💡 Proposition 2.3.6 (Distributive Law) For any natural numbers $a$, $b$, $c$, we have $a(b + c) = ab + ac$ and $(b + c)a = ba + ca$. From: tao-analysis-1 Learn more:

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📐 Whitney A graph $G$ with at least 3 vertices is 2-connected if and only if every pair of edges lies on a common cycle. Proof: If 2-connected, any two edges $e = xy$ and $f = uv$ can be put on a cycle: by Menger, there are 2 internally disjoint $x$-$u$ paths. Together with edges $e$ and $f$ and paths from $y$ and $v$, we can form a cycle through both edges. Conversely, if every pair of edges lies on a cycle, there\ From: Introduction to Graph Theory Learn more:

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📐 Schur (1) Any $G$-homomorphism between irreducible representations is either zero or an isomorphism. (2) If $k$ is algebraically closed, any $G$-endomorphism of an irreducible representation is a scalar multiple of the identity. From: df-course Learn more:

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📐 Division Algorithm For any integers $a$ and $b$ with $b > 0$, there exist unique integers $q$ (quotient) and $r$ (remainder) such that $a = bq + r$ and $0 \\le r < b$. Proof: \\textbf{Existence:} Let $S = \\{a - bk : k \\in \\mathbb{Z}, a - bk \\geq 0\\}$. Since $S$ is non-empty (choose $k$ sufficiently negative) and bounded below by 0, by the Well-Ordering Principle, $S$ has a minimum element $r = a - bq$ for some $q$. If $r \\geq b$, then $a - b(q+1) = r - b \\geq 0... From: df-course Learn more:

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💡 Proposition I.20 (Triangle Inequality) In any triangle two sides taken together in any manner are greater than the remaining one. From: Euclid's Elements Learn more:

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💡 Proposition VII.31 Any composite number is measured by some prime number. From: Euclid's Elements Learn more:

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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: tontines Learn more:

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📖 Legendre Symbol For odd prime $p$: $(a/p) = 0$ if $p|a$; $(a/p) = 1$ if $a$ is QR mod $p$; $(a/p) = -1$ if $a$ is QNR mod $p$. From: Algebraic Number Theory Learn more:

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📐 Rationality of Action Action is necessarily always rational in the sense that it involves selecting means believed suitable for attaining ends. Irrational action does not exist—people can be mistaken about which means achieve their ends, but error is not irrationality. From: Human Action Learn more:

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Ludwig von Mises

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📐 Theorem 2.47 (Connected Subsets of ℝ) A subset $E$ of $\\mathbb{R}$ is connected if and only if: whenever $x, y \\in E$ and $x < z < y$, we have $z \\in E$. That is, the connected subsets of $\\mathbb{R}$ are precisely the intervals. From: rudin Learn more:

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📐 Theorem 10.5.1 (L Let $f, g : (a, b) \\to \\mathbb{R}$ be differentiable with $g\ From: tao-analysis-1 Learn more:

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📖 Euler $\\phi(n)$ is the number of integers from 1 to $n$ that are relatively prime to $n$. From: intro-discrete Learn more:

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📖 Definition 5.5.1 (Upper Bound) Let $E$ be a subset of $\\mathbb{R}$, and let $M$ be a real number. We say $M$ is an **upper bound** for $E$ iff $x \\leq M$ for every $x \\in E$. From: tao-analysis-1 Learn more:

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📐 Distance Formula The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Proof: By the Pythagorean theorem applied to the right triangle formed by the horizontal and vertical displacements: $d^2 = (x_2-x_1)^2 + (y_2-y_1)^2$ Therefore $d = \\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ From: Men of Mathematics Learn more:

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📖 Definition 4.3.4 (ε-Closeness for Rationals) Let $\\varepsilon > 0$ be a rational. We say rationals $x$ and $y$ are **$\\varepsilon$-close** iff $|x - y| \\leq \\varepsilon$. From: tao-analysis-1 Learn more:

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📖 Exponential Distribution A random variable $X$ has an \\textbf{exponential distribution} with parameter $\\lambda > 0$ if its distribution function is $F(t) = 1 - e^{-\\lambda t}$ for $t \\geq 0$, and $F(t) = 0$ for $t < 0$. The density function is $f(t) = \\lambda e^{-\\lambda t}$ for $t \\geq 0$. From: calc2 Learn more:

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💡 Proposition III.12 If two circles touch one another externally, the straight line joining their centres will pass through the point of contact. From: Euclid's Elements Learn more:

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💡 The Principle of Correlation Between Planes Every being, force, and form on the physical plane has its counterpart on the astral and devachanic planes. Physical matter is the condensed expression of astral forces, which are in turn the densified expression of devachanic archetypes. What appears as a physical object to the senses appears as a living, luminous form on the astral plane and as a sounding, creative thought-being on the devach... From: steiner-GA90a Learn more:

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📐 Theorem 4.14 (Image of Compact) If $f: X \\to Y$ is continuous and $K \\subset X$ is compact, then $f(K)$ is compact. Proof: Let $\\{V_\\alpha\\}$ be an open cover of $f(K)$. Since $f$ is continuous, each $f^{-1}(V_\\alpha)$ is open. These sets cover $K$: $K \\subset \\bigcup_\\alpha f^{-1}(V_\\alpha)$. Since $K$ is compact, finitely many suffice: $K \\subset f^{-1}(V_{\\alpha_1}) \\cup \\cdots \\cup f^{-1}(V_{\\alp... From: rudin Learn more:

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📖 Ends and Means Every action aims at an end (goal) and employs means to achieve it. Means are always scarce relative to ends; otherwise no action would be needed. From: Man, Economy, and State Learn more:

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