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- 22:25, 6 May 2024 Knot (Knot Theory)/Examples/Right-Handed Trefoil (hist | edit) [216 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Knot == <onlyinclude> '''Right-handed trefoil''': :300px </onlyinclude> Category:Examples of Knots")
- 22:25, 6 May 2024 Knot (Knot Theory)/Examples/Left-Handed Trefoil (hist | edit) [213 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Knot == <onlyinclude> '''Left-handed trefoil''': :300px </onlyinclude> Category:Examples of Knots")
- 22:21, 6 May 2024 Knot (Knot Theory)/Examples/Trefoil Knot (hist | edit) [508 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Knot == <onlyinclude> These are the planar diagrams of the two types of trefoil knot: === Left-Handed Trefoil === {{:Definition:Left-Handed Trefoil Knot}} === Right-Handed Trefoil === {{:Definition:Right-Handed Trefoil Knot}}</onlyinclude> Category:Examples of Knots")
- 08:09, 6 May 2024 Knot (Knot Theory)/Examples (hist | edit) [229 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Knots == <onlyinclude> === Trefoil Knot === {{:Knot (Knot Theory)/Examples/Trefoil Knot}}</onlyinclude> Category:Examples of Knots")
- 08:02, 6 May 2024 Planar Diagram/Examples (hist | edit) [237 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Planar Diagrams == <onlyinclude> === Trefoil Knot === {{:Planar Diagram/Examples/Trefoil Knot}}</onlyinclude> Category:Examples of Planar Diagrams")
- 06:14, 6 May 2024 Outer Jordan Content of Right Triangle (hist | edit) [2,809 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == Let $T \subseteq \R^2$ be defined as: :$T = \set {\tuple {x, y} \in \R^2 : x \ge 0 \land y \ge 0 \land x + y \le 1}$ Then: :$\map {m^*} T = \dfrac 1 2$ == Proof == Let $\epsilon > 0$ be arbitrary. By the Axiom of Archimedes, let $n \in \N$ such that: :$n > 2 \epsilon$ Define $C \subseteq \powerset {\R^2}$ as: :$C = \set {\closedint {\dfrac p n} {\dfrac {p + 1} n} \times \closedint {\dfrac q n} {\dfrac {q + 1} n} : p, q \in \set {0, 1, \dotsc, n...")
- 10:11, 4 May 2024 Kinetic Energy of Body at Constant Angular Speed (hist | edit) [996 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $B$ be a body rotating at an angular speed $\omega$ about some axis of rotation $R$. Let $I$ denote the moment of inertia of $B$ about $R$. Then the kinetic energy $T$ of $B$ brought about by this rotation is given by: :$T = \dfrac {I \omega^2} 2$...")
- 10:06, 4 May 2024 Kinetic Energy of Body at Constant Speed (hist | edit) [925 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $B$ be a body of mass $m$ moving at a speed of $v$. Let $v$ be considerably less than the speed of light. Then the kinetic energy $T$ of $B$ is given by: :$T \approx \dfrac {m v^2} 2$ {{expand|Add a subpage giving the relativistic KE of $B$}} == Proof == {{ProofWanted}} == Sources == * {{BookReference|...")
- 00:20, 4 May 2024 Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces (hist | edit) [1,584 bytes] Hbghlyj (talk | contribs) (add a theorem needed in the proof of primitive element theorem)
- 22:56, 3 May 2024 Equivalence of Definitions of Separable Degree (hist | edit) [3,321 bytes] Hbghlyj (talk | contribs) (Add proof)
- 20:26, 3 May 2024 Steinitz's Theorem (hist | edit) [3,255 bytes] Hbghlyj (talk | contribs) (A theorem needed in the proof of Primitive Element Theorem. name based on Wikipedia page https://en.wikipedia.org/wiki/Steinitz%27s_theorem_(field_theory))
- 19:53, 3 May 2024 Separable Degree of Field Extensions is Multiplicative (hist | edit) [1,529 bytes] Hbghlyj (talk | contribs) (Creating the page based on a theorem in Lang's book, which in needed in Transitivity of Separable Field Extensions)
- 19:51, 3 May 2024 Separable Degree is At Most Equal To Degree (hist | edit) [704 bytes] Hbghlyj (talk | contribs) (Creating the page based on a theorem in Lang's book, which in needed in Transitivity of Separable Field Extensions)
- 19:01, 3 May 2024 Transitivity of Separable Field Extensions (hist | edit) [801 bytes] Hbghlyj (talk | contribs) (Create the page which is a non-existent link in Subextensions of Separable Field Extension are Separable)
- 16:14, 3 May 2024 Equivalence of Definitions of Purely Inseparable Extension (hist | edit) [4,800 bytes] Hbghlyj (talk | contribs) (Created page with "== Theorem == Let $E/F$ be an algebraic field extension. {{TFAE|def = Purely Inseparable Field Extension}} === Definition 1 === {{:Definition:Purely Inseparable Field Extension/Definition 1}} === Definition 2 === {{:Definition:Purely Inseparable Field Extension/Definition 2}} === Definition:Purely In...")
- 14:03, 3 May 2024 Pi Squared is Irrational/Proof 3 (hist | edit) [4,344 bytes] Robkahn131 (talk | contribs) (Cosine version of proof 1)
- 14:02, 3 May 2024 Pi Squared is Irrational/Proof 3/Lemma (hist | edit) [6,988 bytes] Robkahn131 (talk | contribs) (Created page with "== Pi Squared is Irrational: Lemma == <onlyinclude> Let $n \in \Z_{\ge 0}$ be a positive integer. Let it be supposed that $\pi^2$ is irrational, so that: :$\pi^2 = \dfrac p q$ where $p$ and $q$ are integers and $q \ne 0$. Let $A_n$ be defined as: :$\ds A_n = \frac \pi 2 \frac {p^n} {n!} \int_0^1 \paren {1 - x^2 }^n \map \cos {\dfrac {\pi x} 2} \rd x$ Then: :$A_n = \paren {16...")
- 10:12, 2 May 2024 Kepler's Conjecture/Also known as (hist | edit) [285 bytes] Prime.mover (talk | contribs) (Created page with "== Kepler's Conjecture: Also known as == <onlyinclude> A '''Kepler's Conjecture''' is also known as: :the '''Kepler conjecture''' :the '''Kepler's problem'''. </onlyinclude> == Sources == {{Mathworld|Kepler Conjecture|KeplerConjecture}} Category:Kepler's Conjecture")
- 10:08, 2 May 2024 Kepler's Conjecture/Historical Note/Mistake (hist | edit) [935 bytes] Prime.mover (talk | contribs) (Created page with "== Source Work == {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition}}: :'''Kepler's conjecture'' (J. Kepler, 1611) == Mistake == <onlyinclude> :''The conjecture was proved in $2006$ by {{AuthorRef|Thomas Callister Hales|T.C. Hales}}.'' </onlyinclude> == Correction == Technically incorrect. {{JournalLink|name = Annals of Mathematics}} accepted the proof presented by {{AuthorRef|Thomas Callister Hales}} in $2005$...")
- 06:17, 2 May 2024 Kappa Curve has Double Cusp at Origin (hist | edit) [960 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\KK$ be a kappa curve expressed in Cartesian coordinates as: :$x^4 + x^2 y^2 = a^2 y^2$ $\KK$ has a double cusp at the origin. This double cusp is a double cusp of the first kind. </onlyinclude> :520px == Proof == {{ProofWante...")
- 06:08, 2 May 2024 Kelvin-Stokes Theorem/Also known as (hist | edit) [1,211 bytes] Prime.mover (talk | contribs) (Created page with "== Kelvin-Stokes Theorem: Also known as == <onlyinclude> The '''Kelvin-Stokes Theorem''' is also known as the '''Classical Stokes' Theorem'''. It is also known as just '''Stokes's Theorem''', or '''Stokes' Theorem'''. </onlyinclude> == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Divergence Theorem|next = Kelvin-Stokes Theorem}}: $\S 22$: Integrals involving Vectors: $22.60$ * {{BookRefere...")
- 05:54, 2 May 2024 Asymptotes of Kappa Curve (hist | edit) [880 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\KK$ be a kappa curve expressed in Cartesian coordinates as: :$x^4 + x^2 y^2 = a^2 y^2$ $\KK$ has two asymptotes: :the line $x = a$ :the line $x = -a$. </onlyinclude> :520px == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Penguin Dictionary of Ma...")
- 05:51, 2 May 2024 Axes of Symmetry of Kappa Curve (hist | edit) [828 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\KK$ be a kappa curve expressed in Cartesian coordinates as: :$x^4 + x^2 y^2 = a^2 y^2$ $\KK$ has two axes of symmetry: :the $x$-axis :the $y$-axis. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage...")
- 12:03, 1 May 2024 Group of Units Ring of Integers Modulo p^2 is Cyclic (hist | edit) [3,561 bytes] Hbghlyj (talk | contribs) (Created page with "== Theorem == Let $p$ be a prime. Let $\struct {\Z / p^2 \Z, +, \times}$ be the ring of integers modulo $p^2$. Let $U = \struct {\paren {\Z / p^2 \Z}^\times, \times}$ denote the group of units of $\struct {\Z / p^2 \Z, +, \times}$. Then $U$ is cyclic. == Proof == The case $p = 2$ follows from Isomorp...")
- 11:00, 1 May 2024 Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Lemma (hist | edit) [952 bytes] Hbghlyj (talk | contribs) (Created page with "=== Lemma === There are only $5$ numbers $n$ with the property that $\map \phi n = 8$, and they are $15$, $16$, $20$, $24$ and $30$. === Proof of lemma === Let $p$ be a prime factor of $n$. By Euler Phi Function is Multiplicative: :$p - 1 = \map \phi p \divides \map \phi n = 8$ so $p \in \set{2, 3, 5}$. Let $i,j,k \in \Z^{\ge 0}$ such that $n = 2^i 3^j 5^k$. By Euler Phi Function is Multiplicative: :$8 = \map \phi n = \map \phi {...")
- 10:55, 1 May 2024 Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Proof 2 (hist | edit) [1,032 bytes] Prime.mover (talk | contribs) (Created page with "{{MissingLinks}} {{tidy}} == Theorem == {{:Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n}} == Proof == <onlyinclude> {{AimForCont}} $U$ and $C_8$ are isomorphic. :$\order U = \order {C_8} = 8$ From Order of Group of Units of Integers Modulo n we have that :$8 = \order U = \map \phi n$ where $\phi$ denotes the Euler $\phi$-function. $\map \phi 1 = \map \phi 2 = 1$, so $n > 2$. By Cyclicity...")
- 10:53, 1 May 2024 Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n/Proof 1 (hist | edit) [3,727 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Cyclic Group of Order 8 is not isomorphic to Group of Units of Integers Modulo n}} == Proof == {{AimForCont}} $U$ and $C_8$ are isomorphic. :$\order U = \order {C_8} = 8$ From Order of Group of Units of Integers Modulo n we have that :$8 = \order U = \map \phi n$ where $\phi$ denotes the Euler $\phi$-function. There are $5$ numbers $n$ with the property that $\map \phi n = 8$, and they are $15$, $16$, $20$, $2...")
- 10:49, 1 May 2024 Euler Phi Function of 5 (hist | edit) [385 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> :$\map \phi 5 = 4$ </onlyinclude> where $\phi$ denotes the Euler $\phi$ function. == Proof == From Euler Phi Function of Prime: :$\map \phi p = p - 1$ As $3$ is a prime number it follows that: :$\map \phi 5 = 5 - 1 = 4$ {{qed}} Category:Examples of Euler Phi Function Category:5")
- 10:26, 1 May 2024 Condition for Connectedness of Julia Set of z^2 + c (hist | edit) [1,005 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $J$ be the Julia set of the rational function on $\overline C$ defined as: :$\forall z \in \overline C: z \mapsto z^2 + c$ for some constant $c \in \overline C$. Then $J$ is connected in $\overline C$ {{iff}} $c$ is an element of the Mandelbrot set. </onlyinclude>...")
- 10:21, 1 May 2024 Julia Set for Square Function is Unit Circle (hist | edit) [774 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> The Julia set for the square function: :$\forall z \in \overline C: z \mapsto z^2$ is the unit circle. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Julia Set/Examples/Filled for -0.75/Mistake|next = Condition for Connectedness of Jul...")
- 09:20, 1 May 2024 Kakeya Problem/Historical Note (hist | edit) [526 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on the Kakeya Problem == <onlyinclude> The Kakeya Problem was raised by {{AuthorRef|Soichi Kakeya}} in $1917$. It remained unanswered until $1928$, at which time {{AuthorRef|Abram Samoilovitch Besicovitch}} solved it. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Kakeya Problem|next = Definition:Kappa Curve|entry = Kakeya's problem|subentry = S....")
- 09:02, 1 May 2024 Kakeya Problem/Also known as (hist | edit) [420 bytes] Prime.mover (talk | contribs) (Created page with "== Kakeya Problem: Also known as == <onlyinclude> The '''Kakeya Problem''' can also be seen rendered as '''Kakeya's Problem'''. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Jump Discontinuity|next = Kakeya Problem|entry = Kakeya's problem|subentry = S. Kakeya, 1917}} Category:Kakeya Problem")
- 09:01, 1 May 2024 Kakeya Problem (hist | edit) [1,316 bytes] Prime.mover (talk | contribs) (Created page with "== Problem == <onlyinclude> The Kakeya Problem is the question: :What is the smallest possible area of a set in the plane inside which a needle of length $1$ can be moved continuously in order to reverse its direction? </onlyinclude> == Solution == There is no such smallest area. That is, let $\epsilon \i...")
- 23:11, 30 April 2024 Julia Set/Examples/Filled for -0.75/Mistake (hist | edit) [1,539 bytes] Prime.mover (talk | contribs) (Created page with "== Source Work == {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition}} {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition}}: :'''Julia set''' == Mistake == <onlyinclude> :(b) Filled Julia set for $c = -0.75$ ::500px </onlyinclude> == Correction == The screenshot given is actually the Definitio...")
- 21:34, 30 April 2024 Julia Set/Examples/Filled for -0.75 (hist | edit) [944 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Julia Set == <onlyinclude> The below is a graphical representation of the '''filled Julia set''' for the point $-0.75$: :500px </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Julia Set/Examples/-0.13 + 0.75i|next = Julia Set/Examples/Filled for -0.7...")
- 21:32, 30 April 2024 Julia Set/Examples/-0.13 + 0.75i (hist | edit) [912 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Julia Set == <onlyinclude> The below is a graphical representation of the '''Julia set''' for the point $-0.13 + 0.75i$: :500px </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Condition for Connectedness of Julia Set|next = Julia Set/Examples/Filled for -0.75|en...")
- 21:00, 30 April 2024 Julia Set/Examples (hist | edit) [463 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Julia Sets == <onlyinclude> === Julia Set for $-0.13 + 0.75 i$ === {{:Julia Set/Examples/-0.13 + 0.75i}} === Filled Julia Set for $-0.75 i$ === {{:Julia Set/Examples/Filled for -0.75}}</onlyinclude> Category:Examples of Julia Sets")
- 19:52, 30 April 2024 Non-Trivial Jordan Matrix is not Diagonalizable (hist | edit) [789 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\mathbf J$ be a Jordan matrix of order $n$, where $n > 1$. Then $\mathbf J$ is not diagonalizable. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Jordan Matrix/Examples/Arbitrary Example 1|next = Definition:J...")
- 19:48, 30 April 2024 Jordan Matrix/Examples/Arbitrary Example 1 (hist | edit) [886 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Jordan Matrix == <onlyinclude> This is an arbitrary example of an order $ Jordan matrix: :$\quad \begin {pmatrix} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda \end {pmatrix}$ </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev...")
- 19:44, 30 April 2024 Jordan Matrix/Examples (hist | edit) [253 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Jordan Matrices == <onlyinclude> === Arbitrary Example === {{:Jordan Matrix/Examples/Arbitrary Example 1}}</onlyinclude> Category:Examples of Jordan Matrices")
- 17:28, 30 April 2024 Jordan-Hölder Theorem/Historical Note (hist | edit) [547 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Jordan-Hölder Theorem == <onlyinclude> The '''Jordan-Hölder Theorem''' was proved by {{AuthorRef|Camille Jordan}} in $1869$, and then independently rediscovered by {{AuthorRef|Otto Ludwig Hölder}} in $1889$. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Jordan-Hölder Theorem|next = Definition:Jordan Matrix|entry = Jordan-Hölder theorem|sub...")
- 19:03, 29 April 2024 Jack-Knife Resampling/Examples/Standard Errors (hist | edit) [862 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Jack-Knife Resampling == <onlyinclude> '''Jack-knifing''' can be used to obtain estimates of standard errors. While the '''jack-knife''' involves only $n$ samplings of the original data, each of a specified form, the bootstrapping method often uses $N$...")
- 18:46, 29 April 2024 Jack-Knife Resampling/Examples/Arbitrary Example 1 (hist | edit) [1,417 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Jack-Knife Resampling == <onlyinclude> Let us attempt to determine the bias in the correlation coefficient $r$ of a sample of $n$ pairs used as an estimator of the correlation coefficient $\rho$ of the population...")
- 16:57, 29 April 2024 Jack-Knife Resampling/Examples (hist | edit) [417 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Jack-Knife Resampling == <onlyinclude> === Arbitrary Example === {{:Jack-Knife Resampling/Examples/Arbitrary Example 1}}</onlyinclude> Category:Examples of Jack-Knife Resampling")
- 09:07, 29 April 2024 Equivalence of Definitions of Trapezium (hist | edit) [2,146 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{TFAE|def = Trapezium}} === Definition $1$ === {{:Definition:Trapezium/Definition 1}} === Definition $2$ === {{:Definition:Trapezium/Definition 2}} == Proof == === Definition $(1)$ implies Definition $(2)$ === Let $T$ be a trapezium by definition $1$. Then by definition $T$ has exactly one pair of Definitio...")
- 00:27, 29 April 2024 Inner Jordan Content is Well-Defined (hist | edit) [2,927 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space. Let: :$\ds R = \prod_{i \mathop = 1}^n \closedint {a_i} {b_i}$ :$\ds R' = \prod_{i \mathop = 1}^n \closedint {a'_i} {b'_i}$ be closed $n$-rectangles that contain $M$. Let $V_R, V_{R'} \in \R_{\ge 0}$ be defined as: :$\ds V_R = \prod_...")
- 23:56, 28 April 2024 Direct Iteration/Examples/x = cos x (hist | edit) [887 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Direct Iteration == <onlyinclude> The following is a graph of the '''direct iteration''' $x_{n + 1} = \map cos {x_n}$ for $n = 0, 1, 2, 3, 4$ where $x_0$ is an arbitrary point such that $1 < x_0 < \dfrac \pi 2$: :500px </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Seco...")
- 09:50, 28 April 2024 Trapezium Rule for Definite Integrals/Also known as (hist | edit) [1,733 bytes] Prime.mover (talk | contribs) (Created page with "== Trapezium Rule for Definite Integrals: Also known as == <onlyinclude> US sources refer to this rule as the '''trapezoidal formula''', as a result of the fact that, in the US, the terms '''trapezoid''' and '''trapezium''' have the opposite definitions. </onlyinclude> == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|p...")
- 06:43, 28 April 2024 Outer Jordan Content of Dilation (hist | edit) [4,454 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == Let $M \subseteq \R^n$ be a bounded subspace of Euclidean $n$-space. Let $c_1, c_2, \dotsc, c_n \in \R_{\ge 0}$ be non-negative real numbers. Let $M' \subseteq \R^n$ be defined as: :$M' = \set {\tuple {c_1 x_1, c_2 x_2, \dotsc, c_n x_n} : \tuple {x_1, x_2, \dotsc, x_n} \in \R^n}$ Then: :$\map {m^*} {M'} = c_1 c_2 \dotsm c...")
- 04:32, 28 April 2024 Set of Intersections with Superset is Cover (hist | edit) [670 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == Let $S$ be a set. Let $\CC$ be a cover of $S$. Let $T \supseteq S$ be a superset of $S$. Then: :$\set {C \cap T : C \in \CC}$ is a cover of $S$. == Proof == Let $x \in S$ be arbitrary. By definition of cover, there is some $C \in \CC$ such that: :$x \in C$ By definition of superset: :$x \in T$ Ther...")