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📐 Cantor\ For any set $A$, $|A| < |\\mathcal{P}(A)|$. The power set has strictly greater cardinality. From: Real Analysis Learn more:

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A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 Proposition VII.37 If a number be measured by any number, the number which is measured will have a part called by the same name as the measuring number. 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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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 Proposition II.4 (Square of a Sum) If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments. 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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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

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

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 Proposition VII.8 If a number be the same parts of a number that a number subtracted is of a number subtracted, the remainder will also be the same parts of the remainder that the whole is of the whole. 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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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Lagrange If $H \\leq G$ and $G$ is finite, then $|H|$ divides $|G|$. The index is $[G:H] = |G|/|H|$. Proof: Cosets partition $G$ into $[G:H]$ sets, each of size $|H|$. Thus $|G| = [G:H] \\cdot |H|$. From: intro-discrete Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 Diagonalizability via Minimal Polynomial $T$ is diagonalizable if and only if its minimal polynomial is a product of distinct linear factors. Proof: **(⇒)** If $T$ is diagonalizable with eigenvalues $\\lambda_1, \\ldots, \\lambda_k$, then $(T - \\lambda_1 I) \\cdots (T - \\lambda_k I) = 0$ on each eigenbasis vector, hence on all of $V$. The minimal polynomial divides $(x - \\lambda_1) \\cdots (x - \\lambda_k)$. **(⇐)** If $m(x) = (x - \\lamb... From: Advanced Linear Algebra Learn more:

Advanced Linear Algebra | Magic Internet Math

A rigorous treatment of linear algebra based on Hoffman & Kunze, covering vector spaces, linear transformations, and canonical forms.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📖 Partition A partition $P$ of $[a,b]$ is a finite set $\\{a = x_0 < x_1 < \\cdots < x_n = b\\}$. From: Real Analysis Learn more:

Real Analysis | Math Academy

A rigorous introduction to real analysis covering limits, continuity, differentiation, and integration through formal mathematical proofs.

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

💡 The Socialist Roots of Nazism Nazism emerged not as a reaction against socialism but as its evolution. The Nazis were recruited from socialist ranks, adopted socialist methods, and fulfilled socialist predictions about capitalism\ From: The Road to Serfdom Learn more:

The Road to Serfdom | Magic Internet Math

An interactive course exploring F.A. Hayek

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

📐 GCD and LCM from Prime Factorisations If $a = \\prod p_i^{\\alpha_i}$ and $b = \\prod p_i^{\\beta_i}$, then $\\gcd(a,b) = \\prod p_i^{\\min(\\alpha_i, \\beta_i)}$ and $\\text{lcm}(a,b) = \\prod p_i^{\\max(\\alpha_i, \\beta_i)}$. From: intro-discrete Learn more:

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Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.

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Magic Internet Math

Interactive courses covering the mathematics that powers modern technology, from foundational algebra to the cryptography securing the internet.