📖 Group Presentation A \\textbf{presentation} $\\langle S | R \\rangle$ defines a group as $F(S)/N$ where $N$ is the normal closure of the relators $R$. From: df-course Learn more:

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📐 Third Isomorphism Theorem If $N \\trianglelefteq G$ and $M \\trianglelefteq G$ with $N \\le M$, then $(G/N)/(M/N) \\cong G/M$. From: df-course Learn more:

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📐 Two Vectors Span a Plane Two non-parallel vectors in $\\mathbb{R}^n$ are linearly independent and span a 2-dimensional subspace (a plane through the origin). From: Four Pillars of Geometry Learn more:

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Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📖 Definition II.2 (Gnomon) In any parallelogrammic area let any one whatever of the parallelograms about its diagonal with the two complements be called a gnomon. From: Euclid's Elements Learn more:

Euclid's Elements | Math Academy

The foundational text of Western mathematics. Study all 13 books of Euclid

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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: Man, Economy, and State Learn more:

Man, Economy, and State | Math Academy

Murray Rothbard

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💡 Parallel Line Construction Given line $L$ and point $P$ not on $L$, we can construct a line through $P$ parallel to $L$. From: Four Pillars of Geometry Learn more:

The Four Pillars of Geometry | Magic Internet Math

Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: Beginner Calculus Learn more:

Introductory Calculus | Math Academy

Calculus Made Easy - An interactive introduction to calculus

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📐 Fundamental Theorem of Arithmetic Every integer $n > 1$ can be written uniquely as a product of prime powers: $n = p_1^{a_1} p_2^{a_2} \\cdots p_k^{a_k}$ where $p_1 < p_2 < \\cdots < p_k$ are primes and $a_i \\geq 1$. From: df-course Learn more:

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📖 Coset For $H \\le G$ and $a \\in G$, the \\textbf{left coset} of $H$ containing $a$ is $aH = \\{ah : h \\in H\\}$. The \\textbf{right coset} is $Ha = \\{ha : h \\in H\\}$. From: df-course Learn more:

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💡 Proposition III.32 (Tangent-Chord Angle) If a straight line touch a circle, and from the point of contact there be drawn across, in the circle, a straight line cutting the circle, the angles which it makes with the tangent will be equal to the angles in the alternate segments of the circle. From: Euclid's Elements Learn more:

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The foundational text of Western mathematics. Study all 13 books of Euclid

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📐 Converse of Pythagorean Theorem (Euclid I.48) If in a triangle $a^2 + b^2 = c^2$, then the angle opposite the side of length $c$ is a right angle. Proof: Given $\\triangle ABC$ with $|AC|^2 + |BC|^2 = |AB|^2$. Construct right triangle $\\triangle DEF$ with legs $|DE| = |AC|$, $|EF| = |BC|$, right angle at $E$. By the Pythagorean theorem, $|DF|^2 = |DE|^2 + |EF|^2 = |AC|^2 + |BC|^2 = |AB|^2$. So $|DF| = |AB|$. By SSS congruence, $\\triangle ABC \\c... From: Four Pillars of Geometry Learn more:

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Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📐 Cyclic Quadrilateral Theorem A quadrilateral is inscribed in a circle if and only if its opposite angles sum to $180°$. Proof: Forward: Let $ABCD$ be inscribed in a circle. $\\angle A$ subtends arc $BCD$, $\\angle C$ subtends arc $DAB$. Together these arcs form the whole circle ($360°$ central angle). So $\\angle A + \\angle C = \\frac{1}{2}(360°) = 180°$. Similarly $\\angle B + \\angle D = 180°$. From: Four Pillars of Geometry Learn more:

The Four Pillars of Geometry | Magic Internet Math

Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📐 Second Sylow Theorem All Sylow $p$-subgroups are conjugate to each other. From: df-course Learn more:

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📐 Thales If parallel lines cut two transversals, then the segments cut off on one transversal are proportional to the corresponding segments on the other. That is, if $PQ \\parallel RS$ and they cut transversals at $A, P, B$ and $A, R, C$, then $\\frac{|AP|}{|PB|} = \\frac{|AR|}{|RC|}$. Proof: Consider triangles $\\triangle APR$ and $\\triangle BPS$. The parallel lines create equal corresponding angles. By similarity of triangles, the sides are proportional: $\\frac{|AP|}{|PB|} = \\frac{|AR|}{|RC|}$. From: Four Pillars of Geometry Learn more:

The Four Pillars of Geometry | Magic Internet Math

Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📖 Sample Definition A function $f: A \\to B$ is a mapping from set $A$ to set $B$. From: tao2 Learn more:

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📐 Matroid Greedy Algorithm A set system $(E, \\mathcal{I})$ is a matroid if and only if the greedy algorithm finds a maximum-weight independent set for all weight functions. Proof: If matroid: greedy builds an independent set $I$ by adding highest-weight elements that maintain independence. By the exchange property, $|I|$ equals the rank. Any independent set of same size has $\\leq$ weight (greedy chose highest weights). Conversely, if greedy works for all weights, the exch... From: Introduction to Graph Theory Learn more:

Introduction to Graph Theory | Math Academy

Interactive course based on Douglas B. West

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📖 Euclid If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two lines, if extended indefinitely, meet on that side. From: Four Pillars of Geometry Learn more:

The Four Pillars of Geometry | Magic Internet Math

Explore geometry from four perspectives: Euclidean constructions, coordinates, projective geometry, and transformations. Based on John Stillwell

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📖 Definition VII.10 (Odd-Times Odd) An odd-times odd number is that which is measured by an odd number according to an odd number. From: Euclid's Elements Learn more:

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📖 Family-Wise Error Rate (FWER) FWER = $\\Pr(\\text{at least one Type I error})$. When testing $m$ hypotheses at level $\\alpha$, FWER can be much larger than $\\alpha$. From: Intro to Statistical Learning Learn more:

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An interactive course based on James et al.

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📐 Sample Theorem If $A \\subseteq B$ and $B \\subseteq A$, then $A = B$ Proof: Let $x \\in A$. Since $A \\subseteq B$, we have $x \\in B$ by definition of subset. Therefore, every element of $A$ is in $B$. Now, let $y \\in B$. Since $B \\subseteq A$, we have $y \\in A$ by definition. Therefore, every element of $B$ is in $A$. Since $A \\subseteq B$ and $B \\subseteq A... From: Algebraic Number Theory Learn more:

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Interactive course for learning Number Theory and Cryptography

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