = X + Y$.
Then:
:$\map {\Pi_Z} s = \map {\Pi_X} s \, \map {\Pi_Y} s$
where $\map {\Pi_Z} s$ is the probability gener... | {{begin-eqn}}
{{eqn | l = \map {\Pi_Z} s
| r = \expect {s^Z}
| c = {{Defof|Probability Generating Function}}
}}
{{eqn | r = \expect {s^{X + Y} }
| c = Definition of $Z$ (see above)
}}
{{eqn | r = \expect {s^X s^Y}
| c = Exponential of Sum
}}
{{eqn | r = \expect {s^X} \expect {s^Y}
| c = Co... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Independent Random Variables|independent]] [[Definition:Discrete Random Variable|discrete random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $Z$ be a [[Definition:Discrete Random Variab... | {{begin-eqn}}
{{eqn | l = \map {\Pi_Z} s
| r = \expect {s^Z}
| c = {{Defof|Probability Generating Function}}
}}
{{eqn | r = \expect {s^{X + Y} }
| c = Definition of $Z$ (see above)
}}
{{eqn | r = \expect {s^X s^Y}
| c = [[Exponential of Sum]]
}}
{{eqn | r = \expect {s^X} \expect {s^Y}
| c ... | PGF of Sum of Independent Discrete Random Variables | https://proofwiki.org/wiki/PGF_of_Sum_of_Independent_Discrete_Random_Variables | https://proofwiki.org/wiki/PGF_of_Sum_of_Independent_Discrete_Random_Variables | [
"Probability Generating Functions"
] | [
"Definition:Probability Space",
"Definition:Independent Random Variables",
"Definition:Random Variable/Discrete",
"Definition:Random Variable/Discrete",
"Definition:Probability Generating Function"
] | [
"Exponential of Sum",
"Condition for Independence from Product of Expectations"
] |
proofwiki-2709 | Brahmagupta Theorem | If a cyclic quadrilateral has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.
Specifically:
Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ are perpendicular, crossing at $M$.
Let $EF$ be a line pass... | :420px | If a [[Definition:Cyclic Quadrilateral|cyclic quadrilateral]] has [[Definition:Diagonal of Quadrilateral|diagonals]] which are [[Definition:Perpendicular|perpendicular]], then the perpendicular to a side from the point of intersection of the diagonals always [[Definition:Bisect|bisects]] the [[Definition:Opposite Sides... | :[[File:BrahmaguptaTheorem.png|420px]] | Brahmagupta Theorem | https://proofwiki.org/wiki/Brahmagupta_Theorem | https://proofwiki.org/wiki/Brahmagupta_Theorem | [
"Cyclic Quadrilaterals"
] | [
"Definition:Cyclic Quadrilateral",
"Definition:Diameter of Quadrilateral",
"Definition:Right Angle/Perpendicular",
"Definition:Bisection",
"Definition:Polygon/Opposite",
"Definition:Cyclic Quadrilateral",
"Definition:Diameter of Quadrilateral",
"Definition:Right Angle/Perpendicular",
"Definition:Pol... | [
"File:BrahmaguptaTheorem.png"
] |
proofwiki-2710 | Crossbar Theorem | Let $\triangle ABC$ be a triangle.
Let $D$ be a point in the interior of $\triangle ABC$.
Then there exists a point $E$ such that $E$ lies on both $AD$ and $BC$. | :380px
{{AimForCont}} $BC$ does not meet ray $\overrightarrow {AD}$.
Either $BC$ meets line $\overleftrightarrow {AD}$ or it does not.
If it meets $\overleftrightarrow {AD}$, by the Line Separation Property it must meet the ray opposite to $\overrightarrow {AD}$ at a point $E \ne A$.
According to Proposition 3.8 (b), $... | Let $\triangle ABC$ be a [[Definition:Triangle (Geometry)|triangle]].
Let $D$ be a point in the interior of $\triangle ABC$.
Then there exists a [[Definition:Point|point]] $E$ such that $E$ lies on both $AD$ and $BC$. | :[[File:CrossbarTheorem.png|380px]]
{{AimForCont}} $BC$ does not meet ray $\overrightarrow {AD}$.
Either $BC$ meets line $\overleftrightarrow {AD}$ or it does not.
If it meets $\overleftrightarrow {AD}$, by the [[Line Separation Property]] it must meet the ray opposite to $\overrightarrow {AD}$ at a point $E \ne A$... | Crossbar Theorem | https://proofwiki.org/wiki/Crossbar_Theorem | https://proofwiki.org/wiki/Crossbar_Theorem | [
"Triangles",
"Named Theorems"
] | [
"Definition:Triangle (Geometry)",
"Definition:Point"
] | [
"File:CrossbarTheorem.png",
"Line Separation Property",
"Proposition 3.8 (b)",
"Proposition 3.7",
"Definition:Contradiction",
"B-2",
"Lemma 3.2.2",
"B-4(iii)",
"Proposition 3.8(c)",
"Definition:Contradiction",
"Category:Triangles",
"Category:Named Theorems"
] |
proofwiki-2711 | Clavius's Law | If, from the negation of a proposition $p$ we can derive $p$, we may conclude $p$:
=== Formulation 1 ===
{{:Clavius's Law/Formulation 1}}
=== Formulation 2 ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\neg p \implies p \vdash p}}
{{Premise|1|\neg p \implies p}}
{{ExcludedMiddle|2|p \lor \neg p}}
{{Assumption|3|\neg p|Either $p$ is false ...}}
{{ModusPonens|4|1, 3|p|1|3}}
{{Assumption|5|p|... or $p$ is true}}
{{ProofByCases|6|1|p|2|3|4|5|5}}
{{EndTableau|qed}}
{{LEM||3}} | If, from the [[Definition:Logical Not|negation]] of a [[Definition:Proposition|proposition]] $p$ we can derive $p$, we may conclude $p$:
=== [[Clavius's Law/Formulation 1|Formulation 1]] ===
{{:Clavius's Law/Formulation 1}}
=== [[Clavius's Law/Formulation 2|Formulation 2]] ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\neg p \implies p \vdash p}}
{{Premise|1|\neg p \implies p}}
{{ExcludedMiddle|2|p \lor \neg p}}
{{Assumption|3|\neg p|Either $p$ is false ...}}
{{ModusPonens|4|1, 3|p|1|3}}
{{Assumption|5|p|... or $p$ is true}}
{{ProofByCases|6|1|p|2|3|4|5|5}}
{{EndTableau|qed}}
{{LEM||3}} | Clavius's Law/Formulation 1/Proof 1 | https://proofwiki.org/wiki/Clavius's_Law | https://proofwiki.org/wiki/Clavius's_Law/Formulation_1/Proof_1 | [
"Clavius's Law",
"Conditional",
"Reductio ad Absurdum"
] | [
"Definition:Logical Not",
"Definition:Proposition",
"Clavius's Law/Formulation 1",
"Clavius's Law/Formulation 2"
] | [] |
proofwiki-2712 | Clavius's Law | If, from the negation of a proposition $p$ we can derive $p$, we may conclude $p$:
=== Formulation 1 ===
{{:Clavius's Law/Formulation 1}}
=== Formulation 2 ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\neg p \implies p \vdash p}}
{{Premise|1|\neg p \implies p}}
{{Assumption|2|p \implies \bot}}
{{SequentIntro|3|2|\neg p|2|Negation as Implication of Bottom}}
{{ModusPonens|4|1,2|p|1|3}}
{{Implication|5|1|(p \implies \bot) \implies p|2|4}}
{{SequentIntro|6|1|p|5|Peirce's Law}}
{{EndTableau|qed}}
{{LEM|Pei... | If, from the [[Definition:Logical Not|negation]] of a [[Definition:Proposition|proposition]] $p$ we can derive $p$, we may conclude $p$:
=== [[Clavius's Law/Formulation 1|Formulation 1]] ===
{{:Clavius's Law/Formulation 1}}
=== [[Clavius's Law/Formulation 2|Formulation 2]] ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\neg p \implies p \vdash p}}
{{Premise|1|\neg p \implies p}}
{{Assumption|2|p \implies \bot}}
{{SequentIntro|3|2|\neg p|2|[[Negation as Implication of Bottom]]}}
{{ModusPonens|4|1,2|p|1|3}}
{{Implication|5|1|(p \implies \bot) \implies p|2|4}}
{{SequentIntro|6|1|p|5|[[Peirce's Law/Formulation 1|Peirce's L... | Clavius's Law/Formulation 1/Proof 2 | https://proofwiki.org/wiki/Clavius's_Law | https://proofwiki.org/wiki/Clavius's_Law/Formulation_1/Proof_2 | [
"Clavius's Law",
"Conditional",
"Reductio ad Absurdum"
] | [
"Definition:Logical Not",
"Definition:Proposition",
"Clavius's Law/Formulation 1",
"Clavius's Law/Formulation 2"
] | [
"Negation as Implication of Bottom",
"Peirce's Law/Formulation 1"
] |
proofwiki-2713 | Clavius's Law | If, from the negation of a proposition $p$ we can derive $p$, we may conclude $p$:
=== Formulation 1 ===
{{:Clavius's Law/Formulation 1}}
=== Formulation 2 ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\vdash \paren {\neg p \implies p} \implies p}}
{{Premise|1|\neg p \implies p}}
{{SequentIntro|2|1|p|1|Clavius's Law: Formulation 1}}
{{Implication|3||\paren {\neg p \implies p} \implies p|1|2}}
{{EndTableau|qed}}
{{LEM|Clavius's Law/Formulation 1|3}} | If, from the [[Definition:Logical Not|negation]] of a [[Definition:Proposition|proposition]] $p$ we can derive $p$, we may conclude $p$:
=== [[Clavius's Law/Formulation 1|Formulation 1]] ===
{{:Clavius's Law/Formulation 1}}
=== [[Clavius's Law/Formulation 2|Formulation 2]] ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\vdash \paren {\neg p \implies p} \implies p}}
{{Premise|1|\neg p \implies p}}
{{SequentIntro|2|1|p|1|[[Clavius's Law/Formulation 1|Clavius's Law: Formulation 1]]}}
{{Implication|3||\paren {\neg p \implies p} \implies p|1|2}}
{{EndTableau|qed}}
{{LEM|Clavius's Law/Formulation 1|3}} | Clavius's Law/Formulation 2/Proof 1 | https://proofwiki.org/wiki/Clavius's_Law | https://proofwiki.org/wiki/Clavius's_Law/Formulation_2/Proof_1 | [
"Clavius's Law",
"Conditional",
"Reductio ad Absurdum"
] | [
"Definition:Logical Not",
"Definition:Proposition",
"Clavius's Law/Formulation 1",
"Clavius's Law/Formulation 2"
] | [
"Clavius's Law/Formulation 1"
] |
proofwiki-2714 | Clavius's Law | If, from the negation of a proposition $p$ we can derive $p$, we may conclude $p$:
=== Formulation 1 ===
{{:Clavius's Law/Formulation 1}}
=== Formulation 2 ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\vdash \paren {\neg p \implies p} \implies p}}
{{Premise|1|\neg p \implies p}}
{{ExcludedMiddle|2|p \lor \neg p}}
{{Assumption|3|\neg p|Either $p$ is false ...}}
{{ModusPonens|4|1, 3|p|1|3}}
{{Assumption|5|p|... or $p$ is true}}
{{ProofByCases|6|1|p|2|3|4|5|5}}
{{Implication|7||\paren {\neg p \implies p... | If, from the [[Definition:Logical Not|negation]] of a [[Definition:Proposition|proposition]] $p$ we can derive $p$, we may conclude $p$:
=== [[Clavius's Law/Formulation 1|Formulation 1]] ===
{{:Clavius's Law/Formulation 1}}
=== [[Clavius's Law/Formulation 2|Formulation 2]] ===
{{:Clavius's Law/Formulation 2}} | {{BeginTableau|\vdash \paren {\neg p \implies p} \implies p}}
{{Premise|1|\neg p \implies p}}
{{ExcludedMiddle|2|p \lor \neg p}}
{{Assumption|3|\neg p|Either $p$ is false ...}}
{{ModusPonens|4|1, 3|p|1|3}}
{{Assumption|5|p|... or $p$ is true}}
{{ProofByCases|6|1|p|2|3|4|5|5}}
{{Implication|7||\paren {\neg p \implies p... | Clavius's Law/Formulation 2/Proof 2 | https://proofwiki.org/wiki/Clavius's_Law | https://proofwiki.org/wiki/Clavius's_Law/Formulation_2/Proof_2 | [
"Clavius's Law",
"Conditional",
"Reductio ad Absurdum"
] | [
"Definition:Logical Not",
"Definition:Proposition",
"Clavius's Law/Formulation 1",
"Clavius's Law/Formulation 2"
] | [] |
proofwiki-2715 | Clavius's Law | If, from the negation of a proposition $p$ we can derive $p$, we may conclude $p$:
=== Formulation 1 ===
{{:Clavius's Law/Formulation 1}}
=== Formulation 2 ===
{{:Clavius's Law/Formulation 2}} | We apply the Method of Truth Tables.
As can be seen by inspection, the truth value under the main connective is true for all boolean interpretations.
:<nowiki>$\begin{array}{|cccc|c|c|} \hline
(\neg & p & \implies & p) & \implies & p \\
\hline
\T & \F & \F & \F & \T & \F \\
\F & \T & \T & \T & \T & \T \\
\hline
\end{ar... | If, from the [[Definition:Logical Not|negation]] of a [[Definition:Proposition|proposition]] $p$ we can derive $p$, we may conclude $p$:
=== [[Clavius's Law/Formulation 1|Formulation 1]] ===
{{:Clavius's Law/Formulation 1}}
=== [[Clavius's Law/Formulation 2|Formulation 2]] ===
{{:Clavius's Law/Formulation 2}} | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth value]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|c... | Clavius's Law/Formulation 2/Proof by Truth Table | https://proofwiki.org/wiki/Clavius's_Law | https://proofwiki.org/wiki/Clavius's_Law/Formulation_2/Proof_by_Truth_Table | [
"Clavius's Law",
"Conditional",
"Reductio ad Absurdum"
] | [
"Definition:Logical Not",
"Definition:Proposition",
"Clavius's Law/Formulation 1",
"Clavius's Law/Formulation 2"
] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-2716 | Viète's Formulas | Let $P_n$ be a polynomial of degree $n$ with real or complex coefficients:
{{begin-eqn}}
{{eqn | l = \map {P_n} x
| r = \sum_{i \mathop = 0}^n a_i x^i
}}
{{eqn | r = a_n x^n + a_{n - 1} x^{n - 1} + \dotsb + a_1 x + a_0
}}
{{end-eqn}}
where $a_n \ne 0$.
Let $z_1, \ldots, z_n$ be the roots of $P_n$ (be they real or... | First we note that from the Polynomial Factor Theorem:
:$\ds \map {P_n} x = a_n \prod_{k \mathop = 1}^n \paren {x - z_k}$
To shorten the exposition, let us define:
:$E_{\tuple {r, s} } := \map {e_r} {\set {z_1, \ldots, z_s} }$
for $r, s \in \Z_{\ge 1}$.
It is sufficient to consider the case $a_n = 1$, in which case:
:$... | Let $P_n$ be a [[Definition:Polynomial over Ring|polynomial]] of [[Definition:Degree of Polynomial|degree]] $n$ with [[Definition:Real Number|real]] or [[Definition:Complex Number|complex]] [[Definition:Polynomial Coefficient|coefficients]]:
{{begin-eqn}}
{{eqn | l = \map {P_n} x
| r = \sum_{i \mathop = 0}^n a_i... | First we note that from the [[Polynomial Factor Theorem]]:
:$\ds \map {P_n} x = a_n \prod_{k \mathop = 1}^n \paren {x - z_k}$
To shorten the exposition, let us define:
:$E_{\tuple {r, s} } := \map {e_r} {\set {z_1, \ldots, z_s} }$
for $r, s \in \Z_{\ge 1}$.
It is sufficient to consider the case $a_n = 1$, in which... | Viète's Formulas | https://proofwiki.org/wiki/Viète's_Formulas | https://proofwiki.org/wiki/Viète's_Formulas | [
"Viète's Formulas",
"Polynomial Theory",
"Algebra",
"Elementary Symmetric Functions",
"Proofs by Induction"
] | [
"Definition:Polynomial over Ring",
"Definition:Degree of Polynomial",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Coefficient of Polynomial",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Definition:Complex Number",
"Definition:Distinct/Plural",
"Definition:Sym... | [
"Polynomial Factor Theorem",
"Principle of Mathematical Induction",
"Definition:Statement",
"Polynomial Factor Theorem",
"Principle of Mathematical Induction"
] |
proofwiki-2717 | Inscribed Angle Theorem | An inscribed angle is equal to half the angle that is subtended by that arc.
:300px
Thus, in the figure above:
:$\angle ABC = \frac 1 2 \angle ADC$
{{EuclidSaid}}
:''In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.''
{{EuclidPropRef|II... | :380px
Let $ABC$ be a circle, let $\angle BEC$ be an angle at its center, and let $\angle BAC$ be an angle at the circumference.
Let these angles have the same arc $BC$ at their base.
Let $AE$ be joined and drawn through to $F$.
Since $EA = EB$, then from Isosceles Triangle has Two Equal Angles we have that $\angle EBA... | An [[Definition:Angle Inscribed in Circle|inscribed angle]] is equal to half the [[Definition:Angle Subtended by Arc|angle that is subtended]] by that [[Definition:Arc of Circle|arc]].
:[[File:InscribedAngleTheorem.png|300px]]
Thus, in the figure above:
:$\angle ABC = \frac 1 2 \angle ADC$
{{EuclidSaid}}
:''In a [[D... | :[[File:Euclid-III-20.png|380px]]
Let $ABC$ be a [[Definition:Circle|circle]], let $\angle BEC$ be an [[Definition:Angle|angle]] at its [[Definition:Center of Circle|center]], and let $\angle BAC$ be an angle at the [[Definition:Circumference of Circle|circumference]].
Let these angles have the same [[Definition:Arc ... | Inscribed Angle Theorem/Proof 1 | https://proofwiki.org/wiki/Inscribed_Angle_Theorem | https://proofwiki.org/wiki/Inscribed_Angle_Theorem/Proof_1 | [
"Circles",
"Named Theorems",
"Inscribed Angle Theorem"
] | [
"Definition:Angle Inscribed in Circle",
"Definition:Circle/Arc/Subtend",
"Definition:Circle/Arc",
"File:InscribedAngleTheorem.png",
"Definition:Circle",
"Definition:Angle",
"Definition:Circle/Center",
"Definition:Angle",
"Definition:Circle/Circumference",
"Definition:Angle",
"Definition:Circle/C... | [
"File:Euclid-III-20.png",
"Definition:Circle",
"Definition:Angle",
"Definition:Circle/Center",
"Definition:Circle/Circumference",
"Definition:Circle/Arc",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Triangle equals Two Right Angles"
] |
proofwiki-2718 | Inscribed Angle Theorem | An inscribed angle is equal to half the angle that is subtended by that arc.
:300px
Thus, in the figure above:
:$\angle ABC = \frac 1 2 \angle ADC$
{{EuclidSaid}}
:''In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.''
{{EuclidPropRef|II... | Consider the simplest case that occurs when $AC$ is a diameter of the circle:
:300px
Because all lines radiating from $D$ to the circumference are radii and thus equal:
:$AD = BD = CD$
Hence the triangles $\triangle ADB$ and $\triangle BDC$ are isosceles.
Therefore from Isosceles Triangle has Two Equal Angles:
:$\angle... | An [[Definition:Angle Inscribed in Circle|inscribed angle]] is equal to half the [[Definition:Angle Subtended by Arc|angle that is subtended]] by that [[Definition:Arc of Circle|arc]].
:[[File:InscribedAngleTheorem.png|300px]]
Thus, in the figure above:
:$\angle ABC = \frac 1 2 \angle ADC$
{{EuclidSaid}}
:''In a [[D... | Consider the simplest case that occurs when $AC$ is a [[Definition:Diameter of Circle|diameter of the circle]]:
:[[File:InscribedAngleTheorem1.png|300px]]
Because all [[Definition:Line Segment|lines]] radiating from $D$ to the [[Definition:Circumference of Circle|circumference]] are [[Definition:Radius of Circle|radi... | Inscribed Angle Theorem/Proof 2 | https://proofwiki.org/wiki/Inscribed_Angle_Theorem | https://proofwiki.org/wiki/Inscribed_Angle_Theorem/Proof_2 | [
"Circles",
"Named Theorems",
"Inscribed Angle Theorem"
] | [
"Definition:Angle Inscribed in Circle",
"Definition:Circle/Arc/Subtend",
"Definition:Circle/Arc",
"File:InscribedAngleTheorem.png",
"Definition:Circle",
"Definition:Angle",
"Definition:Circle/Center",
"Definition:Angle",
"Definition:Circle/Circumference",
"Definition:Angle",
"Definition:Circle/C... | [
"Definition:Circle/Diameter",
"File:InscribedAngleTheorem1.png",
"Definition:Line/Segment",
"Definition:Circle/Circumference",
"Definition:Circle/Radius",
"Definition:Triangle (Geometry)",
"Definition:Triangle (Geometry)/Isosceles",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Triang... |
proofwiki-2719 | Linearly Independent Solutions to 1st Order Systems | The 1st-order homogeneous linear system of differential equations $x' = A \left({t}\right) x$, expressed with the vectors $x', x: \R \to \R^n$ and the matrix function $A: \R \to M_{n \times n} \left({\R}\right)$, has $n$ linearly independent solutions, and if $ \phi_1 , \phi_2, \dots, \phi_n$ are $n$ linearly independe... | Let $v_1, v_2, \dots, v_n$ be linearly independent vectors in $\R^n$, and let $\phi_i$ be solutions to the IVPs $x' = A \left({t}\right) x, \, x \left({t_0}\right) = v_i$ for $i = 1, 2, \dots, n$.
Suppose the solutions are not independent, i.e. $c_1 \phi_1 + c_2 \phi_2 + \cdots + c_n \phi_n = 0$ for some constants $c_i... | The 1st-order homogeneous linear system of differential equations $x' = A \left({t}\right) x$, expressed with the vectors $x', x: \R \to \R^n$ and the matrix function $A: \R \to M_{n \times n} \left({\R}\right)$, has $n$ linearly independent solutions, and if $ \phi_1 , \phi_2, \dots, \phi_n$ are $n$ linearly independe... | Let $v_1, v_2, \dots, v_n$ be linearly independent vectors in $\R^n$, and let $\phi_i$ be solutions to the IVPs $x' = A \left({t}\right) x, \, x \left({t_0}\right) = v_i$ for $i = 1, 2, \dots, n$.
Suppose the solutions are not independent, i.e. $c_1 \phi_1 + c_2 \phi_2 + \cdots + c_n \phi_n = 0$ for some constants $c_... | Linearly Independent Solutions to 1st Order Systems | https://proofwiki.org/wiki/Linearly_Independent_Solutions_to_1st_Order_Systems | https://proofwiki.org/wiki/Linearly_Independent_Solutions_to_1st_Order_Systems | [
"Ordinary Differential Equations"
] | [] | [
"Category:Ordinary Differential Equations"
] |
proofwiki-2720 | Existence and Uniqueness Theorem for 1st Order IVPs | Let $x' = \map f {t, x}$, $\map x {t_0} = x_0$ be an explicit ODE of dimension $n$.
Let there exist an open ball $V = \sqbrk {t_0 - \ell_0, t_0 + \ell_0} \times \map {\overline B} {x_0, \epsilon}$ of $\tuple {t_0, x_0}$ in phase space $\R \times \R^n$ such that $f$ is Lipschitz continuous on $V$.
{{explain|Notation nee... | For $0 < \ell < \ell_0$, let $\XX = \map \CC {\closedint {t_0 - \ell_0} {t_0 + \ell_0}; \R^n}$ endowed with the sup norm be the Banach Space of Continuous Functions on Compact Space $\closedint {t_0 - \ell_0} {t_0 + \ell_0} \to \R^n$.
By Fixed Point Formulation of Explicit ODE it is sufficient to find a fixed point of ... | Let $x' = \map f {t, x}$, $\map x {t_0} = x_0$ be an [[Definition:Explicit ODE|explicit ODE]] of [[Definition:Dimension of Differential Equation|dimension]] $n$.
Let there exist an [[Definition:Open Ball|open ball]] $V = \sqbrk {t_0 - \ell_0, t_0 + \ell_0} \times \map {\overline B} {x_0, \epsilon}$ of $\tuple {t_0, x_... | For $0 < \ell < \ell_0$, let $\XX = \map \CC {\closedint {t_0 - \ell_0} {t_0 + \ell_0}; \R^n}$ endowed with the [[Definition:Supremum Norm|sup norm]] be the [[Banach Space of Continuous Functions on Compact Space]] $\closedint {t_0 - \ell_0} {t_0 + \ell_0} \to \R^n$.
By [[Fixed Point Formulation of Explicit ODE]] it i... | Existence and Uniqueness Theorem for 1st Order IVPs | https://proofwiki.org/wiki/Existence_and_Uniqueness_Theorem_for_1st_Order_IVPs | https://proofwiki.org/wiki/Existence_and_Uniqueness_Theorem_for_1st_Order_IVPs | [
"Differential Equations"
] | [
"Definition:Differential Equation/Explicit",
"Definition:Differential Equation/Order",
"Definition:Open Ball",
"Definition:Phase Space",
"Definition:Lipschitz Continuity/Real Function",
"Definition:Real Interval/Closed",
"Definition:Real Interval/Notation/Wirth",
"Definition:Real Interval/Closed",
"... | [
"Definition:Supremum Norm",
"Continuous Functions on Compact Space form Banach Space",
"Fixed Point Formulation of Explicit ODE",
"Definition:Fixed Point",
"Definition:Mapping",
"Closed Subset of Complete Metric Space is Complete",
"Banach Fixed-Point Theorem",
"Definition:Subset",
"Definition:Close... |
proofwiki-2721 | Liouville's Theorem (Differential Equations) | Let $\map \Phi t$ be a solution to the matrix differential equation:
:$X' = \map A t X$
with $\map A t$ continuous on the interval $I$ such that $t_0 \in I$.
Then:
:$\ds \det \map \Phi t = \map \exp {\int_{t_0}^t \map \tr {\map A s} \rd s} \det \map \Phi {t_0}$ | {{ProofWanted}}
{{Namedfor|Joseph Liouville|cat = Liouville}}
Category:Differential Equations
Category:Liouville's Theorem
k2u4k0nv2vl7jnxo5h4qryjh3xu8658 | Let $\map \Phi t$ be a solution to the matrix differential equation:
:$X' = \map A t X$
with $\map A t$ continuous on the interval $I$ such that $t_0 \in I$.
Then:
:$\ds \det \map \Phi t = \map \exp {\int_{t_0}^t \map \tr {\map A s} \rd s} \det \map \Phi {t_0}$ | {{ProofWanted}}
{{Namedfor|Joseph Liouville|cat = Liouville}}
[[Category:Differential Equations]]
[[Category:Liouville's Theorem]]
k2u4k0nv2vl7jnxo5h4qryjh3xu8658 | Liouville's Theorem (Differential Equations) | https://proofwiki.org/wiki/Liouville's_Theorem_(Differential_Equations) | https://proofwiki.org/wiki/Liouville's_Theorem_(Differential_Equations) | [
"Differential Equations",
"Liouville's Theorem"
] | [] | [
"Category:Differential Equations",
"Category:Liouville's Theorem"
] |
proofwiki-2722 | Maximum Modulus Principle | Let $D$ be an open region of the complex plane $\C$.
Let $f: D \to \C$ be a non-constant holomorphic function.
Then $\cmod f$ does not have any maximum points in the interior of $D$.
That is, for each $z \in D$ and $\delta > 0$, there exists some $\omega \in \map {B_\delta} z \cap D$, such that:
:$\cmod {\map f \omega}... | Pick some $r > 0$ such that $\map {B_r} z \subset D$.
By the Mean Value Theorem for Holomorphic Functions:
:$\ds \map f z = \dfrac 1 {2 \pi} \int_0^{2 \pi} \map f {z + r e^{i \theta} } \rd \theta$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\map f z}
| o = \le
| r = \frac 1 {2 \pi} \int_0^{2 \pi} \cmod {\map f {... | Let $D$ be an [[Definition:Open Region of Complex Plane|open region]] of the [[Definition:Complex Plane|complex plane]] $\C$.
Let $f: D \to \C$ be a non-[[Definition:Constant Function|constant]] [[Definition:Holomorphic Function|holomorphic function]].
Then $\cmod f$ does not have any [[Definition:Maximum Value of Re... | Pick some $r > 0$ such that $\map {B_r} z \subset D$.
By the [[Mean Value Theorem for Holomorphic Functions]]:
:$\ds \map f z = \dfrac 1 {2 \pi} \int_0^{2 \pi} \map f {z + r e^{i \theta} } \rd \theta$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {\map f z}
| o = \le
| r = \frac 1 {2 \pi} \int_0^{2 \pi} \cmod {\... | Maximum Modulus Principle | https://proofwiki.org/wiki/Maximum_Modulus_Principle | https://proofwiki.org/wiki/Maximum_Modulus_Principle | [
"Maximum Modulus Principle",
"Complex Analysis",
"Named Theorems"
] | [
"Definition:Open Region/Complex",
"Definition:Complex Number/Complex Plane",
"Definition:Constant Mapping",
"Definition:Holomorphic Function",
"Definition:Maximum Value of Real Function",
"Definition:Interior (Complex Analysis)"
] | [
"Mean Value Theorem for Holomorphic Functions",
"Darboux's Theorem",
"Definition:Circle",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Constant Mapping",
"Definition:Constant Mapping",
"Definition:Constant Mapping"
] |
proofwiki-2723 | Euler Polyhedron Formula | For any convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces:
:$V - E + F = 2$ | {{ProofWanted|There should be a proof that the net of the polyhedron is a planar graph. The result then follows from Euler's Theorem for Planar Graphs.}} | For any [[Definition:Convex Polyhedron|convex polyhedron]] with $V$ [[Definition:Vertex of Polyhedron|vertices]], $E$ [[Definition:Edge of Polyhedron|edges]], and $F$ [[Definition:Face of Polyhedron|faces]]:
:$V - E + F = 2$ | {{ProofWanted|There should be a proof that the net of the polyhedron is a planar graph. The result then follows from [[Euler's Theorem for Planar Graphs]].}} | Euler Polyhedron Formula | https://proofwiki.org/wiki/Euler_Polyhedron_Formula | https://proofwiki.org/wiki/Euler_Polyhedron_Formula | [
"Euler Polyhedron Formula",
"Graph Theory"
] | [
"Definition:Convex Polyhedron",
"Definition:Polyhedron/Vertex",
"Definition:Polyhedron/Edge",
"Definition:Polyhedron/Face"
] | [
"Euler's Theorem for Planar Graphs"
] |
proofwiki-2724 | Line Joining Centers of Two Circles Touching Internally | Let two circles touch internally.
Then the straight line joining their centers passes through the point where they touch.
{{:Euclid:Proposition/III/11}} | Let the circles $ABC$ and $ADE$ touch internally at $A$.
Let $F$ be the center of $ABC$ and let $G$ be the center of $ADE$.
We are to show that the straight line joining $F$ to $G$ passes through $A$.
:400px
Suppose, as in the diagram above, that it does not.
Let $FG$ fall on $H$ instead.
It will also pass through $D$ ... | Let two [[Definition:Circle|circles]] touch internally.
Then the [[Definition:Straight Line|straight line]] joining their [[Definition:Center of Circle|centers]] passes through the [[Definition:Point|point]] where they touch.
{{:Euclid:Proposition/III/11}} | Let the [[Definition:Circle|circles]] $ABC$ and $ADE$ touch internally at $A$.
Let $F$ be the [[Definition:Center of Circle|center of $ABC$]] and let $G$ be the [[Definition:Center of Circle|center of $ADE$]].
We are to show that the [[Definition:Straight Line|straight line]] joining $F$ to $G$ passes through $A$.
... | Line Joining Centers of Two Circles Touching Internally | https://proofwiki.org/wiki/Line_Joining_Centers_of_Two_Circles_Touching_Internally | https://proofwiki.org/wiki/Line_Joining_Centers_of_Two_Circles_Touching_Internally | [
"Circles"
] | [
"Definition:Circle",
"Definition:Line/Straight Line",
"Definition:Circle/Center",
"Definition:Point"
] | [
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Center",
"Definition:Line/Straight Line",
"File:Euclid-III-11.png",
"Definition:Circle",
"Axiom:Euclid's First Postulate",
"Sum of Two Sides of Triangle Greater than Third Side",
"Definition:Circle/Center",
"Definition:Circle/Radi... |
proofwiki-2725 | PGF of Sum of Random Number of Independent Discrete Random Variables | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let:
:$N, X_1, X_2, \ldots$
be independent discrete random variables such that the $X$'s have the same probability distribution.
Let:
:$\map {\Pi_N} s$ be the PGF of $N$
:$\map {\Pi_X} s$ be the PGF of each of the $X$'s.
Let:
:$Z = X_1 + X_2 + \ldots + X_N$
Th... | {{begin-eqn}}
{{eqn | l = \map {\Pi_Z} s
| r = \expect {s^{X_1 + X_2 + \cdots + X_N} }
| c = {{Defof|Probability Generating Function}}
}}
{{eqn | r = \sum_{n \mathop \ge 0} \expect {s^{X_1 + X_2 + \cdots + X_N} \mid N = n} \map \Pr {N = n}
| c = Total Expectation Theorem
}}
{{eqn | r = \sum_{n \mathop... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let:
:$N, X_1, X_2, \ldots$
be [[Definition:Independent Random Variables|independent]] [[Definition:Discrete Random Variable|discrete random variables]] such that the $X$'s have the same [[Definition:Discrete Probability Distr... | {{begin-eqn}}
{{eqn | l = \map {\Pi_Z} s
| r = \expect {s^{X_1 + X_2 + \cdots + X_N} }
| c = {{Defof|Probability Generating Function}}
}}
{{eqn | r = \sum_{n \mathop \ge 0} \expect {s^{X_1 + X_2 + \cdots + X_N} \mid N = n} \map \Pr {N = n}
| c = [[Total Expectation Theorem]]
}}
{{eqn | r = \sum_{n \ma... | PGF of Sum of Random Number of Independent Discrete Random Variables | https://proofwiki.org/wiki/PGF_of_Sum_of_Random_Number_of_Independent_Discrete_Random_Variables | https://proofwiki.org/wiki/PGF_of_Sum_of_Random_Number_of_Independent_Discrete_Random_Variables | [
"Probability Generating Functions"
] | [
"Definition:Probability Space",
"Definition:Independent Random Variables",
"Definition:Random Variable/Discrete",
"Definition:Discrete Probability Distribution",
"Definition:Probability Generating Function",
"Definition:Probability Generating Function"
] | [
"Total Expectation Theorem",
"PGF of Sum of Independent Discrete Random Variables"
] |
proofwiki-2726 | Floquet's Theorem | Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$.
Let $\Phi \left({t}\right)$ be a fundamental matrix of the Floquet system $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$.
Then $\Phi \left({t + T}\right)$ is also a fundamental matrix.
Moreover, there exists:
: A nonsingular, continu... | We assume the two hypotheses of the theorem.
We have that:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d t} } {\map \Phi {t + T} }
| r = \map {\Phi'} {t + T}
| c =
}}
{{eqn | r = \map {\mathbf A} {t + T} \map \Phi {t + T}
| c =
}}
{{eqn | r = \map {\mathbf A} t \map \Phi {t + T}
| c =
}}
{{end... | Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$.
Let $\Phi \left({t}\right)$ be a [[Definition:Fundamental Matrix|fundamental matrix]] of the [[Definition:Floquet System|Floquet system]] $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$.
Then $\Phi \left({t + T}\right)$ is also a [... | We assume the two hypotheses of the theorem.
We have that:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d t} } {\map \Phi {t + T} }
| r = \map {\Phi'} {t + T}
| c =
}}
{{eqn | r = \map {\mathbf A} {t + T} \map \Phi {t + T}
| c =
}}
{{eqn | r = \map {\mathbf A} t \map \Phi {t + T}
| c =
}}
{{... | Floquet's Theorem/Proof 1 | https://proofwiki.org/wiki/Floquet's_Theorem | https://proofwiki.org/wiki/Floquet's_Theorem/Proof_1 | [
"Linear Algebra",
"Differential Equations",
"Floquet's Theorem"
] | [
"Definition:Fundamental Matrix",
"Definition:Floquet System",
"Definition:Fundamental Matrix"
] | [
"Definition:Fundamental Matrix",
"Definition:Matrix",
"Existence of Matrix Logarithm",
"Definition:Matrix"
] |
proofwiki-2727 | Floquet's Theorem | Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$.
Let $\Phi \left({t}\right)$ be a fundamental matrix of the Floquet system $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$.
Then $\Phi \left({t + T}\right)$ is also a fundamental matrix.
Moreover, there exists:
: A nonsingular, continu... | Let $\map S t = \map \Phi {t + T} {\map \Phi T}^{-1}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d t} } {\map S t}
| r = \map {\Phi'} {t + T} {\map \Phi T}^{-1}
| c =
}}
{{eqn | r = \map {\mathbf A} {t + T} \map \Phi {t + T} {\map \Phi T}^{-1}
| c =
}}
{{eqn | r = \map {\mathbf A} t \map S t
... | Let $\mathbf A \left({t}\right)$ be a continuous matrix function with period $T$.
Let $\Phi \left({t}\right)$ be a [[Definition:Fundamental Matrix|fundamental matrix]] of the [[Definition:Floquet System|Floquet system]] $\mathbf x' = \mathbf A \left({t}\right) \mathbf x$.
Then $\Phi \left({t + T}\right)$ is also a [... | Let $\map S t = \map \Phi {t + T} {\map \Phi T}^{-1}$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d t} } {\map S t}
| r = \map {\Phi'} {t + T} {\map \Phi T}^{-1}
| c =
}}
{{eqn | r = \map {\mathbf A} {t + T} \map \Phi {t + T} {\map \Phi T}^{-1}
| c =
}}
{{eqn | r = \map {\mathbf A} t \map S ... | Floquet's Theorem/Proof 2 | https://proofwiki.org/wiki/Floquet's_Theorem | https://proofwiki.org/wiki/Floquet's_Theorem/Proof_2 | [
"Linear Algebra",
"Differential Equations",
"Floquet's Theorem"
] | [
"Definition:Fundamental Matrix",
"Definition:Floquet System",
"Definition:Fundamental Matrix"
] | [
"Definition:Fundamental Matrix",
"Existence of Matrix Logarithm",
"Definition:Matrix",
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period"
] |
proofwiki-2728 | Lyapunov's Stability Theorem | Let $V$ be a Lyapunov function on an open set containing an equilibrium point $x_0$.
{{Disambiguate|Definition:Open Set}}
Then $x_0$ is stable.
If $V$ is strict, then $x_0$ is asymptotically stable.
{{explain|strict}} | {{ProofWanted}}
{{Namedfor|Aleksandr Mikhailovich Lyapunov|cat = Lyapunov}}
Category:Differential Equations
Category:Lyapunov Functions
mpigav5ix0mmnldur1vtkr6u6msegch | Let $V$ be a [[Definition:Lyapunov Function|Lyapunov function]] on an [[Definition:Open Set|open set]] containing an equilibrium point $x_0$.
{{Disambiguate|Definition:Open Set}}
Then $x_0$ is [[Definition:Stability (Differential Equations)|stable]].
If $V$ is [[Definition:Strict Lyapunov Function|strict]], then $x... | {{ProofWanted}}
{{Namedfor|Aleksandr Mikhailovich Lyapunov|cat = Lyapunov}}
[[Category:Differential Equations]]
[[Category:Lyapunov Functions]]
mpigav5ix0mmnldur1vtkr6u6msegch | Lyapunov's Stability Theorem | https://proofwiki.org/wiki/Lyapunov's_Stability_Theorem | https://proofwiki.org/wiki/Lyapunov's_Stability_Theorem | [
"Differential Equations",
"Lyapunov Functions"
] | [
"Definition:Lyapunov Function",
"Definition:Open Set",
"Definition:Stability (Differential Equations)",
"Definition:Lyapunov Function/Strict",
"Definition:Asymptotic Stability"
] | [
"Category:Differential Equations",
"Category:Lyapunov Functions"
] |
proofwiki-2729 | Bendixson-Dulac Theorem | Suppose there exists a continuously differentiable function $\map \alpha {x, y}$ on a simply connected domain.
{{Explain|What ''is'' the domain? Reals, complex, or what?}}
Suppose that:
:$\nabla \cdot \paren {\alpha F}$
is either always positive or always negative.
Then the two-dimensional autonomous system:
:$\tuple {... | {{ProofWanted}}
{{Namedfor|Ivar Otto Bendixson|name2 = Henri Claudius Rosaris Dulac|cat = Bendixson|cat2 = Dulac}}
Category:Differential Equations
ic5qrm85zv86t44pyutgh5gq7c0o48o | Suppose there exists a [[Definition:Continuously Differentiable|continuously differentiable function]] $\map \alpha {x, y}$ on a [[Definition:Simply Connected|simply connected]] [[Definition:Domain of Mapping|domain]].
{{Explain|What ''is'' the domain? Reals, complex, or what?}}
Suppose that:
:$\nabla \cdot \paren {\... | {{ProofWanted}}
{{Namedfor|Ivar Otto Bendixson|name2 = Henri Claudius Rosaris Dulac|cat = Bendixson|cat2 = Dulac}}
[[Category:Differential Equations]]
ic5qrm85zv86t44pyutgh5gq7c0o48o | Bendixson-Dulac Theorem | https://proofwiki.org/wiki/Bendixson-Dulac_Theorem | https://proofwiki.org/wiki/Bendixson-Dulac_Theorem | [
"Differential Equations"
] | [
"Definition:Continuously Differentiable",
"Definition:Simply Connected",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Differential Equation/System/Autonomous",
"Definition:Cycle (Periodic Solution)"
] | [
"Category:Differential Equations"
] |
proofwiki-2730 | General Vector Solution of Fundamental Matrix | Let $\map \Phi t$ be a fundamental matrix of the system $x' = \map A t x$.
Then:
:$\map \Phi t c$ is a general solution of $x' = \map A t x$. | By definition, $\map \Phi t$ is non-singular, and therefore has an inverse $\map {\Phi^{-1} } t$.
If $z$ is an arbitrary solution, then $\map \Phi t \, \map {\Phi^{-1} } {t_0} \, \map z {t_0}$ also solves the system and has the same initial condition.
Hence by Existence and Uniqueness Theorem for 1st Order IVPs $\map \... | Let $\map \Phi t$ be a [[Definition:Fundamental Matrix|fundamental matrix]] of the [[Definition:System of Differential Equations|system]] $x' = \map A t x$.
Then:
:$\map \Phi t c$ is a [[Definition:General Solution to Differential Equation|general solution]] of $x' = \map A t x$. | By [[Definition:Fundamental Matrix|definition]], $\map \Phi t$ is [[Definition:Singular Matrix|non-singular]], and therefore has an [[Definition:Inverse Matrix|inverse]] $\map {\Phi^{-1} } t$.
If $z$ is an arbitrary solution, then $\map \Phi t \, \map {\Phi^{-1} } {t_0} \, \map z {t_0}$ also solves the [[Definition:Sy... | General Vector Solution of Fundamental Matrix | https://proofwiki.org/wiki/General_Vector_Solution_of_Fundamental_Matrix | https://proofwiki.org/wiki/General_Vector_Solution_of_Fundamental_Matrix | [
"Differential Equations"
] | [
"Definition:Fundamental Matrix",
"Definition:Differential Equation/System",
"Definition:Differential Equation/Solution/General Solution"
] | [
"Definition:Fundamental Matrix",
"Definition:Singular Matrix",
"Definition:Inverse Matrix",
"Definition:Differential Equation/System",
"Definition:Initial Condition",
"Existence and Uniqueness Theorem for 1st Order IVPs",
"Category:Differential Equations"
] |
proofwiki-2731 | General Fundamental Matrix | Let $\map \Phi t$ be a fundamental matrix of the system $x' = \map A t x$.
Then:
:$\map \Phi t C$
is a general fundamental matrix of $x' = \map A t x$, where $C$ is ''any'' nonsingular matrix. | $\map \Phi t C$ is a fundamental matrix as follows:
:$\dfrac \d {\d t} \map \Phi t C = \map {\Phi'} t C = \map A t \map \Phi t C$
:$\map \det {\map \Phi t C} = \map \det {\map \Phi t} \map \det C \ne 0$
Let $\map \Psi t$ be an arbitrary fundamental matrix.
Then from General Vector Solution of Fundamental Matrix $\map \... | Let $\map \Phi t$ be a [[Definition:Fundamental Matrix|fundamental matrix]] of the [[Definition:System of Differential Equations|system]] $x' = \map A t x$.
Then:
:$\map \Phi t C$
is a general [[Definition:Fundamental Matrix|fundamental matrix]] of $x' = \map A t x$, where $C$ is ''any'' [[Definition:Singular Matrix|n... | $\map \Phi t C$ is a [[Definition:Fundamental Matrix|fundamental matrix]] as follows:
:$\dfrac \d {\d t} \map \Phi t C = \map {\Phi'} t C = \map A t \map \Phi t C$
:$\map \det {\map \Phi t C} = \map \det {\map \Phi t} \map \det C \ne 0$
Let $\map \Psi t$ be an arbitrary [[Definition:Fundamental Matrix|fundamental m... | General Fundamental Matrix | https://proofwiki.org/wiki/General_Fundamental_Matrix | https://proofwiki.org/wiki/General_Fundamental_Matrix | [
"Differential Equations"
] | [
"Definition:Fundamental Matrix",
"Definition:Differential Equation/System",
"Definition:Fundamental Matrix",
"Definition:Singular Matrix"
] | [
"Definition:Fundamental Matrix",
"Definition:Fundamental Matrix",
"General Vector Solution of Fundamental Matrix",
"Existence and Uniqueness Theorem for 1st Order IVPs",
"Category:Differential Equations"
] |
proofwiki-2732 | Condition for Composite Mapping on Left | Let $A, B, C$ be sets.
Suppose that $C$ is non-empty.
Let $f: A \to B$ and $g: A \to C$ be mappings.
Let $\RR: B \to C$ be a relation such that $g = \RR \circ f$ is the composite of $f$ and $\RR$.
Then $\RR$ may be a mapping {{iff}}:
:$\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$
That is:
:$\fo... | === Sufficient Condition ===
Suppose $\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$.
Consider the subset $G \subseteq \Img f \times C$ defined by:
:$G = \set {\tuple {y, z}: \exists x \in A: y = \map f x, z =\map g x}$
Clearly $G \ne \O$ because for any $x \in A$ we have $\tuple {\map f x, \map ... | Let $A, B, C$ be [[Definition:Set|sets]].
Suppose that $C$ is non-empty.
Let $f: A \to B$ and $g: A \to C$ be [[Definition:Mapping|mappings]].
Let $\RR: B \to C$ be a [[Definition:Relation|relation]] such that $g = \RR \circ f$ is the [[Definition:Composition of Relations|composite of $f$ and $\RR$]].
Then $\RR$ m... | === Sufficient Condition ===
Suppose $\forall x, y \in A: \map f x = \map f y \implies \map g x = \map g y$.
Consider the subset $G \subseteq \Img f \times C$ defined by:
:$G = \set {\tuple {y, z}: \exists x \in A: y = \map f x, z =\map g x}$
Clearly $G \ne \O$ because for any $x \in A$ we have $\tuple {\map f x, \m... | Condition for Composite Mapping on Left | https://proofwiki.org/wiki/Condition_for_Composite_Mapping_on_Left | https://proofwiki.org/wiki/Condition_for_Composite_Mapping_on_Left | [
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Relation",
"Definition:Composition of Relations",
"Definition:Mapping",
"Definition:Mapping"
] | [
"Definition:Graph of Mapping",
"Definition:Graph of Mapping",
"Law of Excluded Middle",
"Definition:Mapping",
"Definition:Mapping"
] |
proofwiki-2733 | Condition for Composite Mapping on Right | Let $A, B, C$ be sets.
Let $f: B \to A$ and $g: C \to A$ be mappings.
Let $\RR: C \to B$ be a relation such that $g = f \circ \RR$ is the composite of $\RR$ and $f$.
Then $\RR$ may be a mapping {{iff}}:
:$\Img g \subseteq \Img f$
That is:
:$\Img g \subseteq \Img f$
{{iff}}:
:$\exists h: C \to B$ such that $h$ is a mapp... | === Sufficient Condition ===
Suppose $\Img g \subseteq \Img f$.
That is:
:$\forall x \in C: \map g x \in \Img f$
and so:
:$\forall x \in C: \exists y \in B: \map g x = \map f y$
Take any $x \in C$.
Consider the set $Y_x = \set {y \in B: \map g x = \map f y}$.
We know from above that $Y_x \ne \O$.
So, using the Axiom of... | Let $A, B, C$ be [[Definition:Set|sets]].
Let $f: B \to A$ and $g: C \to A$ be [[Definition:Mapping|mappings]].
Let $\RR: C \to B$ be a [[Definition:Relation|relation]] such that $g = f \circ \RR$ is the [[Definition:Composition of Relations|composite of $\RR$ and $f$]].
Then $\RR$ may be a [[Definition:Mapping|map... | === Sufficient Condition ===
Suppose $\Img g \subseteq \Img f$.
That is:
:$\forall x \in C: \map g x \in \Img f$
and so:
:$\forall x \in C: \exists y \in B: \map g x = \map f y$
Take any $x \in C$.
Consider the set $Y_x = \set {y \in B: \map g x = \map f y}$.
We know from above that $Y_x \ne \O$.
So, using the [... | Condition for Composite Mapping on Right | https://proofwiki.org/wiki/Condition_for_Composite_Mapping_on_Right | https://proofwiki.org/wiki/Condition_for_Composite_Mapping_on_Right | [
"Composite Mappings"
] | [
"Definition:Set",
"Definition:Mapping",
"Definition:Relation",
"Definition:Composition of Relations",
"Definition:Mapping",
"Definition:Mapping"
] | [
"Axiom:Axiom of Choice"
] |
proofwiki-2734 | Set Union Preserves Subsets | Let $A, B, S, T$ be sets.
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | Let $A \subseteq B$, and let $S$ be any set.
From Set Union Preserves Subsets, substituting $S$ for $T$:
:$A \subseteq B, \ S \subseteq S \implies A \cup S \subseteq B \cup S$
From Set is Subset of Itself, $S \subseteq S$ for all sets $S$.
Hence the first result:
:$A \subseteq B \implies A \cup S \subseteq B \cup S$
Th... | Let $A, B, S, T$ be [[Definition:Set|sets]].
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | Let $A \subseteq B$, and let $S$ be any set.
From [[Set Union Preserves Subsets]], substituting $S$ for $T$:
:$A \subseteq B, \ S \subseteq S \implies A \cup S \subseteq B \cup S$
From [[Set is Subset of Itself]], $S \subseteq S$ for all sets $S$.
Hence the first result:
:$A \subseteq B \implies A \cup S \subseteq ... | Set Union Preserves Subsets/Corollary/Proof 1 | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/Corollary/Proof_1 | [
"Set Union",
"Subsets",
"Compatible Relations",
"Set Union Preserves Subsets"
] | [
"Definition:Set"
] | [
"Set Union Preserves Subsets",
"Set is Subset of Itself",
"Union is Commutative"
] |
proofwiki-2735 | Set Union Preserves Subsets | Let $A, B, S, T$ be sets.
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | Let $A$, $B$, and $S$ be sets.
Let $A \subseteq B$.
Let $x \in A \cup S$.
By the definition of union:
:$x \in A$ or $x \in S$
Suppose $x \in A$.
Then by the definition of subset:
:$x \in B$
Thus by the definition of union:
:$x \in B \cup S$
Suppose instead that $x \in S$.
Then by the definition of union:
:$x \in B \cup... | Let $A, B, S, T$ be [[Definition:Set|sets]].
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | Let $A$, $B$, and $S$ be [[Definition:Set|sets]].
Let $A \subseteq B$.
Let $x \in A \cup S$.
By the definition of [[Definition:Set Union|union]]:
:$x \in A$ or $x \in S$
Suppose $x \in A$.
Then by the definition of [[Definition:Subset|subset]]:
:$x \in B$
Thus by the definition of [[Definition:Set Union|union]]:
... | Set Union Preserves Subsets/Corollary/Proof 2 | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/Corollary/Proof_2 | [
"Set Union",
"Subsets",
"Compatible Relations",
"Set Union Preserves Subsets"
] | [
"Definition:Set"
] | [
"Definition:Set",
"Definition:Set Union",
"Definition:Subset",
"Definition:Set Union",
"Definition:Set Union",
"Union is Commutative"
] |
proofwiki-2736 | Set Union Preserves Subsets | Let $A, B, S, T$ be sets.
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | Let $A \subseteq B$ and $S \subseteq T$.
Then:
{{begin-eqn}}
{{eqn | l = x \in A
| o = \leadsto
| r = x \in B
| c = {{Defof|Subset}}
}}
{{eqn | l = x \in S
| o = \leadsto
| r = x \in T
| c = {{Defof|Subset}}
}}
{{end-eqn}}
Now we invoke the Constructive Dilemma of propositional logic... | Let $A, B, S, T$ be [[Definition:Set|sets]].
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | Let $A \subseteq B$ and $S \subseteq T$.
Then:
{{begin-eqn}}
{{eqn | l = x \in A
| o = \leadsto
| r = x \in B
| c = {{Defof|Subset}}
}}
{{eqn | l = x \in S
| o = \leadsto
| r = x \in T
| c = {{Defof|Subset}}
}}
{{end-eqn}}
Now we invoke the [[Constructive Dilemma]] of [[Definitio... | Set Union Preserves Subsets/Proof 1 | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/Proof_1 | [
"Set Union",
"Subsets",
"Compatible Relations",
"Set Union Preserves Subsets"
] | [
"Definition:Set"
] | [
"Constructive Dilemma",
"Definition:Propositional Logic",
"Definition:Set Union",
"Definition:Subset"
] |
proofwiki-2737 | Set Union Preserves Subsets | Let $A, B, S, T$ be sets.
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | By Subset Relation is Transitive, $\subseteq$ is a transitive relation.
By the corollary to Set Union Preserves Subsets (Proof 2), $\subseteq$ is compatible with $\cup$.
Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.
{{qed}} | Let $A, B, S, T$ be [[Definition:Set|sets]].
Then:
:$A \subseteq B, \ S \subseteq T \implies A \cup S \subseteq B \cup T$ | By [[Subset Relation is Transitive]], $\subseteq$ is a [[Definition:Transitive Relation|transitive relation]].
By the [[Set Union Preserves Subsets/Corollary/Proof 2|corollary to Set Union Preserves Subsets (Proof 2)]], $\subseteq$ is [[Definition:Relation Compatible with Operation|compatible]] with $\cup$.
Thus the ... | Set Union Preserves Subsets/Proof 2 | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets | https://proofwiki.org/wiki/Set_Union_Preserves_Subsets/Proof_2 | [
"Set Union",
"Subsets",
"Compatible Relations",
"Set Union Preserves Subsets"
] | [
"Definition:Set"
] | [
"Subset Relation is Transitive",
"Definition:Transitive Relation",
"Set Union Preserves Subsets/Corollary/Proof 2",
"Definition:Relation Compatible with Operation",
"Operating on Transitive Relationships Compatible with Operation"
] |
proofwiki-2738 | Set Complement inverts Subsets | :$S \subseteq T \iff \map \complement T \subseteq \map \complement S$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = S \cap T
| r = S
| c = Intersection with Subset is Subset
}}
{{eqn | ll= \leadstoandfrom
| l = \map \complement {S \cap T}
| r = \map \complement S
| c = Complement of Co... | :$S \subseteq T \iff \map \complement T \subseteq \map \complement S$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = S \cap T
| r = S
| c = [[Intersection with Subset is Subset]]
}}
{{eqn | ll= \leadstoandfrom
| l = \map \complement {S \cap T}
| r = \map \complement S
| c = [[Complement... | Set Complement inverts Subsets/Proof 1 | https://proofwiki.org/wiki/Set_Complement_inverts_Subsets | https://proofwiki.org/wiki/Set_Complement_inverts_Subsets/Proof_1 | [
"Subsets",
"Set Complement",
"Set Complement inverts Subsets"
] | [] | [
"Intersection with Subset is Subset",
"Complement of Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection",
"Union with Superset is Superset"
] |
proofwiki-2739 | Set Complement inverts Subsets | :$S \subseteq T \iff \map \complement T \subseteq \map \complement S$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = (x \in S
| o = \implies
| r = x \in T)
| c = {{Defof|Subset}}
}}
{{eqn | ll= \leadstoandfrom
| l = (x \notin T
| o = \implies
| r = x \notin S)
| c = Rule of... | :$S \subseteq T \iff \map \complement T \subseteq \map \complement S$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = (x \in S
| o = \implies
| r = x \in T)
| c = {{Defof|Subset}}
}}
{{eqn | ll= \leadstoandfrom
| l = (x \notin T
| o = \implies
| r = x \notin S)
| c = [[Rule ... | Set Complement inverts Subsets/Proof 2 | https://proofwiki.org/wiki/Set_Complement_inverts_Subsets | https://proofwiki.org/wiki/Set_Complement_inverts_Subsets/Proof_2 | [
"Subsets",
"Set Complement",
"Set Complement inverts Subsets"
] | [] | [
"Rule of Transposition"
] |
proofwiki-2740 | Set Complement inverts Subsets | :$S \subseteq T \iff \map \complement T \subseteq \map \complement S$ | By definition of set complement:
:$\map \complement T := \relcomp {\mathbb U} T$
where:
:$\mathbb U$ is the universal set
:$\relcomp {\mathbb U} T$ denotes the complement of $T$ relative to $\mathbb U$.
Thus the statement can be expressed as:
:$S \subseteq T \iff \relcomp {\mathbb U} T \subseteq \relcomp {\mathbb U} S$... | :$S \subseteq T \iff \map \complement T \subseteq \map \complement S$ | By definition of [[Definition:Set Complement|set complement]]:
:$\map \complement T := \relcomp {\mathbb U} T$
where:
:$\mathbb U$ is the [[Definition:Universal Set|universal set]]
:$\relcomp {\mathbb U} T$ denotes the [[Definition:Relative Complement|complement of $T$ relative to $\mathbb U$]].
Thus the statement ca... | Set Complement inverts Subsets/Proof 3 | https://proofwiki.org/wiki/Set_Complement_inverts_Subsets | https://proofwiki.org/wiki/Set_Complement_inverts_Subsets/Proof_3 | [
"Subsets",
"Set Complement",
"Set Complement inverts Subsets"
] | [] | [
"Definition:Set Complement",
"Definition:Universal Set",
"Definition:Relative Complement",
"Relative Complement inverts Subsets"
] |
proofwiki-2741 | Set Difference with Superset is Empty Set | :$S \subseteq T \iff S \setminus T = \O$ | {{begin-eqn}}
{{eqn | o =
| r = S \setminus T = \O
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {\exists x: x \in S \land x \notin T}
| c = {{Defof|Empty Set}}
}}
{{eqn | o = \leadstoandfrom
| r = \forall x: \neg \paren {x \in S \land x \notin T}
| c = De Morgan's Laws (Predicate Logi... | :$S \subseteq T \iff S \setminus T = \O$ | {{begin-eqn}}
{{eqn | o =
| r = S \setminus T = \O
}}
{{eqn | o = \leadstoandfrom
| r = \neg \paren {\exists x: x \in S \land x \notin T}
| c = {{Defof|Empty Set}}
}}
{{eqn | o = \leadstoandfrom
| r = \forall x: \neg \paren {x \in S \land x \notin T}
| c = [[De Morgan's Laws (Predicate Lo... | Set Difference with Superset is Empty Set | https://proofwiki.org/wiki/Set_Difference_with_Superset_is_Empty_Set | https://proofwiki.org/wiki/Set_Difference_with_Superset_is_Empty_Set | [
"Set Difference",
"Subsets",
"Empty Set"
] | [] | [
"De Morgan's Laws (Predicate Logic)",
"De Morgan's Laws (Logic)/Disjunction of Negations",
"Rule of Material Implication"
] |
proofwiki-2742 | Intersection with Subset is Subset | :$S \subseteq T \iff S \cap T = S$ | Let $S \cap T = S$.
Then by the definition of set equality, $S \subseteq S \cap T$.
Thus:
{{begin-eqn}}
{{eqn | l = S \cap T
| o = \subseteq
| r = T
| c = Intersection is Subset
}}
{{eqn | ll= \leadsto
| l = S
| o = \subseteq
| r = T
| c = Subset Relation is Transitive
}}
... | :$S \subseteq T \iff S \cap T = S$ | Let $S \cap T = S$.
Then by the definition of [[Definition:Set Equality|set equality]], $S \subseteq S \cap T$.
Thus:
{{begin-eqn}}
{{eqn | l = S \cap T
| o = \subseteq
| r = T
| c = [[Intersection is Subset]]
}}
{{eqn | ll= \leadsto
| l = S
| o = \subseteq
| r = T
| c ... | Intersection with Subset is Subset/Proof 1 | https://proofwiki.org/wiki/Intersection_with_Subset_is_Subset | https://proofwiki.org/wiki/Intersection_with_Subset_is_Subset/Proof_1 | [
"Intersection with Subset is Subset",
"Subsets",
"Set Intersection"
] | [] | [
"Definition:Set Equality",
"Intersection is Subset",
"Subset Relation is Transitive",
"Intersection is Subset",
"Set Intersection Preserves Subsets",
"Set Intersection is Idempotent",
"Intersection is Commutative",
"Definition:Set Equality",
"Definition:Biconditional"
] |
proofwiki-2743 | Intersection with Subset is Subset | :$S \subseteq T \iff S \cap T = S$ | {{begin-eqn}}
{{eqn | o =
| r = S \cap T = S
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \land x \in T \iff x \in S}
| c = {{Defof|Set Equality}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \implies x \in T}
| c = Conditional iff Biconditional ... | :$S \subseteq T \iff S \cap T = S$ | {{begin-eqn}}
{{eqn | o =
| r = S \cap T = S
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \land x \in T \iff x \in S}
| c = {{Defof|Set Equality}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \implies x \in T}
| c = [[Conditional iff Biconditiona... | Intersection with Subset is Subset/Proof 2 | https://proofwiki.org/wiki/Intersection_with_Subset_is_Subset | https://proofwiki.org/wiki/Intersection_with_Subset_is_Subset/Proof_2 | [
"Intersection with Subset is Subset",
"Subsets",
"Set Intersection"
] | [] | [
"Conditional iff Biconditional of Antecedent with Conjunction"
] |
proofwiki-2744 | Union with Superset is Superset | :$S \subseteq T \iff S \cup T = T$ | Let $S \cup T = T$.
Then by definition of set equality:
:$S \cup T \subseteq T$
Thus:
{{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = S \cup T
| c = Subset of Union
}}
{{eqn | ll= \leadsto
| l = S
| o = \subseteq
| r = T
| c = Subset Relation is Transitive
}}
{{end-eqn}}
Now ... | :$S \subseteq T \iff S \cup T = T$ | Let $S \cup T = T$.
Then by definition of [[Definition:Set Equality/Definition 2|set equality]]:
:$S \cup T \subseteq T$
Thus:
{{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = S \cup T
| c = [[Subset of Union]]
}}
{{eqn | ll= \leadsto
| l = S
| o = \subseteq
| r = T
| c = [... | Union with Superset is Superset/Proof 1 | https://proofwiki.org/wiki/Union_with_Superset_is_Superset | https://proofwiki.org/wiki/Union_with_Superset_is_Superset/Proof_1 | [
"Union with Superset is Superset",
"Subsets",
"Set Union"
] | [] | [
"Definition:Set Equality/Definition 2",
"Set is Subset of Union",
"Subset Relation is Transitive",
"Set is Subset of Union",
"Set Union Preserves Subsets",
"Set Union is Idempotent",
"Definition:Set Equality/Definition 2",
"Definition:Biconditional"
] |
proofwiki-2745 | Union with Superset is Superset | :$S \subseteq T \iff S \cup T = T$ | {{begin-eqn}}
{{eqn | o =
| r = S \cup T = T
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \lor x \in T \iff x \in T}
| c = {{Defof|Set Equality}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \implies x \in T}
| c = Conditional iff Biconditional of... | :$S \subseteq T \iff S \cup T = T$ | {{begin-eqn}}
{{eqn | o =
| r = S \cup T = T
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \lor x \in T \iff x \in T}
| c = {{Defof|Set Equality}}
}}
{{eqn | ll= \leadstoandfrom
| o =
| r = \paren {x \in S \implies x \in T}
| c = [[Conditional iff Biconditional ... | Union with Superset is Superset/Proof 2 | https://proofwiki.org/wiki/Union_with_Superset_is_Superset | https://proofwiki.org/wiki/Union_with_Superset_is_Superset/Proof_2 | [
"Union with Superset is Superset",
"Subsets",
"Set Union"
] | [] | [
"Conditional iff Biconditional of Consequent with Disjunction"
] |
proofwiki-2746 | Intersection with Complement is Empty iff Subset | :$S \subseteq T \iff S \cap \map \complement T = \O$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
}}
{{eqn | ll= \leadstoandfrom
| l = S \setminus T
| r = \O
| c = Set Difference with Superset is Empty Set
}}
{{eqn | ll= \leadstoandfrom
| l = S \cap \map \complement T
| r = \O
| c = Set Difference as Intersection wit... | :$S \subseteq T \iff S \cap \map \complement T = \O$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
}}
{{eqn | ll= \leadstoandfrom
| l = S \setminus T
| r = \O
| c = [[Set Difference with Superset is Empty Set]]
}}
{{eqn | ll= \leadstoandfrom
| l = S \cap \map \complement T
| r = \O
| c = [[Set Difference as Intersecti... | Intersection with Complement is Empty iff Subset | https://proofwiki.org/wiki/Intersection_with_Complement_is_Empty_iff_Subset | https://proofwiki.org/wiki/Intersection_with_Complement_is_Empty_iff_Subset | [
"Subsets",
"Set Intersection",
"Set Complement",
"Empty Set"
] | [] | [
"Set Difference with Superset is Empty Set",
"Set Difference as Intersection with Complement"
] |
proofwiki-2747 | Complement Union with Superset is Universe | :$S \subseteq T \iff \map \complement S \cup T = \mathbb U$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
}}
{{eqn | ll= \leadstoandfrom
| l = S \cap \map \complement T
| r = \O
| c = Intersection with Complement is Empty iff Subset
}}
{{eqn | ll= \leadstoandfrom
| l = \map \complement {S \cap \map \complement T}
| r = \mathbb U
... | :$S \subseteq T \iff \map \complement S \cup T = \mathbb U$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
}}
{{eqn | ll= \leadstoandfrom
| l = S \cap \map \complement T
| r = \O
| c = [[Intersection with Complement is Empty iff Subset]]
}}
{{eqn | ll= \leadstoandfrom
| l = \map \complement {S \cap \map \complement T}
| r = \mathbb... | Complement Union with Superset is Universe | https://proofwiki.org/wiki/Complement_Union_with_Superset_is_Universe | https://proofwiki.org/wiki/Complement_Union_with_Superset_is_Universe | [
"Subsets",
"Set Union",
"Set Complement",
"Empty Set"
] | [] | [
"Intersection with Complement is Empty iff Subset",
"Complement of Empty Set is Universal Set",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection",
"Complement of Complement"
] |
proofwiki-2748 | Set Difference is not Associative | Let $R, S, T$ be sets.
The expression:
:$\paren {R \setminus S} \setminus T = R \setminus \paren {S \setminus T}$
holds exactly when $R \cap T = \O$.
Here $R \setminus S$ denotes set difference.
Thus, set difference is not associative. | We assume a universal set $\mathbb U$ such that $R, S, T \subseteq \mathbb U$.
We have the identity Set Difference as Intersection with Complement:
:$R \setminus S = R \cap \overline S$
where $\overline S$ is the set complement of $S$:
:$\overline S = \relcomp {\Bbb U} S$
Thus we can represent the two expressions as fo... | Let $R, S, T$ be [[Definition:Set|sets]].
The expression:
:$\paren {R \setminus S} \setminus T = R \setminus \paren {S \setminus T}$
holds exactly when $R \cap T = \O$.
Here $R \setminus S$ denotes [[Definition:Set Difference|set difference]].
Thus, [[Definition:Set Difference|set difference]] is not [[Definition... | We assume a [[Definition:Universal Set|universal set]] $\mathbb U$ such that $R, S, T \subseteq \mathbb U$.
We have the identity [[Set Difference as Intersection with Complement]]:
:$R \setminus S = R \cap \overline S$
where $\overline S$ is the [[Definition:Set Complement|set complement]] of $S$:
:$\overline S = \... | Set Difference is not Associative | https://proofwiki.org/wiki/Set_Difference_is_not_Associative | https://proofwiki.org/wiki/Set_Difference_is_not_Associative | [
"Set Difference"
] | [
"Definition:Set",
"Definition:Set Difference",
"Definition:Set Difference",
"Definition:Associative Operation"
] | [
"Definition:Universal Set",
"Set Difference as Intersection with Complement",
"Definition:Set Complement",
"Set Difference as Intersection with Complement",
"Intersection is Associative",
"Set Difference as Intersection with Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Inters... |
proofwiki-2749 | Associativity on Four Elements | Let $\struct {S, \circ}$ be a semigroup.
Let $a, b, c, d \in S$.
Then:
:$a \circ b \circ c \circ d$
gives a unique answer no matter how the elements are associated. | As $\struct {S, \circ}$ is a semigroup:
: it is closed
: $\circ$ is associative
From Parenthesization of Word of $4$ Elements, there are exactly $5$ different ways of inserting brackets in the expression $a \circ b \circ c \circ d$.
As $\circ$ is associative, we have that:
:$\forall s_1, s_2, s_3 \in S: \paren {s_1 \ci... | Let $\struct {S, \circ}$ be a [[Definition:Semigroup|semigroup]].
Let $a, b, c, d \in S$.
Then:
:$a \circ b \circ c \circ d$
gives a unique answer no matter how the elements are associated. | As $\struct {S, \circ}$ is a [[Definition:Semigroup|semigroup]]:
: it is [[Definition:Closed Algebraic Structure|closed]]
: $\circ$ is [[Definition:Associative Operation|associative]]
From [[Parenthesization of Word of 4 Elements|Parenthesization of Word of $4$ Elements]], there are exactly $5$ different ways of inse... | Associativity on Four Elements | https://proofwiki.org/wiki/Associativity_on_Four_Elements | https://proofwiki.org/wiki/Associativity_on_Four_Elements | [
"Associativity"
] | [
"Definition:Semigroup"
] | [
"Definition:Semigroup",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative Operation",
"Parenthesization/Examples/4",
"Definition:Associative Operation",
"Definition:Closure (Abstract Algebra)/Algebraic Structure",
"Definition:Associative Operation"
] |
proofwiki-2750 | Group Isomorphism Preserves Identity | Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism.
Let:
:$e_G$ be the identity of $\struct {G, \circ}$
:$e_H$ be the identity of $\struct {H, *}$.
Then:
:$\map \phi {e_G} = e_H$ | An group isomorphism is by definition a group epimorphism.
The result follows from Epimorphism Preserves Identity.
{{Qed}} | Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a [[Definition:Group Isomorphism|group isomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $\struct {G, \circ}$
:$e_H$ be the [[Definition:Identity Element|identity]] of $\struct {H, *}$.
Then:
:$\map \phi {e_G} = e_H$ | An [[Definition:Group Isomorphism|group isomorphism]] is by definition a [[Definition:Group Epimorphism|group epimorphism]].
The result follows from [[Epimorphism Preserves Identity]].
{{Qed}} | Group Isomorphism Preserves Identity/Proof 1 | https://proofwiki.org/wiki/Group_Isomorphism_Preserves_Identity | https://proofwiki.org/wiki/Group_Isomorphism_Preserves_Identity/Proof_1 | [
"Group Isomorphisms",
"Identity Elements",
"Group Isomorphism Preserves Identity"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Group Epimorphism",
"Epimorphism Preserves Identity"
] |
proofwiki-2751 | Group Isomorphism Preserves Identity | Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group isomorphism.
Let:
:$e_G$ be the identity of $\struct {G, \circ}$
:$e_H$ be the identity of $\struct {H, *}$.
Then:
:$\map \phi {e_G} = e_H$ | {{begin-eqn}}
{{eqn | l = \map \phi {e_G}
| r = \map \phi {e_G \circ e_G}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \map \phi {e_G} * \map \phi {e_G}
| c = {{Defof|Group Isomorphism}}
}}
{{eqn | r = \map \phi {e_G} * \map \phi {e_G}
| c = {{Defof|Group Isomorphism}}
}}
{{end-eqn}}
It follo... | Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a [[Definition:Group Isomorphism|group isomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $\struct {G, \circ}$
:$e_H$ be the [[Definition:Identity Element|identity]] of $\struct {H, *}$.
Then:
:$\map \phi {e_G} = e_H$ | {{begin-eqn}}
{{eqn | l = \map \phi {e_G}
| r = \map \phi {e_G \circ e_G}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \map \phi {e_G} * \map \phi {e_G}
| c = {{Defof|Group Isomorphism}}
}}
{{eqn | r = \map \phi {e_G} * \map \phi {e_G}
| c = {{Defof|Group Isomorphism}}
}}
{{end-eqn}}
It foll... | Group Isomorphism Preserves Identity/Proof 2 | https://proofwiki.org/wiki/Group_Isomorphism_Preserves_Identity | https://proofwiki.org/wiki/Group_Isomorphism_Preserves_Identity/Proof_2 | [
"Group Isomorphisms",
"Identity Elements",
"Group Isomorphism Preserves Identity"
] | [
"Definition:Isomorphism (Abstract Algebra)/Group Isomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Identity is only Idempotent Element in Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] |
proofwiki-2752 | Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group | Let $\Q_{> 0}$ be the set of strictly positive rational numbers, i.e. $\Q_{> 0} = \set {x \in \Q: x > 0}$.
The structure $\struct {\Q_{> 0}, \times}$ is a countably infinite abelian group. | From Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers we have that $\struct {\Q_{> 0}, \times}$ is a subgroup of $\struct {\Q_{\ne 0}, \times}$, where $\Q_{\ne 0}$ is the set of rational numbers without zero: $\Q_{\ne 0} = \Q \setminus \set 0$.
From Subgroup of Abelian ... | Let $\Q_{> 0}$ be the set of [[Definition:Strictly Positive Rational Number|strictly positive rational numbers]], i.e. $\Q_{> 0} = \set {x \in \Q: x > 0}$.
The [[Definition:Algebraic Structure with One Operation|structure]] $\struct {\Q_{> 0}, \times}$ is a [[Definition:Countably Infinite Group|countably infinite]] [[... | From [[Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers]] we have that $\struct {\Q_{> 0}, \times}$ is a [[Definition:Subgroup|subgroup]] of $\struct {\Q_{\ne 0}, \times}$, where $\Q_{\ne 0}$ is the set of [[Definition:Rational Number|rational numbers]] without [[Defini... | Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group | https://proofwiki.org/wiki/Strictly_Positive_Rational_Numbers_under_Multiplication_form_Countably_Infinite_Abelian_Group | https://proofwiki.org/wiki/Strictly_Positive_Rational_Numbers_under_Multiplication_form_Countably_Infinite_Abelian_Group | [
"Rational Multiplication",
"Examples of Abelian Groups",
"Examples of Infinite Groups"
] | [
"Definition:Strictly Positive/Rational Number",
"Definition:Algebraic Structure/One Operation",
"Definition:Infinite Group/Countable",
"Definition:Abelian Group"
] | [
"Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers",
"Definition:Subgroup",
"Definition:Rational Number",
"Definition:Zero (Number)",
"Subgroup of Abelian Group is Abelian",
"Definition:Abelian Group",
"Positive Rational Numbers are Countably Infinite",
... |
proofwiki-2753 | Multinomial Theorem | Let $x_1, x_2, \ldots, x_k \in F$, where $F$ is a field.
Then:
:$\ds \paren {x_1 + x_2 + \cdots + x_m}^n = \sum_{k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} {x_1}^{k_1} {x_2}^{k_2} \cdots {x_m}^{k_m}$
where:
:$m \in \Z_{> 0}$ is a positive integer
:$n \in \Z_{\... | The proof proceeds by induction on $m$.
For each $m \in \N_{\ge 1}$, let $\map P m$ be the proposition:
:$\ds \forall n \in \N: \paren {x_1 + x_2 + \cdots + x_m}^n = \sum_{k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} {x_1}^{k_1} {x_2}^{k_2} \cdots {x_m}^{k_m}$ | Let $x_1, x_2, \ldots, x_k \in F$, where $F$ is a [[Definition:Field (Abstract Algebra)|field]].
Then:
:$\ds \paren {x_1 + x_2 + \cdots + x_m}^n = \sum_{k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} {x_1}^{k_1} {x_2}^{k_2} \cdots {x_m}^{k_m}$
where:
:$m \in \... | The proof proceeds by [[Principle of Mathematical Induction|induction]] on $m$.
For each $m \in \N_{\ge 1}$, let $\map P m$ be the proposition:
:$\ds \forall n \in \N: \paren {x_1 + x_2 + \cdots + x_m}^n = \sum_{k_1 \mathop + k_2 \mathop + \mathop \cdots \mathop + k_m \mathop = n} \binom n {k_1, k_2, \ldots, k_m} {x_... | Multinomial Theorem | https://proofwiki.org/wiki/Multinomial_Theorem | https://proofwiki.org/wiki/Multinomial_Theorem | [
"Multinomial Theorem",
"Multinomial Coefficients",
"Binomial Coefficients",
"Algebra",
"Discrete Mathematics",
"Proofs by Induction"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Positive/Integer",
"Definition:Positive/Integer",
"Definition:Multinomial Coefficient",
"Definition:Summation",
"Definition:Positive/Integer"
] | [
"Principle of Mathematical Induction",
"Principle of Mathematical Induction"
] |
proofwiki-2754 | Mapping is Constant iff Increasing and Decreasing | Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping.
Then $\phi$ is a constant mapping {{iff}} $\phi$ is both increasing and decreasing. | === Necessary Condition ===
Suppose $\phi$ is a constant mapping.
Then:
: $\forall x, y \in S: \map \phi x = \map \phi y$
So:
: $\forall x, y \in S: \map \phi x \mathop{\preceq_2} \map \phi y$
: $\forall x, y \in S: \map \phi y \mathop{\preceq_2} \map \phi x$
and so $\phi$ is both increasing and decreasing.
{{qed|lemma... | Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be [[Definition:Ordered Set|ordered sets]].
Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a [[Definition:Mapping|mapping]].
Then $\phi$ is a [[Definition:Constant Mapping|constant mapping]] {{iff}} $\phi$ is both [[Definition:Increasing Map... | === Necessary Condition ===
Suppose $\phi$ is a [[Definition:Constant Mapping|constant mapping]].
Then:
: $\forall x, y \in S: \map \phi x = \map \phi y$
So:
: $\forall x, y \in S: \map \phi x \mathop{\preceq_2} \map \phi y$
: $\forall x, y \in S: \map \phi y \mathop{\preceq_2} \map \phi x$
and so $\phi$ is both [... | Mapping is Constant iff Increasing and Decreasing | https://proofwiki.org/wiki/Mapping_is_Constant_iff_Increasing_and_Decreasing | https://proofwiki.org/wiki/Mapping_is_Constant_iff_Increasing_and_Decreasing | [
"Increasing Mappings",
"Decreasing Mappings",
"Constant Mappings"
] | [
"Definition:Ordered Set",
"Definition:Mapping",
"Definition:Constant Mapping",
"Definition:Increasing/Mapping",
"Definition:Decreasing/Mapping"
] | [
"Definition:Constant Mapping",
"Definition:Increasing/Mapping",
"Definition:Decreasing/Mapping",
"Definition:Increasing/Mapping",
"Definition:Decreasing/Mapping",
"Definition:Constant Mapping"
] |
proofwiki-2755 | Powers Drown Logarithms | Let $r \in \R_{>0}$ be a (strictly) positive real number.
Then:
:$\ds \lim_{x \mathop \to \infty} x^{-r} \ln x = 0$ | From Upper Bound of Natural Logarithm:
When $x > 1$:
: $\forall s \in \R: s > 0: \ln x \le \dfrac {x^s} s$
Given that $r > 0$, we can plug $s = \dfrac r 2$ in:
{{begin-eqn}}
{{eqn | l = x^{-r} \ln x
| r = x^{-r/2} \paren {x^{-s} \ln x}
| c =
}}
{{eqn | o = \le
| r = \frac {x^{-r/2} } s
| c =
}... | Let $r \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Then:
:$\ds \lim_{x \mathop \to \infty} x^{-r} \ln x = 0$ | From [[Upper Bound of Natural Logarithm]]:
When $x > 1$:
: $\forall s \in \R: s > 0: \ln x \le \dfrac {x^s} s$
Given that $r > 0$, we can plug $s = \dfrac r 2$ in:
{{begin-eqn}}
{{eqn | l = x^{-r} \ln x
| r = x^{-r/2} \paren {x^{-s} \ln x}
| c =
}}
{{eqn | o = \le
| r = \frac {x^{-r/2} } s
|... | Powers Drown Logarithms | https://proofwiki.org/wiki/Powers_Drown_Logarithms | https://proofwiki.org/wiki/Powers_Drown_Logarithms | [
"Logarithms",
"Powers"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Upper Bound of Natural Logarithm",
"Sequence of Powers of Reciprocals is Null Sequence",
"Squeeze Theorem/Sequences/Real Numbers"
] |
proofwiki-2756 | Equivalence of Well-Ordering Principle and Induction | The Well-Ordering Principle, the Principle of Finite Induction and the Principle of Complete Finite Induction are logically equivalent.
That is:
:Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties:
:: $0 \in S$
:: $n \in S \implies n + 1 \in S$
:then $S = \N... | To save space, we will refer to:
: The Well-Ordering Principle as '''WOP'''
: The Principle of Finite Induction as '''PFI'''
: The Principle of Complete Finite Induction as '''PCI'''. | The [[Well-Ordering Principle]], the [[Principle of Finite Induction]] and the [[Principle of Complete Finite Induction]] are [[Definition:Logical Equivalence|logically equivalent]].
That is:
:[[Principle of Finite Induction]]: Given a [[Definition:Subset|subset]] $S \subseteq \N$ of the [[Definition:Natural Numbers|... | To save space, we will refer to:
: The [[Well-Ordering Principle]] as '''[[Well-Ordering Principle|WOP]]'''
: The [[Principle of Finite Induction]] as '''[[Principle of Finite Induction|PFI]]'''
: The [[Principle of Complete Finite Induction]] as '''[[Principle of Complete Finite Induction|PCI]]'''. | Equivalence of Well-Ordering Principle and Induction | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction | https://proofwiki.org/wiki/Equivalence_of_Well-Ordering_Principle_and_Induction | [
"Proofs by Induction",
"Well-Ordering Principle",
"Equivalence of Well-Ordering Principle and Induction"
] | [
"Well-Ordering Principle",
"Principle of Finite Induction",
"Second Principle of Finite Induction",
"Definition:Logical Equivalence",
"Principle of Finite Induction",
"Definition:Subset",
"Definition:Natural Numbers",
"Second Principle of Finite Induction",
"Definition:Subset",
"Definition:Natural... | [
"Well-Ordering Principle",
"Well-Ordering Principle",
"Principle of Finite Induction",
"Principle of Finite Induction",
"Second Principle of Finite Induction",
"Second Principle of Finite Induction",
"Well-Ordering Principle",
"Well-Ordering Principle"
] |
proofwiki-2757 | Lower Bound for Subset | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $L$ be a lower bound for $S$.
Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$.
Then $L$ is a lower bound for $T$. | By definition of lower bound:
:$\forall x \in S: L \preceq x$
But as $\forall y \in T: y \in S$ by definition of subset, it follows that:
:$\forall y \in T: L \preceq y$.
Hence the result, again by definition of lower bound.
{{qed}} | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $L$ be a [[Definition:Lower Bound of Set|lower bound]] for $S$.
Let $\left({T, \preceq}\right)$ be a [[Definition:Subset|subset]] of $\left({S, \preceq}\right)$.
Then $L$ is a [[Definition:Lower Bound of Set|lower bound]] for $T$. | By definition of [[Definition:Lower Bound of Set|lower bound]]:
:$\forall x \in S: L \preceq x$
But as $\forall y \in T: y \in S$ by definition of [[Definition:Subset|subset]], it follows that:
:$\forall y \in T: L \preceq y$.
Hence the result, again by definition of [[Definition:Lower Bound of Set|lower bound]].
{{q... | Lower Bound for Subset | https://proofwiki.org/wiki/Lower_Bound_for_Subset | https://proofwiki.org/wiki/Lower_Bound_for_Subset | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Lower Bound of Set",
"Definition:Subset",
"Definition:Lower Bound of Set"
] | [
"Definition:Lower Bound of Set",
"Definition:Subset",
"Definition:Lower Bound of Set"
] |
proofwiki-2758 | Upper Bound for Subset | Let $\left({S, \preceq}\right)$ be an ordered set.
Let $U$ be an upper bound for $S$.
Let $\left({T, \preceq}\right)$ be a subset of $\left({S, \preceq}\right)$.
Then $U$ is an upper bound for $T$. | By definition of upper bound:
:$\forall x \in S: x \preceq U$
But as $\forall y \in T: y \in S$ by definition of subset, it follows that:
:$\forall y \in T: y \preceq U$.
Hence the result, again by definition of upper bound.
{{qed}} | Let $\left({S, \preceq}\right)$ be an [[Definition:Ordered Set|ordered set]].
Let $U$ be an [[Definition:Upper Bound of Set|upper bound]] for $S$.
Let $\left({T, \preceq}\right)$ be a [[Definition:Subset|subset]] of $\left({S, \preceq}\right)$.
Then $U$ is an [[Definition:Upper Bound of Set|upper bound]] for $T$. | By definition of [[Definition:Upper Bound of Set|upper bound]]:
:$\forall x \in S: x \preceq U$
But as $\forall y \in T: y \in S$ by definition of [[Definition:Subset|subset]], it follows that:
:$\forall y \in T: y \preceq U$.
Hence the result, again by definition of [[Definition:Upper Bound of Set|upper bound]].
{{q... | Upper Bound for Subset | https://proofwiki.org/wiki/Upper_Bound_for_Subset | https://proofwiki.org/wiki/Upper_Bound_for_Subset | [
"Order Theory"
] | [
"Definition:Ordered Set",
"Definition:Upper Bound of Set",
"Definition:Subset",
"Definition:Upper Bound of Set"
] | [
"Definition:Upper Bound of Set",
"Definition:Subset",
"Definition:Upper Bound of Set"
] |
proofwiki-2759 | Common Divisor Divides GCD | Let $a, b \in \Z$ such that not both of $a$ and $b$ are zero.
Let $c$ be any common divisor of $a$ and $b$.
That is, let $c \in \Z: c \divides a, c \divides b$.
Then:
:$c \divides \gcd \set {a, b}$
where $\gcd \set {a, b}$ is the greatest common divisor of $a$ and $b$. | Let $d = \gcd \set {a, b}$.
Then $d \divides a$ and $d \divides b$ by definition.
Then from Bézout's Identity, $\exists u, v \in \Z: d = u a + v b$.
From Common Divisor Divides Integer Combination, $c \divides a \land c \divides b \implies c \divides u a + v b$ for all $u, v \in \Z$.
Thus $c \divides d$.
{{qed}} | Let $a, b \in \Z$ such that not both of $a$ and $b$ are zero.
Let $c$ be any [[Definition:Common Divisor of Integers|common divisor]] of $a$ and $b$.
That is, let $c \in \Z: c \divides a, c \divides b$.
Then:
:$c \divides \gcd \set {a, b}$
where $\gcd \set {a, b}$ is the [[Definition:Greatest Common Divisor of Inte... | Let $d = \gcd \set {a, b}$.
Then $d \divides a$ and $d \divides b$ [[Definition:Greatest Common Divisor of Integers|by definition]].
Then from [[Bézout's Identity]], $\exists u, v \in \Z: d = u a + v b$.
From [[Common Divisor Divides Integer Combination]], $c \divides a \land c \divides b \implies c \divides u a + v... | Common Divisor Divides GCD | https://proofwiki.org/wiki/Common_Divisor_Divides_GCD | https://proofwiki.org/wiki/Common_Divisor_Divides_GCD | [
"Greatest Common Divisor"
] | [
"Definition:Common Divisor/Integers",
"Definition:Greatest Common Divisor/Integers"
] | [
"Definition:Greatest Common Divisor/Integers",
"Bézout's Identity",
"Common Divisor Divides Integer Combination"
] |
proofwiki-2760 | Square Modulo 5 | Let $x \in \Z$ be an integer.
Then one of the following holds:
{{begin-eqn}}
{{eqn | l = x^2
| o = \equiv
| r = 0 \pmod 5
| c =
}}
{{eqn | l = x^2
| o = \equiv
| r = 1 \pmod 5
| c =
}}
{{eqn | l = x^2
| o = \equiv
| r = 4 \pmod 5
| c =
}}
{{end-eqn}} | Let $x$ be an integer.
Using Congruence of Powers throughout, we make use of $x \equiv y \pmod 5 \implies x^2 \equiv y^2 \pmod 5$.
There are five cases to consider:
: $x \equiv 0 \pmod 5$: we have $x^2 \equiv 0^2 \pmod 5 \equiv 0 \pmod 5$.
: $x \equiv 1 \pmod 5$: we have $x^2 \equiv 1^2 \pmod 5 \equiv 1 \pmod 5$.
: $x ... | Let $x \in \Z$ be an [[Definition:Integer|integer]].
Then one of the following holds:
{{begin-eqn}}
{{eqn | l = x^2
| o = \equiv
| r = 0 \pmod 5
| c =
}}
{{eqn | l = x^2
| o = \equiv
| r = 1 \pmod 5
| c =
}}
{{eqn | l = x^2
| o = \equiv
| r = 4 \pmod 5
| c =
}}... | Let $x$ be an [[Definition:Integer|integer]].
Using [[Congruence of Powers]] throughout, we make use of $x \equiv y \pmod 5 \implies x^2 \equiv y^2 \pmod 5$.
There are five cases to consider:
: $x \equiv 0 \pmod 5$: we have $x^2 \equiv 0^2 \pmod 5 \equiv 0 \pmod 5$.
: $x \equiv 1 \pmod 5$: we have $x^2 \equiv 1^2 \... | Square Modulo 5 | https://proofwiki.org/wiki/Square_Modulo_5 | https://proofwiki.org/wiki/Square_Modulo_5 | [
"Modulo Arithmetic",
"Square Numbers"
] | [
"Definition:Integer"
] | [
"Definition:Integer",
"Congruence of Powers"
] |
proofwiki-2761 | Square Modulo 3 | Let $x \in \Z$ be an integer.
Then one of the following holds:
{{begin-eqn}}
{{eqn | l = x^2
| o = \equiv
| r = 0 \pmod 3
| c = when $3 \divides x$
}}
{{eqn | l = x^2
| o = \equiv
| r = 1 \pmod 3
| c = when $3 \nmid x$
}}
{{end-eqn}}
where:
:$\divides$ denotes divisibility
:$\nmid$ d... | Let $x$ be an integer.
Using Congruence of Powers throughout, we make use of:
: $x \equiv y \pmod 3 \implies x^2 \equiv y^2 \pmod 3$
There are three cases to consider:
: $(1): \quad x \equiv 0 \pmod 3$: we have $x^2 \equiv 0^2 \pmod 3 \equiv 0 \pmod 3$
: $(2): \quad x \equiv 1 \pmod 3$: we have $x^2 \equiv 1^2 \pmod 3 ... | Let $x \in \Z$ be an [[Definition:Integer|integer]].
Then one of the following holds:
{{begin-eqn}}
{{eqn | l = x^2
| o = \equiv
| r = 0 \pmod 3
| c = when $3 \divides x$
}}
{{eqn | l = x^2
| o = \equiv
| r = 1 \pmod 3
| c = when $3 \nmid x$
}}
{{end-eqn}}
where:
:$\divides$ denot... | Let $x$ be an [[Definition:Integer|integer]].
Using [[Congruence of Powers]] throughout, we make use of:
: $x \equiv y \pmod 3 \implies x^2 \equiv y^2 \pmod 3$
There are three cases to consider:
: $(1): \quad x \equiv 0 \pmod 3$: we have $x^2 \equiv 0^2 \pmod 3 \equiv 0 \pmod 3$
: $(2): \quad x \equiv 1 \pmod 3$: w... | Square Modulo 3 | https://proofwiki.org/wiki/Square_Modulo_3 | https://proofwiki.org/wiki/Square_Modulo_3 | [
"Modulo Arithmetic",
"Square Numbers",
"Square Modulo 3"
] | [
"Definition:Integer",
"Definition:Divisor (Algebra)/Integer",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Integer",
"Congruence of Powers"
] |
proofwiki-2762 | Additive Group of Integers is Countably Infinite Abelian Group | The set of integers under addition $\struct {\Z, +}$ forms a countably infinite abelian group. | From Integers under Addition form Abelian Group, $\struct {\Z, +}$ is an abelian group.
From Integers are Countably Infinite, the set of integers can be placed in one-to-one correspondence with the set of natural numbers.
Hence by definition, the underlying set of $\struct {\Z, +}$ is countably infinite.
{{qed}} | The [[Definition:Set|set]] of [[Definition:Integer|integers]] under [[Definition:Integer Addition|addition]] $\struct {\Z, +}$ forms a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Abelian Group|abelian group]]. | From [[Integers under Addition form Abelian Group]], $\struct {\Z, +}$ is an [[Definition:Abelian Group|abelian group]].
From [[Integers are Countably Infinite]], the [[Definition:Integer|set of integers]] can be placed in [[Definition:Bijection|one-to-one correspondence]] with the [[Definition:Natural Numbers|set of ... | Additive Group of Integers is Countably Infinite Abelian Group | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Countably_Infinite_Abelian_Group | https://proofwiki.org/wiki/Additive_Group_of_Integers_is_Countably_Infinite_Abelian_Group | [
"Abelian Groups",
"Infinite Groups",
"Additive Group of Integers"
] | [
"Definition:Set",
"Definition:Integer",
"Definition:Addition/Integers",
"Definition:Countably Infinite/Set",
"Definition:Abelian Group"
] | [
"Integers under Addition form Abelian Group",
"Definition:Abelian Group",
"Integers are Countably Infinite",
"Definition:Integer",
"Definition:Bijection",
"Definition:Natural Numbers",
"Definition:Underlying Set/Abstract Algebra",
"Definition:Countably Infinite/Set"
] |
proofwiki-2763 | Division Laws for Groups | Let $G$ be a group.
Let $a, b, x \in G$.
Then:
:$(1): \quad a x = b \iff x = a^{-1} b$
:$(2): \quad x a = b \iff x = b a^{-1}$ | All derivations can be achieved using applications of the group axioms: | Let $G$ be a [[Definition:Group|group]].
Let $a, b, x \in G$.
Then:
:$(1): \quad a x = b \iff x = a^{-1} b$
:$(2): \quad x a = b \iff x = b a^{-1}$ | All derivations can be achieved using applications of the [[Axiom:Group Axioms|group axioms]]: | Division Laws for Groups | https://proofwiki.org/wiki/Division_Laws_for_Groups | https://proofwiki.org/wiki/Division_Laws_for_Groups | [
"Group Theory"
] | [
"Definition:Group"
] | [
"Axiom:Group Axioms"
] |
proofwiki-2764 | Center is Intersection of Centralizers | The center of a group is the intersection of all the centralizers of the elements of that group:
:$\ds \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$ | Denote $Z = \map Z G$ and $C = \ds \bigcap_{g \mathop \in G} \map {C_G} g$ for simplicity.
By definition of set equality, it suffices to prove $Z \subseteq C$ and $C \subseteq Z$. | The [[Definition:Center of Group|center]] of a [[Definition:Group|group]] is the [[Definition:Set Intersection|intersection]] of all the [[Definition:Centralizer of Group Element|centralizers]] of the elements of that group:
:$\ds \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$ | Denote $Z = \map Z G$ and $C = \ds \bigcap_{g \mathop \in G} \map {C_G} g$ for simplicity.
By definition of [[Definition:Set Equality/Definition 2|set equality]], it suffices to prove $Z \subseteq C$ and $C \subseteq Z$. | Center is Intersection of Centralizers | https://proofwiki.org/wiki/Center_is_Intersection_of_Centralizers | https://proofwiki.org/wiki/Center_is_Intersection_of_Centralizers | [
"Centers of Groups",
"Centralizers"
] | [
"Definition:Center (Abstract Algebra)/Group",
"Definition:Group",
"Definition:Set Intersection",
"Definition:Centralizer/Group Element"
] | [
"Definition:Set Equality/Definition 2",
"Definition:Set Equality/Definition 2"
] |
proofwiki-2765 | Group Homomorphism Preserves Identity | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.
Let:
:$e_G$ be the identity of $G$
:$e_H$ be the identity of $H$.
Then:
:$\map \phi {e_G} = e_H$ | {{begin-eqn}}
{{eqn | l = \map \phi {e_G}
| r = \map \phi {e_G \circ e_G}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \map \phi {e_G} * \map \phi {e_G}
| c = {{Defof|Morphism Property}}
}}
{{end-eqn}}
That is:
{{begin-eqn}}
{{eqn | l = \map \phi {e_G} * e_H
| r = \map \phi {e_G} * \map \phi ... | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $G$
:$e_H$ be the [[Definition:Identity Element|identity]] of $H$.
... | {{begin-eqn}}
{{eqn | l = \map \phi {e_G}
| r = \map \phi {e_G \circ e_G}
| c = {{Defof|Identity Element}}
}}
{{eqn | r = \map \phi {e_G} * \map \phi {e_G}
| c = {{Defof|Morphism Property}}
}}
{{end-eqn}}
That is:
{{begin-eqn}}
{{eqn | l = \map \phi {e_G} * e_H
| r = \map \phi {e_G} * \map \p... | Group Homomorphism Preserves Identity/Proof 1 | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity/Proof_1 | [
"Group Homomorphisms",
"Group Homomorphism Preserves Identity"
] | [
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Cancellation Laws"
] |
proofwiki-2766 | Group Homomorphism Preserves Identity | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.
Let:
:$e_G$ be the identity of $G$
:$e_H$ be the identity of $H$.
Then:
:$\map \phi {e_G} = e_H$ | A direct application of Homomorphism to Group Preserves Identity.
{{qed}} | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $G$
:$e_H$ be the [[Definition:Identity Element|identity]] of $H$.
... | A direct application of [[Homomorphism to Group Preserves Identity]].
{{qed}} | Group Homomorphism Preserves Identity/Proof 2 | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity/Proof_2 | [
"Group Homomorphisms",
"Group Homomorphism Preserves Identity"
] | [
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Homomorphism to Group Preserves Identity"
] |
proofwiki-2767 | Group Homomorphism Preserves Identity | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a group homomorphism.
Let:
:$e_G$ be the identity of $G$
:$e_H$ be the identity of $H$.
Then:
:$\map \phi {e_G} = e_H$ | From Group Homomorphism of Product with Inverse, we have:
:$\forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$
Putting $x = y$ we have:
{{begin-eqn}}
{{eqn | l = \map \phi {e_G}
| r = \map \phi {x \circ x^{-1} }
| c =
}}
{{eqn | r = \map \phi x * \paren {\map \phi x}... | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G, \circ} \to \struct {H, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $G$
:$e_H$ be the [[Definition:Identity Element|identity]] of $H$.
... | From [[Group Homomorphism of Product with Inverse]], we have:
:$\forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$
Putting $x = y$ we have:
{{begin-eqn}}
{{eqn | l = \map \phi {e_G}
| r = \map \phi {x \circ x^{-1} }
| c =
}}
{{eqn | r = \map \phi x * \paren {\map \p... | Group Homomorphism Preserves Identity/Proof 3 | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Identity/Proof_3 | [
"Group Homomorphisms",
"Group Homomorphism Preserves Identity"
] | [
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Group Homomorphism of Product with Inverse"
] |
proofwiki-2768 | Group Homomorphism Preserves Inverses | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a group homomorphism.
Let:
:$e_G$ be the identity of $G$
:$e_H$ be the identity of $H$
Then:
:$\forall x \in G: \map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$ | Let $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi x * \map \phi {x^{-1} }
| r = \map \phi {x \circ x^{-1} }
| c = {{Defof|Group Homomorphism}}
}}
{{eqn | r = \map \phi {e_G}
| c = {{Defof|Inverse Element}}
}}
{{eqn | r = e_H
| c = Group Homomorphism Preserves Identity
}}
{{end-eqn}}
So, b... | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $G$
:$e_H$ be the [[Definition:Identity Element|identity]] of $H$
T... | Let $x \in G$.
Then:
{{begin-eqn}}
{{eqn | l = \map \phi x * \map \phi {x^{-1} }
| r = \map \phi {x \circ x^{-1} }
| c = {{Defof|Group Homomorphism}}
}}
{{eqn | r = \map \phi {e_G}
| c = {{Defof|Inverse Element}}
}}
{{eqn | r = e_H
| c = [[Group Homomorphism Preserves Identity]]
}}
{{end-eqn}}... | Group Homomorphism Preserves Inverses/Proof 1 | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Inverses | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Inverses/Proof_1 | [
"Group Homomorphisms",
"Group Homomorphism Preserves Inverses"
] | [
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Group Homomorphism Preserves Identity",
"Definition:Inverse (Abstract Algebra)/Right Inverse",
"Group Homomorphism Preserves Identity",
"Definition:Inverse (Abstract Algebra)/Left Inverse",
"Definition:Inverse (Abstract Algebra)/Left Inverse",
"Definition:Inverse (Abstract Algebra)/Right Inverse",
"Def... |
proofwiki-2769 | Group Homomorphism Preserves Inverses | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a group homomorphism.
Let:
:$e_G$ be the identity of $G$
:$e_H$ be the identity of $H$
Then:
:$\forall x \in G: \map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$ | A direct application of Homomorphism to Group Preserves Inverses.
{{qed}} | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $G$
:$e_H$ be the [[Definition:Identity Element|identity]] of $H$
T... | A direct application of [[Homomorphism to Group Preserves Inverses]].
{{qed}} | Group Homomorphism Preserves Inverses/Proof 2 | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Inverses | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Inverses/Proof_2 | [
"Group Homomorphisms",
"Group Homomorphism Preserves Inverses"
] | [
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Homomorphism to Group Preserves Inverses"
] |
proofwiki-2770 | Group Homomorphism Preserves Inverses | Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a group homomorphism.
Let:
:$e_G$ be the identity of $G$
:$e_H$ be the identity of $H$
Then:
:$\forall x \in G: \map \phi {x^{-1} } = \paren {\map \phi x}^{-1}$ | From Group Homomorphism of Product with Inverse, we have:
:$\forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$
Putting $x = e_G$ and $y = x$ we have:
{{begin-eqn}}
{{eqn | l = \map \phi {x^{-1} }
| r = \map \phi {e_G \circ x^{-1} }
| c =
}}
{{eqn | r = \map \phi {e_G... | Let $\struct {G, \circ}$ and $\struct {H, *}$ be [[Definition:Group|groups]].
Let $\phi: \struct {G, \circ} \to\struct {H, *}$ be a [[Definition:Group Homomorphism|group homomorphism]].
Let:
:$e_G$ be the [[Definition:Identity Element|identity]] of $G$
:$e_H$ be the [[Definition:Identity Element|identity]] of $H$
T... | From [[Group Homomorphism of Product with Inverse]], we have:
:$\forall x, y \in G: \map \phi {x \circ y^{-1} } = \map \phi x * \paren {\map \phi y}^{-1}$
Putting $x = e_G$ and $y = x$ we have:
{{begin-eqn}}
{{eqn | l = \map \phi {x^{-1} }
| r = \map \phi {e_G \circ x^{-1} }
| c =
}}
{{eqn | r = \map \phi... | Group Homomorphism Preserves Inverses/Proof 3 | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Inverses | https://proofwiki.org/wiki/Group_Homomorphism_Preserves_Inverses/Proof_3 | [
"Group Homomorphisms",
"Group Homomorphism Preserves Inverses"
] | [
"Definition:Group",
"Definition:Group Homomorphism",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity"
] | [
"Group Homomorphism of Product with Inverse",
"Group Homomorphism Preserves Identity"
] |
proofwiki-2771 | Subset Product with Normal Subgroup is Subgroup | Let $G$ be a group whose identity is $e$.
Let:
:$(1): \quad H$ be a subgroup of $G$
:$(2): \quad N$ be a normal subgroup of $G$.
Let $H N$ denote subset product.
Then $H N$ and $N H$ are both subgroups of $G$. | It is clear that $e \in N H$, so $N H \ne \O$.
Suppose $n_1, n_2 \in N$ and $h_1, h_2 \in H$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {n_1 h_1} \paren {n_2 h_2}
| r = n_1 \paren {h_1 n_2 h_1^{-1} h_1} h_2
}}
{{eqn | r = n_1 \paren {h_1 n_2 h_1^{-1} } \paren {h_1 h_2}
}}
{{end-eqn}}
Since $N$ is normal in $G$:
:$\e... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let:
:$(1): \quad H$ be a [[Definition:Subgroup|subgroup]] of $G$
:$(2): \quad N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $H N$ denote [[Definition:Subset Product|subset product]].
Then $H N$ and $... | It is clear that $e \in N H$, so $N H \ne \O$.
Suppose $n_1, n_2 \in N$ and $h_1, h_2 \in H$.
Then:
{{begin-eqn}}
{{eqn | l = \paren {n_1 h_1} \paren {n_2 h_2}
| r = n_1 \paren {h_1 n_2 h_1^{-1} h_1} h_2
}}
{{eqn | r = n_1 \paren {h_1 n_2 h_1^{-1} } \paren {h_1 h_2}
}}
{{end-eqn}}
Since $N$ is [[Definition:... | Subset Product with Normal Subgroup is Subgroup/Proof 1 | https://proofwiki.org/wiki/Subset_Product_with_Normal_Subgroup_is_Subgroup | https://proofwiki.org/wiki/Subset_Product_with_Normal_Subgroup_is_Subgroup/Proof_1 | [
"Subset Product with Normal Subgroup is Subgroup",
"Normal Subgroups",
"Subset Products"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Subset Product",
"Definition:Subgroup"
] | [
"Definition:Normal Subgroup",
"Two-Step Subgroup Test",
"Definition:Subgroup",
"Subset Product of Subgroups"
] |
proofwiki-2772 | Subset Product with Normal Subgroup is Subgroup | Let $G$ be a group whose identity is $e$.
Let:
:$(1): \quad H$ be a subgroup of $G$
:$(2): \quad N$ be a normal subgroup of $G$.
Let $H N$ denote subset product.
Then $H N$ and $N H$ are both subgroups of $G$. | Consider the subset product $N H$.
We have that:
{{begin-eqn}}
{{eqn | l = N H
| r = \set {n h : n \in N , h \in H}
| c = {{Defof|Subset Product}}
}}
{{eqn | r = \bigcup_{h \mathop \in H} \set {n h : n \in N}
| c = {{Defof|Union of Family}}
}}
{{eqn | r = \bigcup_{h \mathop \in H} N h
| c = {{De... | Let $G$ be a [[Definition:Group|group]] whose [[Definition:Identity Element|identity]] is $e$.
Let:
:$(1): \quad H$ be a [[Definition:Subgroup|subgroup]] of $G$
:$(2): \quad N$ be a [[Definition:Normal Subgroup|normal subgroup]] of $G$.
Let $H N$ denote [[Definition:Subset Product|subset product]].
Then $H N$ and $... | Consider the [[Definition:Subset Product|subset product]] $N H$.
We have that:
{{begin-eqn}}
{{eqn | l = N H
| r = \set {n h : n \in N , h \in H}
| c = {{Defof|Subset Product}}
}}
{{eqn | r = \bigcup_{h \mathop \in H} \set {n h : n \in N}
| c = {{Defof|Union of Family}}
}}
{{eqn | r = \bigcup_{h \ma... | Subset Product with Normal Subgroup is Subgroup/Proof 2 | https://proofwiki.org/wiki/Subset_Product_with_Normal_Subgroup_is_Subgroup | https://proofwiki.org/wiki/Subset_Product_with_Normal_Subgroup_is_Subgroup/Proof_2 | [
"Subset Product with Normal Subgroup is Subgroup",
"Normal Subgroups",
"Subset Products"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup",
"Definition:Normal Subgroup",
"Definition:Subset Product",
"Definition:Subgroup"
] | [
"Definition:Subset Product",
"Definition:Normal Subgroup",
"Definition:Permutable Subgroups",
"Subset Product of Subgroups"
] |
proofwiki-2773 | Ring Homomorphism Preserves Zero | Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism.
Let:
:$0_{R_1}$ be the zero of $R_1$
:$0_{R_2}$ be the zero of $R_2$.
Then:
:$\map \phi {0_{R_1} } = 0_{R_2}$ | By definition, if $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ are rings then $\struct {R_1, +_1}$ and $\struct {R_2, +_2}$ are groups.
Again by definition:
:the zero of $\struct {R_1, +_1, \circ_1}$ is the identity of $\struct {R_1, +_1}$
:the zero of $\struct {R_2, +_2, \circ_2}$ is the identity of... | Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let:
:$0_{R_1}$ be the [[Definition:Ring Zero|zero]] of $R_1$
:$0_{R_2}$ be the [[Definition:Ring Zero|zero]] of $R_2$.
Then:
:$\map \phi {0_{R_1} } = 0_{R_2}$ | By definition, if $\struct {R_1, +_1, \circ_1}$ and $\struct {R_2, +_2, \circ_2}$ are [[Definition:Ring (Abstract Algebra)|rings]] then $\struct {R_1, +_1}$ and $\struct {R_2, +_2}$ are [[Definition:Group|groups]].
Again by definition:
:the [[Definition:Ring Zero|zero]] of $\struct {R_1, +_1, \circ_1}$ is the [[Defini... | Ring Homomorphism Preserves Zero | https://proofwiki.org/wiki/Ring_Homomorphism_Preserves_Zero | https://proofwiki.org/wiki/Ring_Homomorphism_Preserves_Zero | [
"Ring Homomorphisms"
] | [
"Definition:Ring Homomorphism",
"Definition:Ring Zero",
"Definition:Ring Zero"
] | [
"Definition:Ring (Abstract Algebra)",
"Definition:Group",
"Definition:Ring Zero",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Ring Zero",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Group Homomorphism Preserves Identity"
] |
proofwiki-2774 | Image of Set Difference under Mapping | Let $f: S \to T$ be a mapping.
The set difference of the images of two subsets of $S$ is a subset of the image of the set difference.
That is:
Let $S_1$ and $S_2$ be subsets of $S$.
Then:
:$f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$
where $\setminus$ denotes set difference. | As $f$, being a mapping, is also a relation, we can apply Image of Set Difference under Relation:
:$\RR \sqbrk {S_1} \setminus \RR \sqbrk {S_2} \subseteq \RR \sqbrk {S_1 \setminus S_2}$
{{qed}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
The [[Definition:Set Difference|set difference]] of the [[Definition:Image of Subset under Mapping|images]] of two [[Definition:Subset|subsets]] of $S$ is a [[Definition:Subset|subset]] of the [[Definition:Image of Subset under Mapping|image]] of the [[Definition:S... | As $f$, being a [[Definition:Mapping|mapping]], is also a [[Definition:Relation|relation]], we can apply [[Image of Set Difference under Relation]]:
:$\RR \sqbrk {S_1} \setminus \RR \sqbrk {S_2} \subseteq \RR \sqbrk {S_1 \setminus S_2}$
{{qed}} | Image of Set Difference under Mapping/Proof 1 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Proof_1 | [
"Image of Set Difference under Mapping",
"Images",
"Set Difference"
] | [
"Definition:Mapping",
"Definition:Set Difference",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Set Difference",
"Definition:Subset",
"Definition:Set Difference"
] | [
"Definition:Mapping",
"Definition:Relation",
"Image of Set Difference under Relation"
] |
proofwiki-2775 | Image of Set Difference under Mapping | Let $f: S \to T$ be a mapping.
The set difference of the images of two subsets of $S$ is a subset of the image of the set difference.
That is:
Let $S_1$ and $S_2$ be subsets of $S$.
Then:
:$f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$
where $\setminus$ denotes set difference. | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = f \sqbrk {S_1} \setminus f \sqbrk {S_2}
| c =
}}
{{eqn | ll= \leadsto
| q = \exists x \in {S_1}: x \notin {S_2}
| l = \tuple {x, y}
| o = \in
| r = f
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | ll= \leadsto
|... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
The [[Definition:Set Difference|set difference]] of the [[Definition:Image of Subset under Mapping|images]] of two [[Definition:Subset|subsets]] of $S$ is a [[Definition:Subset|subset]] of the [[Definition:Image of Subset under Mapping|image]] of the [[Definition:S... | {{begin-eqn}}
{{eqn | l = y
| o = \in
| r = f \sqbrk {S_1} \setminus f \sqbrk {S_2}
| c =
}}
{{eqn | ll= \leadsto
| q = \exists x \in {S_1}: x \notin {S_2}
| l = \tuple {x, y}
| o = \in
| r = f
| c = {{Defof|Image of Subset under Mapping}}
}}
{{eqn | ll= \leadsto
|... | Image of Set Difference under Mapping/Proof 2 | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping | https://proofwiki.org/wiki/Image_of_Set_Difference_under_Mapping/Proof_2 | [
"Image of Set Difference under Mapping",
"Images",
"Set Difference"
] | [
"Definition:Mapping",
"Definition:Set Difference",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Set Difference",
"Definition:Subset",
"Definition:Set Difference"
] | [] |
proofwiki-2776 | Convergence by Multiple of Error Term | Let $\sequence {s_n}$ be a real sequence.
Suppose that $\exists \epsilon \in \R, \epsilon > 0$ such that:
:$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \epsilon$
for any $K \in \R, K > 0$, independent of both $\epsilon$ and $N$.
Then $\sequence {s_n}$ converges to $l$. | Let $\epsilon > 0$.
Then $\dfrac \epsilon K > 0$.
If the condition holds as stated, then:
:$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \paren {\dfrac \epsilon K}$
Hence the result by definition of a convergent sequence.
{{qed}} | Let $\sequence {s_n}$ be a [[Definition:Real Sequence|real sequence]].
Suppose that $\exists \epsilon \in \R, \epsilon > 0$ such that:
:$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \epsilon$
for any $K \in \R, K > 0$, independent of both $\epsilon$ and $N$.
Then $\sequence {s_n}$ [[Definition:Convergent S... | Let $\epsilon > 0$.
Then $\dfrac \epsilon K > 0$.
If the condition holds as stated, then:
:$\exists N \in \N: \forall n \ge N: \size {s_n - l} < K \paren {\dfrac \epsilon K}$
Hence the result by definition of a [[Definition:Convergent Sequence|convergent sequence]].
{{qed}} | Convergence by Multiple of Error Term | https://proofwiki.org/wiki/Convergence_by_Multiple_of_Error_Term | https://proofwiki.org/wiki/Convergence_by_Multiple_of_Error_Term | [
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence"
] | [
"Definition:Convergent Sequence"
] |
proofwiki-2777 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | === Necessary Condition ===
{{:Cauchy's Convergence Criterion/Real Numbers/Necessary Condition}}{{qed|lemma}} | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | === [[Cauchy's Convergence Criterion/Real Numbers/Necessary Condition|Necessary Condition]] ===
{{:Cauchy's Convergence Criterion/Real Numbers/Necessary Condition}}{{qed|lemma}} | Cauchy's Convergence Criterion/Real Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Cauchy's Convergence Criterion/Real Numbers/Necessary Condition"
] |
proofwiki-2778 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | Let $\sequence {x_n}$ be convergent.
Let $\struct {\R, d}$ be the metric space formed from $\R$ and the usual (Euclidean) metric:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the absolute value of $x$.
This is proven to be a metric space in Real Number Line is Metric Space.
From Convergent Sequence in Me... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {x_n}$ be [[Definition:Convergent Sequence|convergent]].
Let $\struct {\R, d}$ be the [[Definition:Metric Space|metric space]] formed from $\R$ and the [[Definition:Euclidean Metric on Real Vector Space|usual (Euclidean) metric]]:
:$\map d {x_1, x_2} = \size {x_1 - x_2}$
where $\size x$ is the [[Definit... | Cauchy's Convergence Criterion/Real Numbers/Necessary Condition/Proof 1 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Necessary_Condition/Proof_1 | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Convergent Sequence",
"Definition:Metric Space",
"Definition:Euclidean Metric/Real Vector Space",
"Definition:Absolute Value",
"Definition:Metric Space",
"Real Number Line is Metric Space",
"Convergent Sequence is Cauchy Sequence/Metric Space",
"Definition:Convergent Sequence",
"Definiti... |
proofwiki-2779 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | Let $\sequence {x_n}$ be a sequence in $\R$ that converges to the limit $l \in \R$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ converges to $l$, we have:
:$\exists N: \forall n > N: \size {x_n - l} < \dfrac \epsilon 2$
So if $m > N$ and $n > N$, then:
{{begin-eqn}}
{{eqn | l = \si... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {x_n}$ be a [[Definition:Sequence|sequence]] in $\R$ that [[Definition:Convergent Real Sequence|converges]] to the [[Definition:Limit of Real Sequence|limit]] $l \in \R$.
Let $\epsilon > 0$.
Then also $\dfrac \epsilon 2 > 0$.
Because $\sequence {x_n}$ [[Definition:Convergent Real Sequence|converges]]... | Cauchy's Convergence Criterion/Real Numbers/Necessary Condition/Proof 2 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Necessary_Condition/Proof_2 | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Triangle Inequality",
"Definition:Cauchy Sequence/Real Numbers"
] |
proofwiki-2780 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
We have the result Real Number Line is Metric Space.
Hence by Convergent Subsequence of Cauchy Sequence, it is sufficient to show that $\sequence {a_n}$ has a convergent subsequence.
Since $\sequence {a_n}$ is Cauchy, by Real Cauchy Sequence is Bounded, it is also bou... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
We have the result [[Real Number Line is Metric Space]].
Hence by [[Convergent Subsequence of Cauchy Sequence/Metric Space|Convergent Subsequence of Cauchy Sequence]], it is sufficient to show that $\sequence {a_n}$ has a [[Defini... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_1 | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Real Number Line is Metric Space",
"Convergent Subsequence of Cauchy Sequence/Metric Space",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy Sequence is Bounded/Real Numbers",
"Defin... |
proofwiki-2781 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
By Real Cauchy Sequence is Bounded, $\sequence {a_n}$ is bounded.
By the Bolzano-Weierstrass Theorem, $\sequence {a_n}$ has a convergent subsequence $\sequence {a_{n_r} }$.
Let $a_{n_r} \to l$ as $r \to \infty$.
It is to be shown that $a_n \to l$ as $n \to \infty$.
Le... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
By [[Real Cauchy Sequence is Bounded]], $\sequence {a_n}$ is [[Definition:Bounded Real Sequence|bounded]].
By the [[Bolzano-Weierstrass Theorem]], $\sequence {a_n}$ has a [[Definition:Convergent Real Sequence|convergent]] [[Defini... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 2 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_2 | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy Sequence is Bounded/Real Numbers",
"Definition:Bounded Sequence/Real",
"Bolzano-Weierstrass Theorem",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Strictly Positive/Real Number",
"Definition:Cauchy Sequence/Real... |
proofwiki-2782 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | The aim is to define two sequences whose elements are respectively upper and lower bounds to subsequences of the sequence $\sequence {a_n}$.
It is then shown that these two sequences converge to the same limit.
This is used to prove that $\sequence {a_n}$ converges.
A sequence $\sequence {\epsilon_i}$ is introduced tha... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | The aim is to define two [[Definition:Real Sequence|sequences]] whose [[Definition:Sequence|elements]] are respectively [[Definition:Upper Bound of Real Sequence|upper]] and [[Definition:Lower Bound of Real Sequence|lower bounds]] to [[Definition:Subsequence|subsequences]] of the [[Definition:Real Sequence|sequence]] $... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 3 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_3 | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Real Sequence",
"Definition:Sequence",
"Definition:Upper Bound of Sequence/Real",
"Definition:Lower Bound of Sequence/Real",
"Definition:Subsequence",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real... |
proofwiki-2783 | Cauchy's Convergence Criterion/Real Numbers | Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is a Cauchy sequence {{iff}} $\sequence {x_n}$ is convergent. | Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.
By Real Cauchy Sequence is Bounded, $\sequence {a_n}$ is bounded.
Thus $\sequence {a_n}$ is both bounded above and bounded below.
Let us create a monotone subsequence $\sequence {b_n}$ of $\sequence {a_n}$ using the following construction:
For each $m \in \N$, let $S_... | Let $\sequence {x_n}$ be a [[Definition:Real Sequence|sequence in $\R$]].
Then $\sequence {x_n}$ is a [[Definition:Real Cauchy Sequence|Cauchy sequence]] {{iff}} $\sequence {x_n}$ is [[Definition:Convergent Real Sequence|convergent]]. | Let $\sequence {a_n}$ be a [[Definition:Real Cauchy Sequence|Cauchy sequence in $\R$]].
By [[Real Cauchy Sequence is Bounded]], $\sequence {a_n}$ is [[Definition:Bounded Real Sequence|bounded]].
Thus $\sequence {a_n}$ is both [[Definition:Bounded Above Real Sequence|bounded above]] and [[Definition:Bounded Below Real... | Cauchy's Convergence Criterion/Real Numbers/Sufficient Condition/Proof 4 | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers | https://proofwiki.org/wiki/Cauchy's_Convergence_Criterion/Real_Numbers/Sufficient_Condition/Proof_4 | [
"Real Analysis",
"Cauchy's Convergence Criterion"
] | [
"Definition:Real Sequence",
"Definition:Cauchy Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Cauchy Sequence/Real Numbers",
"Cauchy Sequence is Bounded/Real Numbers",
"Definition:Bounded Sequence/Real",
"Definition:Bounded Above Sequence/Real",
"Definition:Bounded Below Sequence/Real",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Subsequence",
"Definiti... |
proofwiki-2784 | Wedderburn's Theorem | Every finite division ring $D$ is a field. | Let $D$ be a finite division ring.
If $D$ is shown commutative then, by definition, $D$ is a field.
Let $\map Z D$ be the center of $D$, that is:
:$\map Z D := \set {z \in D: \forall d \in D: z d = d z}$
From Center of Division Ring is Subfield it follows that $\map Z D$ is a Galois field.
Thus from Characteristic of G... | Every [[Definition:Finite Set|finite]] [[Definition:Division Ring|division ring]] $D$ is a [[Definition:Field (Abstract Algebra)|field]]. | Let $D$ be a [[Definition:Finite Ring|finite]] [[Definition:Division Ring|division ring]].
If $D$ is shown [[Definition:Commutative Division Ring|commutative]] then, by definition, $D$ is a [[Definition:Field (Abstract Algebra)|field]].
Let $\map Z D$ be the [[Definition:Center of Ring|center]] of $D$, that is:
:$\m... | Wedderburn's Theorem | https://proofwiki.org/wiki/Wedderburn's_Theorem | https://proofwiki.org/wiki/Wedderburn's_Theorem | [
"Wedderburn's Theorem",
"Division Rings",
"Galois Fields"
] | [
"Definition:Finite Set",
"Definition:Division Ring",
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Finite Ring",
"Definition:Division Ring",
"Definition:Commutative Division Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Center (Abstract Algebra)/Ring",
"Center of Division Ring is Subfield",
"Definition:Galois Field",
"Characteristic of Galois Field is Prime",
"Definition:C... |
proofwiki-2785 | Frobenius's Theorem | An algebraic associative real division algebra $A$ is isomorphic to $\R, \C$ or $\Bbb H$. | Recall that an algebra $A$ is said to be quadratic if it is unital and the set $\set {1, x, x^2}$ is linearly dependent for every $x \in A$.
{{explain|This needs to be reviewed, as Definition:Quadratic Algebra as currently posted up does not match the above.}}
Thus, for every $x \in A$ there exist $\map t x, \map n x \... | An [[Definition:Algebraic Algebra|algebraic]] [[Definition:Associative Algebra|associative]] [[Definition:Real Algebra|real]] [[Definition:Division Algebra|division algebra]] $A$ is [[Definition:Algebra Isomorphism|isomorphic]] to $\R, \C$ or $\Bbb H$. | Recall that an [[Definition:Algebra over Ring|algebra]] $A$ is said to be [[Definition:Quadratic Algebra|quadratic]] if it is [[Definition:Unital Algebra|unital]] and the set $\set {1, x, x^2}$ is [[Definition:Linearly Dependent Set|linearly dependent]] for every $x \in A$.
{{explain|This needs to be reviewed, as [[De... | Frobenius's Theorem | https://proofwiki.org/wiki/Frobenius's_Theorem | https://proofwiki.org/wiki/Frobenius's_Theorem | [
"Frobenius's Theorem",
"Abstract Algebra"
] | [
"Definition:Algebraic Algebra",
"Definition:Associative Algebra",
"Definition:Real Algebra",
"Definition:Division Algebra",
"Definition:Algebra Isomorphism"
] | [
"Definition:Algebra over Ring",
"Definition:Quadratic Algebra",
"Definition:Unital Algebra",
"Definition:Linearly Dependent/Set",
"Definition:Quadratic Algebra",
"Definition:Mapping",
"Definition:Linear Functional",
"Definition:Trace",
"Definition:Complex Modulus",
"Definition:Quadratic Algebra"
] |
proofwiki-2786 | Preimages All Exist iff Surjection | Let $f: S \to T$ be a mapping.
Let $f^{-1}$ be the inverse of $f$.
Let $\map {f^{-1} } t$ be the preimage of $t \in T$.
Then $\map {f^{-1} } t$ is empty for no $t \in T$ {{iff}} $f$ is a surjection. | === Necessary Condition ===
We use a Proof by Contraposition.
To that end, suppose:
:$\exists t \in T: \map {f^{-1} } t = \O$
That is:
:$\neg \paren {\forall t \in T: \exists s \in S: \map f s = t}$
So, by definition, $f: S \to T$ is not a surjection.
From Rule of Transposition it follows that if $f$ is a surjection:
$... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^{-1}$ be the [[Definition:Inverse of Mapping|inverse]] of $f$.
Let $\map {f^{-1} } t$ be the [[Definition:Preimage of Element under Mapping|preimage]] of $t \in T$.
Then $\map {f^{-1} } t$ is [[Definition:Empty Set|empty]] for no $t \in T$ {{iff}} $f$ is ... | === Necessary Condition ===
We use a [[Proof by Contraposition]].
To that end, suppose:
:$\exists t \in T: \map {f^{-1} } t = \O$
That is:
:$\neg \paren {\forall t \in T: \exists s \in S: \map f s = t}$
So, by definition, $f: S \to T$ is not a [[Definition:Surjection|surjection]].
From [[Rule of Transposition]] i... | Preimages All Exist iff Surjection/Proof 1 | https://proofwiki.org/wiki/Preimages_All_Exist_iff_Surjection | https://proofwiki.org/wiki/Preimages_All_Exist_iff_Surjection/Proof_1 | [
"Surjections",
"Inverse Mappings",
"Preimages All Exist iff Surjection"
] | [
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Preimage/Mapping/Element",
"Definition:Empty Set",
"Definition:Surjection"
] | [
"Proof by Contraposition",
"Definition:Surjection",
"Rule of Transposition",
"Proof by Contraposition",
"Definition:Surjection",
"Rule of Transposition",
"Definition:Surjection"
] |
proofwiki-2787 | Preimages All Exist iff Surjection | Let $f: S \to T$ be a mapping.
Let $f^{-1}$ be the inverse of $f$.
Let $\map {f^{-1} } t$ be the preimage of $t \in T$.
Then $\map {f^{-1} } t$ is empty for no $t \in T$ {{iff}} $f$ is a surjection. | Suppose that there is no $t \in T$ such that $\map {f^{-1} } t$ is empty.
By Denial of Existence, this is equivalent to saying that for all $t \in T$, $\map {f^{-1} } t$ is not empty.
This is equivalent to the statement that $\map {f^{-1} } t$ contains at least one element for each $t \in T$.
In other words, for each $... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^{-1}$ be the [[Definition:Inverse of Mapping|inverse]] of $f$.
Let $\map {f^{-1} } t$ be the [[Definition:Preimage of Element under Mapping|preimage]] of $t \in T$.
Then $\map {f^{-1} } t$ is [[Definition:Empty Set|empty]] for no $t \in T$ {{iff}} $f$ is ... | Suppose that there is no $t \in T$ such that $\map {f^{-1} } t$ is empty.
By [[Denial of Existence]], this is equivalent to saying that for all $t \in T$, $\map {f^{-1} } t$ is not empty.
This is equivalent to the statement that $\map {f^{-1} } t$ contains at least one element for each $t \in T$.
In other words, for... | Preimages All Exist iff Surjection/Proof 2 | https://proofwiki.org/wiki/Preimages_All_Exist_iff_Surjection | https://proofwiki.org/wiki/Preimages_All_Exist_iff_Surjection/Proof_2 | [
"Surjections",
"Inverse Mappings",
"Preimages All Exist iff Surjection"
] | [
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Preimage/Mapping/Element",
"Definition:Empty Set",
"Definition:Surjection"
] | [
"De Morgan's Laws (Predicate Logic)/Denial of Existence",
"Definition:Surjection"
] |
proofwiki-2788 | Non-Successor Element of Peano Structure is Unique | Let $\struct {P, s, 0}$ be a Peano structure.
Then:
:$P \setminus s \sqbrk P$ is a singleton
where:
:$\setminus$ denotes set difference
:$s \sqbrk P$ denotes the image of the mapping $s$.
It follows that the non-successor element $0$ is the only element of $P$ with this property. | {{handwaving|It is not explicit which formulation of Peano's axioms is used, while this is very relevant for this elementary result}}
Let $T = P \setminus s \sqbrk P$.
From Axiom $(\text P 4)$ we know that $T \ne \O$.
Now suppose that $t_1 \in T$ and $t_2 \in T$.
{{AimForCont}} $t_1 \ne t_2$.
Define $A = P \setminus \s... | Let $\struct {P, s, 0}$ be a [[Definition:Peano Structure|Peano structure]].
Then:
:$P \setminus s \sqbrk P$ is a [[Definition:Singleton|singleton]]
where:
:$\setminus$ denotes [[Definition:Set Difference|set difference]]
:$s \sqbrk P$ denotes the [[Definition:Image of Mapping|image]] of the [[Definition:Mapping|mapp... | {{handwaving|It is not explicit which formulation of Peano's axioms is used, while this is very relevant for this elementary result}}
Let $T = P \setminus s \sqbrk P$.
From [[Axiom:Peano's Axioms|Axiom $(\text P 4)$]] we know that $T \ne \O$.
Now suppose that $t_1 \in T$ and $t_2 \in T$.
{{AimForCont}} $t_1 \ne t_2$... | Non-Successor Element of Peano Structure is Unique | https://proofwiki.org/wiki/Non-Successor_Element_of_Peano_Structure_is_Unique | https://proofwiki.org/wiki/Non-Successor_Element_of_Peano_Structure_is_Unique | [
"Peano's Axioms"
] | [
"Definition:Peano Structure",
"Definition:Singleton",
"Definition:Set Difference",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Mapping",
"Definition:Non-Successor Element"
] | [
"Axiom:Peano's Axioms",
"Axiom:Peano's Axioms",
"Definition:Contradiction"
] |
proofwiki-2789 | Equivalence of Formulations of Peano's Axioms | Let $P$ be a set.
Let $s: P \to P$ be a mapping.
Let $0 \in P$ be a distinguished element.
{{TFAE|def = Peano's Axioms|view = Peano's Axioms}} | === Formulation 1 implies Formulation 2 ===
For $(\text P 3)$, there is nothing to prove.
Next, axiom $(\text P 4)$ of Formulation 2.
Recall the definition of the image of a mapping as applicable to $s$:
:$\Img s = \set {n \in P: \exists m \in P: \map s m = n}$
From this definition, it follows that $0 \notin \Img s$, a... | Let $P$ be a [[Definition:Set|set]].
Let $s: P \to P$ be a [[Definition:Mapping|mapping]].
Let $0 \in P$ be a distinguished element.
{{TFAE|def = Peano's Axioms|view = Peano's Axioms}} | === Formulation 1 implies Formulation 2 ===
For $(\text P 3)$, there is nothing to prove.
Next, axiom $(\text P 4)$ of [[Axiom:Peano's Axioms/Formulation 2|Formulation 2]].
Recall the definition of the [[Definition:Image of Mapping|image of a mapping]] as applicable to $s$:
:$\Img s = \set {n \in P: \exists m \in ... | Equivalence of Formulations of Peano's Axioms | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Peano's_Axioms | https://proofwiki.org/wiki/Equivalence_of_Formulations_of_Peano's_Axioms | [
"Peano's Axioms"
] | [
"Definition:Set",
"Definition:Mapping"
] | [
"Axiom:Peano's Axioms/Formulation 2",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Axiom:Peano's Axioms/Formulation 2",
"Non-Successor Element of Peano Structure is Unique",
"Definition:Element",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Axiom:Peano's Axioms/Formulation 1",
"Definition:El... |
proofwiki-2790 | Product of Row Sum Unity Matrices | Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix.
Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix.
Let the row sum of $\mathbf A$ and $\mathbf B$ be equal to $1$.
Then the row sum of their (conventional) product is also $1$. | We have that:
:$\ds \sum_{i \mathop = 1}^n a_{i j} = \sum_{i \mathop = 1}^n b_{i j} = 1$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n \paren {\mathbf A \mathbf B}_{i j}
| r = \sum_{i \mathop = 1}^n \paren {\sum_{k \mathop = 1}^n a_{i k} b_{k j} }
| c =
}}
{{eqn | r = \sum_{i, k \mathop = 1}^n a_{... | Let $\mathbf A = \sqbrk a_{m n}$ be an [[Definition:Matrix|$m \times n$ matrix]].
Let $\mathbf B = \sqbrk b_{n p}$ be an [[Definition:Matrix|$n \times p$ matrix]].
Let the [[Definition:Row Sum|row sum]] of $\mathbf A$ and $\mathbf B$ be equal to $1$.
Then the [[Definition:Row Sum|row sum]] of their [[Definition:Mat... | We have that:
:$\ds \sum_{i \mathop = 1}^n a_{i j} = \sum_{i \mathop = 1}^n b_{i j} = 1$
Then:
{{begin-eqn}}
{{eqn | l = \sum_{i \mathop = 1}^n \paren {\mathbf A \mathbf B}_{i j}
| r = \sum_{i \mathop = 1}^n \paren {\sum_{k \mathop = 1}^n a_{i k} b_{k j} }
| c =
}}
{{eqn | r = \sum_{i, k \mathop = 1}^n... | Product of Row Sum Unity Matrices | https://proofwiki.org/wiki/Product_of_Row_Sum_Unity_Matrices | https://proofwiki.org/wiki/Product_of_Row_Sum_Unity_Matrices | [
"Matrix Algebra"
] | [
"Definition:Matrix",
"Definition:Matrix",
"Definition:Row Sum",
"Definition:Row Sum",
"Definition:Matrix Product (Conventional)"
] | [
"Category:Matrix Algebra"
] |
proofwiki-2791 | Image of Successor Mapping forms Peano Structure | Let $\struct {P, s, 0}$ be a Peano structure.
Let $P'$ be the set $s \sqbrk P$, that is:
:$P' = \set {\map s n: n \in P}$
Let $s'$ be the restriction of $s$ to $P'$.
Then $\struct {P', s', \map s 0}$ is also a Peano structure. | {{Improve| Expressions generated by {{TL|PeanoAxiom}} template (starting with Peano's Axiom) should not be confused with the intended ones.}}
We need to check that all of Peano's axioms hold for $\struct {P', s', \map s 0}$.
Now from Restriction of Injection is Injection, because $s$ is an injection then so is $s'$.
So... | Let $\struct {P, s, 0}$ be a [[Definition:Peano Structure|Peano structure]].
Let $P'$ be the [[Definition:Set|set]] $s \sqbrk P$, that is:
:$P' = \set {\map s n: n \in P}$
Let $s'$ be the [[Definition:Restriction of Mapping|restriction]] of $s$ to $P'$.
Then $\struct {P', s', \map s 0}$ is also a [[Definition:Pean... | {{Improve| Expressions generated by {{TL|PeanoAxiom}} template (starting with [[Axiom:Peano's Axioms|Peano's Axiom]]) should not be confused with the intended ones.}}
We need to check that all of [[Axiom:Peano's Axioms|Peano's axioms]] hold for $\struct {P', s', \map s 0}$.
Now from [[Restriction of Injection is Inj... | Image of Successor Mapping forms Peano Structure | https://proofwiki.org/wiki/Image_of_Successor_Mapping_forms_Peano_Structure | https://proofwiki.org/wiki/Image_of_Successor_Mapping_forms_Peano_Structure | [
"Peano's Axioms",
"Peano Structures"
] | [
"Definition:Peano Structure",
"Definition:Set",
"Definition:Restriction/Mapping",
"Definition:Peano Structure"
] | [
"Axiom:Peano's Axioms",
"Axiom:Peano's Axioms",
"Restriction of Injection is Injection",
"Definition:Injection",
"Definition:Restriction/Mapping",
"Definition:Injection",
"Definition:Peano Structure",
"Definition:Set Equality/Definition 2",
"Definition:Peano Structure",
"Category:Peano's Axioms",
... |
proofwiki-2792 | Restriction of Injection is Injection | Let $f: S \to T$ be an injection.
Let $X \subseteq S$ be a subset of $S$.
Let $f \sqbrk X$ denote the image of $X$ under $f$.
Let $Y \subseteq T$ be a subset of $T$ such that $f \sqbrk X \subseteq Y$.
The restriction $f \restriction_{X \times Y}$ of $f$ to $X \times Y$ is an injection from $X$ to $Y$. | First we show that $f \restriction_{X \times Y}$ is a mapping from $X$ to $Y$.
By Restriction of Mapping is Mapping, $f \restriction_{X \times T}$ is a mapping from $X$ to $T$.
If $x \in X$, then by the definition of image:
:$\map f x \in f \sqbrk X$
{{explain|More explanation required.}}
Since $f \sqbrk X \subseteq Y$... | Let $f: S \to T$ be an [[Definition:Injection|injection]].
Let $X \subseteq S$ be a [[Definition:Subset|subset]] of $S$.
Let $f \sqbrk X$ denote the [[Definition:Image of Subset under Mapping|image]] of $X$ under $f$.
Let $Y \subseteq T$ be a [[Definition:Subset|subset]] of $T$ such that $f \sqbrk X \subseteq Y$.
... | First we show that $f \restriction_{X \times Y}$ is a [[Definition:Mapping|mapping]] from $X$ to $Y$.
By [[Restriction of Mapping is Mapping]], $f \restriction_{X \times T}$ is a mapping from $X$ to $T$.
If $x \in X$, then by the definition of [[Definition:Image of Subset under Mapping|image]]:
:$\map f x \in f \sqbr... | Restriction of Injection is Injection | https://proofwiki.org/wiki/Restriction_of_Injection_is_Injection | https://proofwiki.org/wiki/Restriction_of_Injection_is_Injection | [
"Injections"
] | [
"Definition:Injection",
"Definition:Subset",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Subset",
"Definition:Restriction/Mapping",
"Definition:Injection"
] | [
"Definition:Mapping",
"Restriction of Mapping is Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Mapping",
"Definition:Injection",
"Definition:Injection",
"Definition:Injection",
"Category:Injections"
] |
proofwiki-2793 | Diagonal Relation is Many-to-One | The diagonal relation is many-to-one.
That is:
:$\forall x \in \Dom {\Delta_S}: \tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S \implies y_1 = y_2$
where $\Delta_S$ is the diagonal relation on a set $S$. | Let $S$ be a set and let $\Delta_S$ be the diagonal relation on $S$.
Let $\tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S$.
From the definition of the diagonal relation:
:$\tuple {x, y_1} = \tuple {x, x}$
:$\tuple {x, y_2} = \tuple {x, x}$
and so $y_1 = y_2$.
{{qed}} | The [[Definition:Diagonal Relation|diagonal relation]] is [[Definition:Many-to-One Relation|many-to-one]].
That is:
:$\forall x \in \Dom {\Delta_S}: \tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S \implies y_1 = y_2$
where $\Delta_S$ is the [[Definition:Diagonal Relation|diagonal relation]] on a [[Defi... | Let $S$ be a [[Definition:Set|set]] and let $\Delta_S$ be the [[Definition:Diagonal Relation|diagonal relation]] on $S$.
Let $\tuple {x, y_1} \in \Delta_S \land \tuple {x, y_2} \in \Delta_S$.
From the definition of the [[Definition:Diagonal Relation|diagonal relation]]:
:$\tuple {x, y_1} = \tuple {x, x}$
:$\tuple {x... | Diagonal Relation is Many-to-One | https://proofwiki.org/wiki/Diagonal_Relation_is_Many-to-One | https://proofwiki.org/wiki/Diagonal_Relation_is_Many-to-One | [
"Diagonal Relation",
"Many-to-One Relations"
] | [
"Definition:Diagonal Relation",
"Definition:Many-to-One Relation",
"Definition:Diagonal Relation",
"Definition:Set"
] | [
"Definition:Set",
"Definition:Diagonal Relation",
"Definition:Diagonal Relation"
] |
proofwiki-2794 | Absolute Value is Many-to-One | Let $f: \R \to \R$ be the absolute value function:
:$\forall x \in \R: \map f x = \begin{cases}
x & : x \ge 0 \\
-x & : x < 0
\end{cases}$
Then $f$ is a many-to-one relation. | {{AimForCont}} $f$ is not a many-to-one relation.
Then there exists $y_1 \in \R$ and $y_2 \in \R$ where $y_1 \ne y_2$ such that:
:$\exists x \in \R: \map f x = y_1, \map f x = y_2$
There are two possibilities:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \ge
| r = 0
| c =
}}
{{eqn | n = 2
| ... | Let $f: \R \to \R$ be the [[Definition:Absolute Value|absolute value function]]:
:$\forall x \in \R: \map f x = \begin{cases}
x & : x \ge 0 \\
-x & : x < 0
\end{cases}$
Then $f$ is a [[Definition:Many-to-One Relation|many-to-one relation]]. | {{AimForCont}} $f$ is not a [[Definition:Many-to-One Relation|many-to-one relation]].
Then there exists $y_1 \in \R$ and $y_2 \in \R$ where $y_1 \ne y_2$ such that:
:$\exists x \in \R: \map f x = y_1, \map f x = y_2$
There are two possibilities:
{{begin-eqn}}
{{eqn | n = 1
| l = x
| o = \ge
| r = 0... | Absolute Value is Many-to-One | https://proofwiki.org/wiki/Absolute_Value_is_Many-to-One | https://proofwiki.org/wiki/Absolute_Value_is_Many-to-One | [
"Absolute Value Function"
] | [
"Definition:Absolute Value",
"Definition:Many-to-One Relation"
] | [
"Definition:Many-to-One Relation",
"Proof by Cases",
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Many-to-One Relation"
] |
proofwiki-2795 | Subset of Preimage under Relation is Preimage of Subset | Let $\RR \subseteq S \times T$ be a relation.
Let $X \subseteq S, Y \subseteq T$.
Then:
:$X \subseteq \RR^{-1} \sqbrk Y \iff \RR \sqbrk X \subseteq Y$
In the language of direct image mappings, this can be written:
:$X \subseteq \map {\RR^\gets} Y \iff \map {\RR^\to} X \subseteq Y$ | As $\RR$ is a relation, then so is its inverse $\RR^{-1}$.
Let $\RR \sqbrk X \subseteq Y$.
Thus:
{{begin-eqn}}
{{eqn | l = \RR \sqbrk X
| o = \subseteq
| r = Y
| c =
}}
{{eqn | ll= \leadsto
| l = \RR^{-1} \sqbrk {\RR \sqbrk X}
| o = \subseteq
| r = \RR^{-1} \sqbrk Y
| c = {{Co... | Let $\RR \subseteq S \times T$ be a [[Definition:Relation|relation]].
Let $X \subseteq S, Y \subseteq T$.
Then:
:$X \subseteq \RR^{-1} \sqbrk Y \iff \RR \sqbrk X \subseteq Y$
In the language of [[Definition:Direct Image Mapping of Relation|direct image mappings]], this can be written:
:$X \subseteq \map {\RR^\gets... | As $\RR$ is a [[Definition:Relation|relation]], then so is its [[Definition:Inverse Relation|inverse]] $\RR^{-1}$.
Let $\RR \sqbrk X \subseteq Y$.
Thus:
{{begin-eqn}}
{{eqn | l = \RR \sqbrk X
| o = \subseteq
| r = Y
| c =
}}
{{eqn | ll= \leadsto
| l = \RR^{-1} \sqbrk {\RR \sqbrk X}
|... | Subset of Preimage under Relation is Preimage of Subset | https://proofwiki.org/wiki/Subset_of_Preimage_under_Relation_is_Preimage_of_Subset | https://proofwiki.org/wiki/Subset_of_Preimage_under_Relation_is_Preimage_of_Subset | [
"Subsets",
"Preimages under Relations"
] | [
"Definition:Relation",
"Definition:Direct Image Mapping/Relation"
] | [
"Definition:Relation",
"Definition:Inverse Relation",
"Image of Preimage under Relation is Subset",
"Subset Relation is Transitive",
"Image of Subset under Relation is Subset of Image",
"Image of Preimage under Relation is Subset",
"Subset Relation is Transitive",
"Category:Subsets",
"Category:Preim... |
proofwiki-2796 | Composition of Direct Image Mappings of Relations | Let $A, B, C$ be non-empty sets.
Let $\RR_1 \subseteq A \times B, \RR_2 \subseteq B \times C$ be relations.
Let:
:${\RR_1}^\to: \powerset A \to \powerset B$
and
:${\RR_2}^\to: \powerset B \to \powerset C$
be the direct image mappings of $\RR_1$ and $\RR_2$.
Then:
:$\paren {\RR_2 \circ \RR_1}^\to = {\RR_2}^\to \circ {\R... | Let $S \subseteq A, S \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren { {\RR_2}^\to \circ {\RR_1}^\to} } S
| r = \map { {\RR_2}^\to} {\map { {\RR_1}^\to } S}
| c =
}}
{{eqn | r = \set {\map {\RR_2} x: x \in \map { {\RR_1}^\to } S}
| c =
}}
{{eqn | r = \set {\map {\RR_2} x: \exists y \in S: \tu... | Let $A, B, C$ be [[Definition:Non-Empty Set|non-empty]] [[Definition:Set|sets]].
Let $\RR_1 \subseteq A \times B, \RR_2 \subseteq B \times C$ be [[Definition:Relation|relations]].
Let:
:${\RR_1}^\to: \powerset A \to \powerset B$
and
:${\RR_2}^\to: \powerset B \to \powerset C$
be the [[Definition:Direct Image Mapping... | Let $S \subseteq A, S \ne \O$.
Then:
{{begin-eqn}}
{{eqn | l = \map {\paren { {\RR_2}^\to \circ {\RR_1}^\to} } S
| r = \map { {\RR_2}^\to} {\map { {\RR_1}^\to } S}
| c =
}}
{{eqn | r = \set {\map {\RR_2} x: x \in \map { {\RR_1}^\to } S}
| c =
}}
{{eqn | r = \set {\map {\RR_2} x: \exists y \in S: ... | Composition of Direct Image Mappings of Relations | https://proofwiki.org/wiki/Composition_of_Direct_Image_Mappings_of_Relations | https://proofwiki.org/wiki/Composition_of_Direct_Image_Mappings_of_Relations | [
"Composite Relations",
"Composite Mappings",
"Direct Image Mappings"
] | [
"Definition:Non-Empty Set",
"Definition:Set",
"Definition:Relation",
"Definition:Direct Image Mapping/Relation"
] | [
"Category:Composite Relations",
"Category:Composite Mappings",
"Category:Direct Image Mappings"
] |
proofwiki-2797 | Subset equals Preimage of Image iff Mapping is Injection | Let $f: S \to T$ be a mapping.
Let $f^{-1}$ denote the inverse of $f$.
Then:
:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$ {{iff}} $f$ is an injection
where:
:$f \sqbrk A$ denotes the image of $A$ under $f$
:$f^{-1} \circ f$ denotes the composition of $f^{-1}$ and $f$.
This can be expressed in the lang... | === Sufficient Condition ===
Let $g$ be such that:
:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$
Then by Subset equals Preimage of Image implies Injection, $f$ is an injection.
{{qed|lemma}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^{-1}$ denote the [[Definition:Inverse of Mapping|inverse]] of $f$.
Then:
:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$ {{iff}} $f$ is an [[Definition:Injection|injection]]
where:
:$f \sqbrk A$ denotes the [[Definition:Image of Subset under... | === Sufficient Condition ===
Let $g$ be such that:
:$\forall A \subseteq S: A = \paren {f^{-1} \circ f} \sqbrk A$
Then by [[Subset equals Preimage of Image implies Injection]], $f$ is an [[Definition:Injection|injection]].
{{qed|lemma}} | Subset equals Preimage of Image iff Mapping is Injection | https://proofwiki.org/wiki/Subset_equals_Preimage_of_Image_iff_Mapping_is_Injection | https://proofwiki.org/wiki/Subset_equals_Preimage_of_Image_iff_Mapping_is_Injection | [
"Injections",
"Subsets",
"Composite Mappings",
"Preimages under Mappings"
] | [
"Definition:Mapping",
"Definition:Inverse of Mapping",
"Definition:Injection",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Composition of Mappings",
"Definition:Direct Image Mapping",
"Definition:Inverse Image Mapping",
"Definition:Injection"
] | [
"Subset equals Preimage of Image implies Injection",
"Definition:Injection",
"Definition:Injection"
] |
proofwiki-2798 | Subset equals Image of Preimage iff Mapping is Surjection | Let $f: S \to T$ be a mapping.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$.
Similarly, let $f^\gets: \powerset T \to \powerset S$ be the inverse image mapping of $f$.
Then:
:$\forall B \in \powerset T: B = \map {\paren {f^\to \circ f^\gets} } B$
{{iff}} $f$ is a surjection. | === Sufficient Condition ===
Let $f$ be such that:
:$\forall B \in \powerset T: B = \map {\paren {f^\to \circ f^\gets} } B$
From Subset equals Image of Preimage implies Surjection, $f$ is a surjection.
{{qed|lemma}} | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $f^\to: \powerset S \to \powerset T$ be the [[Definition:Direct Image Mapping of Mapping|direct image mapping]] of $f$.
Similarly, let $f^\gets: \powerset T \to \powerset S$ be the [[Definition:Inverse Image Mapping of Mapping|inverse image mapping]] of $f$.
... | === Sufficient Condition ===
Let $f$ be such that:
:$\forall B \in \powerset T: B = \map {\paren {f^\to \circ f^\gets} } B$
From [[Subset equals Image of Preimage implies Surjection]], $f$ is a [[Definition:Surjection|surjection]].
{{qed|lemma}} | Subset equals Image of Preimage iff Mapping is Surjection | https://proofwiki.org/wiki/Subset_equals_Image_of_Preimage_iff_Mapping_is_Surjection | https://proofwiki.org/wiki/Subset_equals_Image_of_Preimage_iff_Mapping_is_Surjection | [
"Surjections"
] | [
"Definition:Mapping",
"Definition:Direct Image Mapping/Mapping",
"Definition:Inverse Image Mapping/Mapping",
"Definition:Surjection"
] | [
"Subset equals Image of Preimage implies Surjection",
"Definition:Surjection",
"Definition:Surjection"
] |
proofwiki-2799 | Inverse of Injection is Many-to-One Relation | Let $f: S \to T$ be an injection.
Let $f^{-1}: T \to S$ be the inverse relation of $f$.
Then $f^{-1}$ is many-to-one. | Let $f: S \to T$ be an injection.
We have by definition of inverse relation that:
:$f^{-1} = \set {\tuple {t, s}: t = \map f s}$
Let $f: S \to T$ be an injection.
Let $\tuple {t, s_1} \in f^{-1}$ and $\tuple {t, s_2} \in f^{-1}$.
By definition, we have that $\map f {s_1} = t = \map f {s_2}$.
But as $f$ is an injection:... | Let $f: S \to T$ be an [[Definition:Injection|injection]].
Let $f^{-1}: T \to S$ be the [[Definition:Inverse of Mapping|inverse relation]] of $f$.
Then $f^{-1}$ is [[Definition:Many-to-One Relation|many-to-one]]. | Let $f: S \to T$ be an [[Definition:Injection|injection]].
We have by definition of [[Definition:Inverse of Mapping|inverse relation]] that:
:$f^{-1} = \set {\tuple {t, s}: t = \map f s}$
Let $f: S \to T$ be an [[Definition:Injection|injection]].
Let $\tuple {t, s_1} \in f^{-1}$ and $\tuple {t, s_2} \in f^{-1}$.
B... | Inverse of Injection is Many-to-One Relation | https://proofwiki.org/wiki/Inverse_of_Injection_is_Many-to-One_Relation | https://proofwiki.org/wiki/Inverse_of_Injection_is_Many-to-One_Relation | [
"Injections",
"Inverse Relations"
] | [
"Definition:Injection",
"Definition:Inverse of Mapping",
"Definition:Many-to-One Relation"
] | [
"Definition:Injection",
"Definition:Inverse of Mapping",
"Definition:Injection",
"Definition:Injection",
"Definition:Many-to-One Relation",
"Category:Injections",
"Category:Inverse Relations"
] |
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