💡 Proposition IV.16 (Inscribed 15-gon)
In a given circle to inscribe a fifteen-angled figure which shall be both equilateral and equiangular.
From: Euclid's Elements
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📖 Survival Function
$S(t) = \\Pr(T > t)$ gives the probability of surviving past time $t$. $S(t)$ is non-increasing with $S(0) = 1$ and $\\lim_{t \\to \\infty} S(t) = 0$.
From: Intro to Statistical Learning
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📖 Cost Complexity Pruning
For each $\\alpha \\geq 0$, find subtree $T$ minimizing $\\sum_{m=1}^{|T|}\\sum_{i: x_i \\in R_m}(y_i - \\hat{y}_{R_m})^2 + \\alpha|T|$ where $|T|$ is number of terminal nodes.
From: Intro to Statistical Learning
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📐 Strong Induction
If $P(1)$ is true and $[P(1) \\land P(2) \\land \\cdots \\land P(k)] \\Rightarrow P(k+1)$, then $P(n)$ is true for all $n$.
From: Real Analysis
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💡 Proposition VI.30 (Golden Ratio)
To cut a given finite straight line in extreme and mean ratio.
From: Euclid's Elements
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📐 Subspace Test
A non-empty subset $W \\subseteq V$ is a subspace if and only if for all $u, v \\in W$ and $c \\in F$: (1) $u + v \\in W$ and (2) $c \\cdot u \\in W$.
Proof: **(⇒)** If $W$ is a subspace, it is closed under addition and scalar multiplication by definition.
**(⇐)** Assume conditions (1) and (2) hold.
- **Zero vector:** Since $W$ is non-empty, there exists $w \\in W$. By (2), $0 \\cdot w = \\mathbf{0} \\in W$.
- **Additive inverse:** For $w \\in W$, by...
From: Advanced Linear Algebra
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📖 Formal Equality
Equality before the law: the liberal principle that the same rules apply to all regardless of status, as opposed to material equality which requires differential treatment and arbitrary power.
From: The Road to Serfdom
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📖 Sample Definition
A function $f: A \\to B$ is a mapping from set $A$ to set $B$.
From: math_history
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💡 Proposition I.23
On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.
From: Euclid's Elements
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📐 Hall
A bipartite graph $G$ with bipartition $(X, Y)$ has a matching saturating $X$ if and only if $|N(S)| \\geq |S|$ for all $S \\subseteq X$.
Proof: Necessity is clear. For sufficiency, induct on $|X|$. If $|N(S)| > |S|$ for all proper $S$, match any $x \\in X$ to a neighbor and apply induction. Otherwise, some $S$ has $|N(S)| = |S|$; by induction match $S$ to $N(S)$. Hall\
From: Introduction to Graph Theory
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📖 Irreducible Polynomial
A polynomial $f$ is irreducible over $F$ if it cannot be factored as $f = gh$ with $\\deg g, \\deg h > 0$.
From: intro-discrete
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🎯 Odd Degree Corollary
Every graph has an even number of vertices with odd degree.
Proof: By the Handshaking Lemma, $\\sum d(v) = 2|E|$ is even. Split the sum into odd-degree and even-degree vertices. The even-degree sum is even, so the odd-degree sum must also be even. Since each odd-degree term is odd, there must be an even number of such terms.
From: Introduction to Graph Theory
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📐 Multiplicity Inequality
For any eigenvalue $\\lambda$: geometric multiplicity $\\leq$ algebraic multiplicity.
From: Advanced Linear Algebra
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💡 Proposition I.27
If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.
From: Euclid's Elements
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💡 Proposition VII.25
If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.
From: Euclid's Elements
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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: Man, Economy, and State
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📖 Integral Domain
An integral domain is a commutative ring with no zero-divisors: $ab = 0$ implies $a = 0$ or $b = 0$.
From: intro-discrete
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📖 Multiplicative Inverse
If $\\gcd(a, m) = 1$, then $a$ has a multiplicative inverse modulo $m$: an integer $a^{-1}$ such that $a \\cdot a^{-1} \\equiv 1 \\pmod{m}$.
From: Disquisitiones Arithmeticae
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📖 Function
A function $f: A \\to B$ assigns to each element of $A$ exactly one element of $B$.
From: Real Analysis
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💡 Proposition VI.12 (Fourth Proportional)
To three given straight lines to find a fourth proportional.
From: Euclid's Elements
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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: coding-theory-course
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