The Garden

Propositional Equivalences

P ≡ P (Identity Law)

P ∧ T ≡ P (Identity Law for Conjunction)

P ∨ F ≡ P (Identity Law for Disjunction)

P ∧ F ≡ F (Domination Law for Conjunction)

P ∨ T ≡ T (Domination Law for Disjunction)

P ∧ P ≡ P (Idempotent Law for Conjunction)

P ∨ P ≡ P (Idempotent Law for Disjunction)

¬(¬P) ≡ P (Double Negation Law)

P ∧ Q ≡ Q ∧ P (Commutative Law for Conjunction)

P ∨ Q ≡ Q ∨ P (Commutative Law for Disjunction)

(P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R) (Associative Law for Conjunction)

(P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R) (Associative Law for Disjunction)

P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R) (Distributive Law of Conjunction over Disjunction)

P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R) (Distributive Law of Disjunction over Conjunction)

¬(P ∧ Q) ≡ ¬P ∨ ¬Q (De Morgan's Law)

¬(P ∨ Q) ≡ ¬P ∧ ¬Q (De Morgan's Law)

P ∧ (P ∨ Q) ≡ P (Absorption Law)

P ∨ (P ∧ Q) ≡ P (Absorption Law)

P ∧ ¬P ≡ F (Law of Non-Contradiction)

P ∨ ¬P ≡ T (Law of Excluded Middle)

Rules of Inference for Propositional Logic

P, P → Q ∴ Q (Modus Ponens)

¬Q, P → Q ∴ ¬P (Modus Tollens)

P → Q, Q → R ∴ P → R (Hypothetical Syllogism)

P ∨ Q, ¬P ∴ Q (Disjunctive Syllogism)

P ∴ P ∨ Q (Addition)

P ∧ Q ∴ P (Simplification)

P, Q ∴ P ∧ Q (Conjunction)

P → Q, R → S, P ∨ R ∴ Q ∨ S (Constructive Dilemma)

P → Q, R → S, ¬Q ∨ ¬S ∴ ¬P ∨ ¬R (Destructive Dilemma)

Implications, Biconditionals, and Quantifiers

P → Q ≡ ¬P ∨ Q (Material Implication)

P → Q ≡ ¬Q → ¬P (Contraposition)

¬(p → Q) ≡ P ∧ ¬Q (Negation of Implication)

(p ∧ Q) → R ≡ P → (Q → R) (Exportation Law)

P ↔ Q ≡ (P → Q) ∧ (Q → P) (Definition of Biconditional)

P ↔ Q ≡ (P ∧ Q) ∨ (¬P ∧ ¬Q) (Alternative Biconditional)

∀x P(x) ∴ P(c) (Universal Instantiation)

P(c) for an arbitrary c ∴ ∀x P(x) (Universal Generalization)

P(c) for some c ∴ ∃x P(x) (Existential Generalization)

∃x P(x) ∴ P(c) for some specific c (Existential Instantiation)

Quantifier Equivalences and Rules

¬∀x P(x) ≡ ∃x ¬P(x) (Quantifier Negation)

¬∃x P(x) ≡ ∀x ¬P(x) (Quantifier Negation)

∀x (P(x) ∧ Q(x)) ≡ ∀x P(x) ∧ ∀x Q(x) (Distribution of Universal over Conjunction)

∃x (P(x) ∨ Q(x)) ≡ ∃x P(x) ∨ ∃x Q(x) (Distribution of Existential over Disjunction)

∀x P(x) ∨ ∀x Q(x) → ∀x (P(x) ∨ Q(x)) (Note: Converse is not valid)

∃x (P(x) ∧ Q(x)) → ∃x P(x) ∧ ∃x Q(x) (Note: Converse is not valid)

∀x ∀y P(x, y) ≡ ∀y ∀x P(x, y) (Order of Universal Quantifiers)

∃x ∃y P(x, y) ≡ ∃y ∃x P(x, y) (Order of Existential Quantifiers)

Meta-Logical Theorems

If Σ ⊨ φ then Σ ⊢ φ (Gödel's Completeness Theorem for First-Order Logic)

A formula is universally valid (a tautology) if and only if it is Provable. (Completeness Theorem for Propositional Logic)

No consistent axiomatic system capable of expressing arithmetic can Prove its own consistency. (Gödel's Second Incompleteness Theorem)

There is no general algorithm to decide the truth of a first-order logic statement. (Church-Turing Undecidability Theorem)

Divisibility and GCD

a | b ⇔ ∃k∈Z: b = ak (Definition of Divisibility)

If a|b and b|c, then a|c (Transitivity of Divisibility)

If a|b and a|c, then a|(bx + cy) ∀x,y∈Z

∀a,b∈Z, b>0, ∃!q,r: a = bq + r, 0≤r (Division Algorithm)

gcd(a,b) = largest d such that d|a and d|b

lcm(a,b) = smallest m>0 such that a|m and b|m

gcd(a,b) × lcm(a,b) = |ab|

gcd(a,b) = gcd(b, a mod b) (Euclidean Algorithm)

If d = gcd(a,b), then ∃x,y∈Z: ax + by = d (Bézout's Identity)

ax + by = d has solutions ⇔ gcd(a,b)|d

Prime Numbers and Factorization

p is prime ⇔ (p|ab ⇒ p|a or p|b) (Euclid's Lemma)

There are infinitely many primes (Euclid's Theorem)

Every n>1 factors uniquely as product of primes (Fundamental Theorem of Arithmetic)

Congruences and Modular Arithmetic

a ≡ b mod n ⇔ n|(a−b) (Definition of Congruence)

a ≡ b mod n and c ≡ d mod n ⇒ a+c ≡ b+d mod n

a ≡ b mod n and c ≡ d mod n ⇒ ac ≡ bd mod n

a ≡ b mod n ⇒ a^k ≡ b^k mod n ∀k≥1

ax ≡ ay (mod n) ⇒ x ≡ y (mod n/d) where d = gcd(a,n). (Modular Cancellation Law)

gcd(a,n)=1 ⇒ ∃b: ab ≡ 1 mod n (Modular Inverse)

gcd(a,n)=1 ⇒ a^ϕ(n) ≡ 1 mod n (Euler's Theorem)

p prime, p∤a ⇒ a^(p−1) ≡ 1 mod p (Fermat's Little Theorem)

p prime ⇒ a^p ≡ a mod p ∀a∈Z

(p−1)! ≡ −1 mod p for prime p (Wilson's Theorem)

(a + b)^p ≡ a^p + b^p (mod p). (Freshman's Dream modulo p)

x≡a mod m, x≡b mod n, gcd(m,n)=1 ⇒ ∃!x mod mn (Chinese Remainder Theorem)

Multiplicative Functions

f multiplicative ⇔ f(m n) = f(m) f(n) when gcd(m,n)=1. (Multiplicative Function)

f completely multiplicative ⇔ f(m n) = f(m) f(n) for all m,n. (Completely Multiplicative)

ϕ(p^k) = p^k − p^(k−1) for prime p

gcd(m,n)=1 ⇒ ϕ(mn) = ϕ(m)ϕ(n)

∑*{d|n} ϕ(d) = n

φ(n) = n ∏*{p|n} (1 − 1/p). (Euler's Totient Product Formula)

σ(n) = ∑*{d|n} d (Sum of Divisors)

τ(n) = ∑*{d|n} 1 (Number of Divisors)

τ(n) = number of divisors = ∏ (α_i + 1) for n = ∏ p_i^{α_i}. (Divisor Count Formula)

σ(n) = sum of divisors = ∏ (p_i^{α_i+1} − 1)/(p_i − 1). (Sum-of-Divisors Formula)

μ(n) = 1 if n=1; 0 if p²|n; (−1)^k if n=p₁⋯pₖ (Möbius Function)

∑*{d|n} μ(d) = 0 for n>1

∑*{d|n} μ(d)(n/d) = ϕ(n) (Möbius Inversion)

(f * g)(n) = ∑_{d|n} f(d) g(n/d). (Dirichlet Convolution)

If g(n) = ∑*{d|n} f(d) then f(n) = ∑*{d|n} μ(d) g(n/d). (Möbius Inversion Formula)

Groups and Orders

(Z/nZ)* has size ϕ(n) (Group of Units)

(Z/pZ)* is cyclic for prime p (Primitive Root Theorem)

If g is primitive mod p then every nonzero residue ≡ g^k for some k. (Generator Property)

gcd(a,n)=1 ⇒ ordₙ(a)|ϕ(n)

n|a^k − 1 ⇒ ordₙ(a)|k

Special Equations and Theorems

x = k(m²−n²), y = k(2mn), z = k(m²+n²) (Pythagorean Triples)

p = x² + y² ⇔ p=2 or p≡1 mod 4 (Two Squares Theorem)

x² − Dy² = 1 has infinitely many solutions for non-square D (Pell's Equation)

xⁿ + yⁿ = zⁿ has no positive integer solutions for n>2 (Fermat's Last Theorem)

σ(n) = 2n (Definition of Perfect Number)

σ(n) = 2n ⇔ n is perfect (even perfect numbers given by n = 2^{p−1}(2^p − 1) when 2^p −1 is prime). (Perfect Number — Euclid–Euler)

Every natural n is sum of four squares. (Lagrange's Four-Square Theorem)

Quadratic Residues and Reciprocity

∃x: x² ≡ a mod p ⇔ a^((p−1)/2) ≡ 1 mod p (Euler's Criterion)

(a/p) = 1 if solvable, −1 if not, 0 if p|a (Legendre Symbol)

(ab/p) = (a/p)(b/p)

(a/p) = (p/a)(−1)^((a−1)(p−1)/4) (Quadratic Reciprocity)

(−1/p) = (−1)^((p−1)/2)

(2/p) = (−1)^((p²−1)/8)

Jacobi symbol (a/n) multiplicative and equals ∏ (a/p_i) over prime factors p_i of n but (a/n)=1 does not guarantee solvability when n composite. (Jacobi Symbol)

Prime Distribution and Zeta

π(n) ~ n/ln(n) (Prime Number Theorem)

π(x) ~ x / log x; p_n ~ n log n. (Prime Number Theorem / nth prime asymptotic)

∑*{p≤n} 1/p ~ ln(ln n)

∏*{p≤n} (1−1/p) ~ e^(−γ)/ln n

ζ(s) = ∑n^(-s) = ∏_p (1−p^(-s))^(-1) for Re(s)>1 (Euler Product)

Coordinate Geometry

d = √((x2 - x1)^2 + (y2 - y1)^2) (Euclidean distance)

d = √((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2) (Euclidean distance in 3D)

M = ((x1 + x2)/2, (y1 + y2)/2) (Midpoint formula)

m = (y2 - y1)/(x2 - x1) (Slope of a line)

y - y1 = m(x - x1) (Point–slope form of a line)

y = mx + b (Slope–intercept form)

Ax + By + C = 0 (Standard form of a line)

d = |Ax + By + C| / √(A^2 + B^2) (Distance from point to line)

Circles

(x - h)^2 + (y - k)^2 = r^2 (Equation of a circle)

C = 2πr (Circumference of a circle)

A = π r^2 (Area of a circle)

L = r θ (Arc length, θ in radians) (Arc length — radians)

A_sector = 1/2 r^2 θ (Sector area, θ in radians) (Sector area — radians)

chord = 2 r sin(θ/2) (Chord length subtending angle θ at center)

∠inscribed = 1/2 ∠central (Inscribed-angle theorem)

tangent ⟂ radius at point of contact (Tangent–radius theorem)

(external secant) PA·PB = PC·PD ; (tangent) (PT)^2 = PA·PB (Power of a point)

Angle subtended by diameter = 90° (Thales' theorem)

Triangles

a^2 + b^2 = c^2 (Pythagorean theorem)

45°–45°–90°: sides 1 : 1 : √2 ; 30°–60°–90°: sides 1 : √3 : 2 (Special right triangles)

c^2 = a^2 + b^2 - 2ab cos C (Law of cosines)

a / sin A = b / sin B = c / sin C = 2R (Law of sines)

A = 1/2 b h (Triangle area — base × height / 2)

A = 1/2 a b sin C (Triangle area — trig form)

A = √(s(s - a)(s - b)(s - c)), s = (a + b + c)/2 (Heron's formula)

a + b > c, b + c > a, c + a > b (Triangle inequality)

m_a = 1/2 √(2b^2 + 2c^2 - a^2) (Median length — Apollonius)

G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3) (Centroid coordinates)

R = abc / (4A) (Circumradius of a triangle)

r = A / s (Inradius of a triangle; s = semiperimeter)

OI^2 = R(R - 2r) (Euler's triangle formula relating circumcenter & incenter)

AB/AC = DB/DC (Angle-bisector theorem; internal bisector dividing opposite side)

G, O, H are collinear (Euler line) (Euler line — triangle centers)

Polygons and Similarity

A = 1/2 |Σ_{i=1..n} (x_i y_{i+1} - x_{i+1} y_i)| (Shoelace formula for polygon area)

Sum interior angles of n-gon = (n - 2) × 180° (Polygon interior angle sum)

A_regular = 1/2 · ap (Area of regular polygon; a = apothem, p = perimeter)

Interior angle (regular n-gon) = (n - 2)·180° / n (Regular polygon interior angle)

Similarity scaling: linear scale k ⇒ lengths ×k, areas ×k^2, volumes ×k^3 (Similarity scaling laws)

Conic Sections and Transformations

Area(parallelogram) = |u × v| (2D determinant / magnitude of cross product) (Vector area)

(x - h)^2 = 4p(y - k) ; (x - h)^2/a^2 + (y - k)^2/b^2 = 1 ; (x - h)^2/a^2 - (y - k)^2/b^2 = 1 (Standard forms of parabola, ellipse, hyperbola)

e = c / a (Eccentricity of ellipse/hyperbola; conic eccentricity)

Rotation (2D): [x'] = [ cos θ -sin θ ; sin θ cos θ ] [x] (2×2 rotation matrix)

Angle of incidence = angle of reflection (Law of reflection)

Quadrilaterals

A_parallelogram = base · height (Area of a parallelogram)

A_trapezoid = 1/2 · (b1 + b2) · h (Area of a trapezoid)

A_rhombus = 1/2 · d1 · d2 (Area of a rhombus via diagonals)

d_rect = √(l^2 + w^2) (Diagonal of a rectangle)

3D Geometry

V_cube = s^3 ; SA_cube = 6 s^2 (Cube — volume and surface area)

V_prism = B · h ; V_pyramid = (1/3) B · h (Volume of prism and pyramid; B = base area)

V_sphere = 4/3 π r^3 ; SA_sphere = 4 π r^2 ; V_cylinder = π r^2 h ; SA_cylinder = 2 π r (h + r) ; V_cone = 1/3 π r^2 h ; SA_cone = π r l + π r^2 (Key volumes & surface areas: sphere, cylinder, cone).

Right Triangle Definitions:

sin(θ) = Opposite / Hypotenuse (Sine Definition)

cos(θ) = Adjacent / Hypotenuse (Cosine Definition)

tan(θ) = Opposite / Adjacent (Tangent Definition)

Unit Circle Definitions:

sin(θ) = y-coordinate (Unit Circle Sine)

cos(θ) = x-coordinate (Unit Circle Cosine)

tan(θ) = sin(θ) / cos(θ) (Tangent Quotient Identity)

Reciprocal Identities:

csc(θ) = 1 / sin(θ) (Cosecant)

sec(θ) = 1 / cos(θ) (Secant)

cot(θ) = 1 / tan(θ) (Cotangent)

Pythagorean Identities:

sin²(θ) + cos²(θ) = 1 (Fundamental Pythagorean Identity)

1 + tan²(θ) = sec²(θ) (Pythagorean Identity)

1 + cot²(θ) = csc²(θ) (Pythagorean Identity)

Periodicity Identities:

sin(θ + 2π) = sin(θ) (Periodicity of Sine)

cos(θ + 2π) = cos(θ) (Periodicity of Cosine)

tan(θ + π) = tan(θ) (Periodicity of Tangent)

Even-Odd Identities:

sin(-θ) = -sin(θ) (Odd Function)

cos(-θ) = cos(θ) (Even Function)

tan(-θ) = -tan(θ) (Odd Function)

Co-function Identities:

sin(π/2 - θ) = cos(θ) (Cofunction Identity)

cos(π/2 - θ) = sin(θ) (Cofunction Identity)

tan(π/2 - θ) = cot(θ) (Cofunction Identity)

Angle Sum Identities:

sin(α + β) = sin(α)cos(β) + cos(α)sin(β) (Sine Addition Formula)

cos(α + β) = cos(α)cos(β) - sin(α)sin(β) (Cosine Addition Formula)

tan(α + β) = (tan(α) + tan(β)) / (1 - tan(α)tan(β)) (Tangent Addition Formula)

Angle Difference Identities:

sin(α - β) = sin(α)cos(β) - cos(α)sin(β) (Sine Subtraction Formula)

cos(α - β) = cos(α)cos(β) + sin(α)sin(β) (Cosine Subtraction Formula)

tan(α - β) = (tan(α) - tan(β)) / (1 + tan(α)tan(β)) (Tangent Subtraction Formula)

Double Angle Formulas:

sin(2θ) = 2sin(θ)cos(θ) (Sine Double Angle)

cos(2θ) = cos²(θ) - sin²(θ) (Cosine Double Angle)

cos(2θ) = 2cos²(θ) - 1 (Cosine Double Angle)

cos(2θ) = 1 - 2sin²(θ) (Cosine Double Angle)

tan(2θ) = 2tan(θ) / (1 - tan²(θ)) (Tangent Double Angle)

Half Angle Formulas:

sin(θ/2) = ±√((1 - cos(θ))/2) (Sine Half Angle)

cos(θ/2) = ±√((1 + cos(θ))/2) (Cosine Half Angle)

tan(θ/2) = sin(θ) / (1 + cos(θ)) (Tangent Half Angle)

Power Reducing Formulas:

sin²(θ) = (1 - cos(2θ))/2 (Power Reduction)

cos²(θ) = (1 + cos(2θ))/2 (Power Reduction)

Product-to-Sum Formulas:

sin(α)sin(β) = ½[cos(α - β) - cos(α + β)] (Product-to-Sum)

cos(α)cos(β) = ½[cos(α - β) + cos(α + β)] (Product-to-Sum)

sin(α)cos(β) = ½[sin(α + β) + sin(α - β)] (Product-to-Sum)

Sum-to-Product Formulas:

sin(α) + sin(β) = 2sin((α + β)/2)cos((α - β)/2) (Sum-to-Product)

sin(α) - sin(β) = 2cos((α + β)/2)sin((α - β)/2) (Sum-to-Product)

cos(α) + cos(β) = 2cos((α + β)/2)cos((α - β)/2) (Sum-to-Product)

cos(α) - cos(β) = -2sin((α + β)/2)sin((α - β)/2) (Sum-to-Product)

Laws of Triangles:

a/sin(A) = b/sin(B) = c/sin(C) = 2R (Law of Sines)

a² = b² + c² - 2bc cos(A) (Law of Cosines)

Area = ½ ab sin(C) (Triangle Area Formula)

Area = √[s(s - a)(s - b)(s - c)] (Heron's Formula)

Inverse Trigonometric Relations:

arcsin(x) + arccos(x) = π/2 (Inverse Complementary Identity)

Euler's Formula:

e^(iθ) = cos(θ) + i sin(θ) (Euler's Formula)

Limits

lim(x→a) k = k (Constant Limit Law)

lim(x→a) x = a (Identity Limit Law)

lim(x→a) [f(x) + g(x)] = lim(x→a) f(x) + lim(x→a) g(x) (Sum Law for Limits)

lim(x→a) [f(x) · g(x)] = lim(x→a) f(x) · lim(x→a) g(x) (Product Law for Limits)

lim(x→a) [f(x)/g(x)] = lim(x→a) f(x) / lim(x→a) g(x), if lim(x→a) g(x) ≠ 0 (Quotient Law for Limits)

lim(x→a) [c·f(x)] = c·lim(x→a) f(x) (Constant Multiple Law for Limits)

If g(x) ≤ f(x) ≤ h(x) and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L (Squeeze Theorem)

lim(x→0) sin(x)/x = 1 (Standard Trigonometric Limit)

lim(x→∞) (1 + 1/x)^x = e (Definition of e)

f is continuous at a if lim(x→a) f(x) = f(a) (Definition of Continuity)

If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c in (a,b) such that f(c) = k (Intermediate Value Theorem)

Derivatives

f'(x) = lim(h→0) [f(x+h) - f(x)]/h (Definition of Derivative)

d/dx[c] = 0 (Constant Rule)

d/dx[x^n] = n·x^(n-1) (Power Rule)

d/dx[c·f(x)] = c·f'(x) (Constant Multiple Rule)

d/dx[f(x) + g(x)] = f'(x) + g'(x) (Sum Rule)

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x) (Product Rule)

d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]² (Quotient Rule)

d/dx[f(g(x))] = f'(g(x))·g'(x) (Chain Rule)

d/dx[e^x] = e^x (Exponential Derivative)

d/dx[a^x] = a^x·ln(a) (General Exponential Derivative)

d/dx[ln(x)] = 1/x (Natural Logarithm Derivative)

d/dx[sin(x)] = cos(x) (Sine Derivative)

d/dx[cos(x)] = -sin(x) (Cosine Derivative)

d/dx[tan(x)] = sec²(x) (Tangent Derivative)

d/dx[arcsin(x)] = 1/√(1-x²) (Arcsine Derivative)

d/dx[arctan(x)] = 1/(1+x²) (Arctangent Derivative)

(f⁻¹)'(x) = 1/f'(f⁻¹(x)) (Inverse Function Theorem)

Mean Value Theorems

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then f'(c) = 0 for some c in (a,b) (Rolle's Theorem)

If f is continuous on [a,b] and differentiable on (a,b), then f'(c) = [f(b) - f(a)]/(b - a) for some c in (a,b) (Mean Value Theorem)

If lim(x→a) f(x)/g(x) is 0/0 or ∞/∞, then lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x) (L'Hôpital's Rule)

Integrals - Fundamental Theorems

d/dx[∫ₐˣ f(t)dt] = f(x) (Fundamental Theorem of Calculus, Part 1)

∫ₐᵇ f(x)dx = F(b) - F(a) where F'(x) = f(x) (Fundamental Theorem of Calculus, Part 2)

∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx (Sum Rule for Integrals)

∫c·f(x)dx = c·∫f(x)dx (Constant Multiple Rule for Integrals)

∫f(g(x))·g'(x)dx = ∫f(u)du where u = g(x) (Integration by Substitution)

∫u dv = uv - ∫v du (Integration by Parts)

Standard Integrals

∫x^n dx = x^(n+1)/(n+1) + C, n ≠ -1 (Power Rule for Integration)

∫(1/x)dx = ln|x| + C (Logarithmic Integration)

∫e^x dx = e^x + C (Exponential Integration)

∫a^x dx = a^x/ln(a) + C (General Exponential Integration)

∫sin(x)dx = -cos(x) + C (Sine Integration)

∫cos(x)dx = sin(x) + C (Cosine Integration)

∫sec²(x)dx = tan(x) + C (Secant Squared Integration)

∫tan(x)dx = -ln|cos(x)| + C (Tangent Integration)

∫sec(x)dx = ln|sec(x) + tan(x)| + C (Secant Integration)

∫1/√(1-x²) dx = arcsin(x) + C (Arcsine Integration)

∫1/(1+x²) dx = arctan(x) + C (Arctangent Integration)

∫ln(x)dx = x·ln(x) - x + C (Logarithm Integration)

Series

∑(n=0 to ∞) ar^n = a/(1-r), for |r| < 1 (Geometric Series)

f(x) = ∑(n=0 to ∞) [f^(n)(a)/n!]·(x-a)^n (Taylor Series)

f(x) = ∑(n=0 to ∞) [f^(n)(0)/n!]·x^n (Maclaurin Series)

e^x = ∑(n=0 to ∞) x^n/n! = 1 + x + x²/2! + x³/3! + ... (Exponential Series)

sin(x) = ∑(n=0 to ∞) (-1)^n·x^(2n+1)/(2n+1)! = x - x³/3! + x⁵/5! - ... (Sine Series)

cos(x) = ∑(n=0 to ∞) (-1)^n·x^(2n)/(2n)! = 1 - x²/2! + x⁴/4! - ... (Cosine Series)

ln(1+x) = ∑(n=1 to ∞) (-1)^(n+1)·x^n/n = x - x²/2 + x³/3 - ..., |x| < 1 (Logarithmic Series)

First-Order Ordinary Differential Equations (ODEs)

dy/dx = f(x, y) (General First-Order ODE)

g(x)dx = h(y)dy (Separable Equation)

∫g(x)dx = ∫h(y)dy + C (Solution for Separable Equation)

y' + p(x)y = q(x) (Linear First-Order ODE)

μ(x) = exp(∫p(x)dx) (Integrating Factor)

y = (1/μ(x)) * [∫μ(x)q(x)dx + C] (Solution via Integrating Factor)

M(x, y)dx + N(x, y)dy = 0 (Exact Equation)

∂M/∂y = ∂N/∂x (Test for Exactness)

y' + p(x)y = q(x)y^n (Bernoulli Equation)

v = y^(1-n) (Bernoulli Substitution)

y' = F(y/x) (Homogeneous (Degree) Equation)

v = y/x (Homogeneous (Degree) Substitution)

Second-Order Linear ODEs

a(x)y'' + b(x)y' + c(x)y = g(x) (General Linear Second-Order ODE)

y = y_c + y_p (General Solution Structure, Non-homogeneous)

y_c = C1*y1 + C2*y2 (Principle of Superposition, Homogeneous)

ay'' + by' + cy = 0 (Homogeneous with Constant Coefficients)

ar^2 + br + c = 0 (Characteristic Equation)

y_c = C1*exp(r1*x) + C2*exp(r2*x) (Solution for Distinct Real Roots)

y_c = (C1 + C2*x)*exp(r*x) (Solution for Repeated Real Roots)

y_c = exp(αx)*(C1*cos(βx) + C2*sin(βx)) (Solution for Complex Roots r = α ± iβ)

W(y1, y2) = y1*y2' - y1'*y2 (Wronskian)

W(x) = C*exp(-∫p(x)dx) (Abel's Theorem)

yp(x) = -y1∫(y2*g/W)dx + y2∫(y1*g/W)dx (Variation of Parameters)

ax^2*y'' + bxy' + cy = 0 (Cauchy-Euler Equation)

ar^2 + (b-a)r + c = 0 (Cauchy-Euler Characteristic Equation)

mx'' + cx' + kx = F(t) (Forced Damped Oscillator)

Laplace Transforms

L{f(t)} = F(s) = ∫[0, ∞] exp(-st)f(t)dt (Definition of Laplace Transform)

L{f'(t)} = sF(s) - f(0) (Laplace Transform of First Derivative)

L{f''(t)} = s^2*F(s) - s*f(0) - f'(0) (Laplace Transform of Second Derivative)

L{exp(at)f(t)} = F(s-a) (First Shifting Theorem)

L{u_c(t)f(t-c)} = exp(-cs)F(s) (Second Shifting Theorem)

L{(f * g)(t)} = F(s)G(s) (Convolution Theorem)

(f * g)(t) = ∫[0, t] f(τ)g(t-τ)dτ (Definition of Convolution)

L{δ(t-c)} = exp(-cs) (Laplace Transform of Dirac Delta)

Systems of ODEs & Stability

**x**' = A**x** (Homogeneous Linear System)

det(A - λI) = 0 (System Characteristic Equation)

**x**(t) = C1*exp(λ1*t)**v1** + C2*exp(λ2*t)**v2** (Solution, Distinct Eigenvalues)

**x**(t) = exp(At)**x**(0) (Matrix Exponential Solution)

exp(At) = I + At + (At)^2/2! + ... (Definition of Matrix Exponential)

J(x, y) = [[∂P/∂x, ∂P/∂y], [∂Q/∂x, ∂Q/∂y]] (Jacobian Matrix for x' = P, y' = Q)

tr(J) < 0, det(J) > 0 (Asymptotic Stability at Critical Point)

V(x) > 0 for x ≠ 0 and dV/dt ≤ 0 (Lyapunov Stability Criteria)

Nonlinear Dynamics & Chaos

A closed trajectory in R² must enclose a critical point. (Poincaré-Bendixson Theorem)

x' = μ - x^2 (Saddle-Node Bifurcation Normal Form)

x' = μx - x^3 (Pitchfork Bifurcation Normal Form)

x_(n+1) = f(x_n) (Discrete Dynamical System / Map)

x_(n+1) = r * x_n * (1 - x_n) (Logistic Map)

δ = (r_n - r_(n-1)) / (r_(n+1) - r_n) ≈ 4.669... (Feigenbaum Constant)

x' = σ(y - x), y' = x(ρ - z) - y, z' = xy - βz (Lorenz Equations)

Partial Differential Equations (PDEs) & Fourier Series

∂u/∂t = k * ∂²u/∂x² (1D Heat Equation)

∂²u/∂t² = c² * ∂²u/∂x² (1D Wave Equation)

∇²u = ∂²u/∂x² + ∂²u/∂y² = 0 (2D Laplace's Equation)

u(x,t) = X(x)T(t) (Separation of Variables Ansatz)

f(x) = a0/2 + Σ[n=1, ∞](an*cos(nπx/L) + bn*sin(nπx/L)) (Fourier Series)

an = (1/L) ∫[-L, L] f(x)cos(nπx/L)dx (Fourier Cosine Coefficient)

bn = (1/L) ∫[-L, L] f(x)sin(nπx/L)dx (Fourier Sine Coefficient)

Basic Vector Operations

a + b = b + a (Commutative Law for Vector Addition)

(a + b) + c = a + (b + c) (Associative Law for Vector Addition)

a + 0 = a (Identity Law for Vector Addition)

a + (-a) = 0 (Inverse Law for Vector Addition)

k(a + b) = ka + kb (Distributive Law for Scalar Multiplication)

(k + m)a = ka + ma (Distributive Law for Scalar Multiplication)

k(ma) = (km)a (Associative Law for Scalar Multiplication)

1·a = a (Identity Law for Scalar Multiplication)

Magnitude and Unit Vectors

|a| = √(a₁² + a₂² + ... + aₙ²) (Euclidean Norm)

|ka| = |k|·|a| (Scalar Multiplication of Magnitude)

â = a/|a| (Unit Vector)

|a| = 0 ⟺ a = 0 (Zero Vector Property)

Dot Product

a · b = |a||b|cos(θ) (Geometric Definition of Dot Product)

a · b = a₁b₁ + a₂b₂ + ... + aₙbₙ (Component Form of Dot Product)

a · b = b · a (Commutative Law for Dot Product)

a · (b + c) = a · b + a · c (Distributive Law for Dot Product)

k(a · b) = (ka) · b = a · (kb) (Scalar Multiplication in Dot Product)

a · a = |a|² (Dot Product with Self)

a · b = 0 ⟺ a ⊥ b (Orthogonality Condition)

|a · b| ≤ |a||b| (Cauchy-Schwarz Inequality)

proj_b(a) = (a · b/|b|²)b (Vector Projection)

comp_b(a) = a · b/|b| (Scalar Component)

Cross Product (3D)

a × b = |a||b|sin(θ)n̂ (Geometric Definition of Cross Product)

a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁) (Component Form of Cross Product)

a × b = -b × a (Anti-commutative Law for Cross Product)

a × (b + c) = a × b + a × c (Distributive Law for Cross Product)

k(a × b) = (ka) × b = a × (kb) (Scalar Multiplication in Cross Product)

a × a = 0 (Cross Product with Self)

a × b = 0 ⟺ a ∥ b (Parallel Vector Condition)

|a × b| = |a||b|sin(θ) (Magnitude of Cross Product)

(a × b) · a = 0, (a × b) · b = 0 (Cross Product Orthogonality)

a × (b × c) = (a · c)b - (a · b)c (Vector Triple Product)

a · (b × c) = b · (c × a) = c · (a × b) (Scalar Triple Product)

|a · (b × c)| = Volume of parallelepiped (Geometric Interpretation of Scalar Triple Product)

Triangle and Parallelogram identities

|a + b|² = |a|² + 2(a · b) + |b|² (Expansion of Sum)

|a - b|² = |a|² - 2(a · b) + |b|² (Expansion of Difference)

|a + b|² + |a - b|² = 2(|a|² + |b|²) (Parallelogram Law)

a · b = (|a + b|² - |a - b|²)/4 (Polarization Identity)

Vector Calculus

d/dt[r(t)] = r'(t) = (dx/dt, dy/dt, dz/dt) (Vector Derivative)

d/dt[u(t) + v(t)] = u'(t) + v'(t) (Sum Rule for Vectors)

d/dt[f(t)u(t)] = f'(t)u(t) + f(t)u'(t) (Scalar Product Rule)

d/dt[u(t) · v(t)] = u'(t) · v(t) + u(t) · v'(t) (Dot Product Rule)

d/dt[u(t) × v(t)] = u'(t) × v(t) + u(t) × v'(t) (Cross Product Rule)

|r'(t)| = ds/dt (Speed as Derivative of Arc Length)

T(t) = r'(t)/|r'(t)| (Unit Tangent Vector)

N(t) = T'(t)/|T'(t)| (Unit Normal Vector)

κ = |T'(t)|/|r'(t)| = |r'(t) × r''(t)|/|r'(t)|³ (Curvature)

B(t) = T(t) × N(t) (Unit Binormal Vector)

Gradient, Divergence, and Curl

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) (Gradient)

∇ · F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z (Divergence)

∇ × F = (∂F₃/∂y - ∂F₂/∂z, ∂F₁/∂z - ∂F₃/∂x, ∂F₂/∂x - ∂F₁/∂y) (Curl)

∇ × (∇f) = 0 (Curl of Gradient is Zero)

∇ · (∇ × F) = 0 (Divergence of Curl is Zero)

∇²f = ∇ · ∇f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² (Laplacian)

Vector Space Axioms

u + v = v + u (Commutativity of Vector Addition)

(u + v) + w = u + (v + w) (Associativity of Vector Addition)

u + 0 = u (Additive Identity)

u + (-u) = 0 (Additive Inverse)

c(u + v) = cu + cv (Distributive Law)

(c + d)u = cu + du (Distributive Law)

c(du) = (cd)u (Associativity of Scalar Multiplication)

1u = u (Scalar Identity)

Inner Products, Norms, & Geometry

⟨u, v⟩ = ⟨v, u⟩ (Symmetry of Real Inner Product)

⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩ (Additivity of Inner Product)

⟨cu, v⟩ = c⟨u, v⟩ (Homogeneity of Inner Product)

⟨v, v⟩ ≥ 0, and ⟨v, v⟩ = 0 ⇔ v = 0 (Positive-Definiteness)

||v|| = sqrt(⟨v, v⟩) (Norm from Inner Product)

|⟨u, v⟩| ≤ ||u|| ||v|| (Cauchy-Schwarz Inequality)

||u + v|| ≤ ||u|| + ||v|| (Triangle Inequality)

d(u, v) = ||u - v|| (Distance Formula)

u . v = u1*v1 + ... + un*vn (Standard Dot Product)

u . v = ||u|| * ||v|| * cos(θ) (Geometric Definition of Dot Product)

u . v = 0 ⇔ u ⊥ v (Orthogonality Condition)

proj_u(v) = (⟨v, u⟩ / ⟨u, u⟩) * u (Orthogonal Projection)

u x v = (u2*v3 - u3*v2, u3*v1 - u1*v3, u1*v2 - u2*v1) (Cross Product in R^3)

u . (v x w) = det([u v w]) (Scalar Triple Product)

Matrix Algebra

(A*B)ij = Σ(k) Aik * Bkj (Definition of Matrix Multiplication)

A + B = B + A (Commutativity of Matrix Addition)

A(BC) = (AB)C (Associativity of Matrix Multiplication)

A(B + C) = AB + AC (Left Distributive Law)

(A + B)C = AC + BC (Right Distributive Law)

A*I = I*A = A (Multiplicative Identity)

(A^T)ij = Aji (Definition of Transpose)

(A + B)^T = A^T + B^T (Transpose of a Sum)

(A*B)^T = B^T * A^T (Transpose of a Product)

tr(A) = Σ(i) Aii (Definition of Trace)

tr(A + B) = tr(A) + tr(B) (Trace Linearity)

tr(A*B) = tr(B*A) (Cyclic Property of Trace)

Determinants & Inverses

det(A) = Σ(j) (-1)^(i+j) * Aij * Mij (Laplace/Cofactor Expansion)

det(A*B) = det(A) * det(B) (Multiplicative Property)

det(A^T) = det(A) (Determinant of Transpose)

det(kA) = k^n * det(A) (Determinant of Scalar Multiple, A is n x n)

det(I) = 1 (Determinant of Identity)

A is invertible ⇔ det(A) ≠ 0 (Invertibility Condition)

A * A^(-1) = A^(-1) * A = I (Definition of Matrix Inverse)

det(A^(-1)) = 1 / det(A) (Determinant of Inverse)

(A*B)^(-1) = B^(-1) * A^(-1) (Inverse of a Product)

(A^T)^(-1) = (A^(-1))^T (Inverse of a Transpose)

Linear Systems, Subspaces & Rank

A*x = b (General Linear System)

A*x = b has solution ⇔ rank(A) = rank([A|b]) (Rouché-Capelli Theorem)

Null(A) = {x | A*x = 0} (Definition of Null Space / Kernel)

Col(A) = {A*x | x in R^n} (Definition of Column Space / Image)

rank(A) = dim(Col(A)) (Definition of Rank)

nullity(A) = dim(Null(A)) (Definition of Nullity)

rank(A) + nullity(A) = n (Rank-Nullity Theorem, A is m x n)

rank(A) = rank(A^T) (Rank Equality)

Col(A) ⊥ Null(A^T) (Fundamental Subspace Orthogonality)

Row(A) ⊥ Null(A) (Fundamental Subspace Orthogonality)

Linear Transformations

T(u + v) = T(u) + T(v) (Additivity)

T(cu) = cT(u) (Homogeneity)

T(x) = A*x (Matrix Representation)

dim(Ker(T)) + dim(Im(T)) = dim(V) (Rank-Nullity Theorem for Transformations)

[x]_C = P_C<-B * [x]_B (Change of Basis Formula)

Eigenvalues & Eigenvectors

A*v = λ*v (Eigenvalue-Eigenvector Equation)

det(A - λ*I) = 0 (Characteristic Equation)

E_λ = Null(A - λ*I) (Eigenspace for λ)

tr(A) = Σ(i) λi (Trace as Sum of Eigenvalues)

det(A) = Π(i) λi (Determinant as Product of Eigenvalues)

A = P*D*P^(-1) (Diagonalization)

p(A) = 0 (Cayley-Hamilton Theorem)

Decompositions & Spectral Theory

Q^T * Q = Q * Q^T = I (Orthogonal Matrix Definition)

A^T * A * x_hat = A^T * b (Normal Equation for Least Squares)

A = Q*Λ*Q^T (Spectral Theorem for Real Symmetric Matrices)

A = L*U (LU Decomposition)

A = Q*R (QR Decomposition)

A = U*Σ*V^T (Singular Value Decomposition - SVD)

Core Algebraic Structures

(G, *) is a Group (Associativity, ∃e Identity, ∃a⁻¹ Inverse)

a * b = b * a (Commutative Law / Abelian Group)

(R, +, *) is a Ring ((R, +) is Abelian Group, (R, \*) is Associative)

a*(b + c) = a*b + a*c and (a + b)*c = a*c + b*c (Distributive Laws)

a * b = 0 ⇒ a = 0 or b = 0 (Integral Domain Property)

(F, +, *) is a Field (Commutative Ring where U(F) = F \\ {0})

U(R) = { u ∈ R | ∃u⁻¹ } (Group of Units of a Ring)

Group Theory

(a * b)⁻¹ = b⁻¹ * a⁻¹ (Inverse of a Product / Socks-Shoes Property)

a * b = a * c ⇒ b = c (Left Cancellation Law)

H ≤ G if a * b⁻¹ ∈ H for all a, b ∈ H (One-Step Subgroup Test)

⟨a⟩ = { aⁿ | n ∈ ℤ } (Cyclic Subgroup generated by a)

ord(a) = |⟨a⟩| (Order of an Element)

aH = { ah | h ∈ H } (Left Coset of H)

|G| = |H| * |G:H| (Lagrange's Theorem)

|G:H| = |G| / |H| (Index of a Subgroup)

ord(a) divides |G| (Corollary to Lagrange's Theorem)

a^|G| = e (Fermat's Little Theorem for Groups)

|G| = p (prime) ⇒ G ≅ ℤ\_p (Groups of Prime Order)

ℤ\_m × ℤ\_n ≅ ℤ\_(mn) if gcd(m, n) = 1 (Chinese Remainder Theorem for Groups)

N ◁ G if g * n * g⁻¹ ∈ N for all g ∈ G, n ∈ N (Normal Subgroup)

gN = Ng for all g ∈ G (Left Cosets equal Right Cosets)

G/N = { gN | g ∈ G } (Quotient Group / Factor Group)

(aN) * (bN) = (ab)N (Operation in G/N)

φ(a * b) = φ(a) * φ(b) (Group Homomorphism)

Ker(φ) = { g ∈ G | φ(g) = e\_H } (Kernel of Homomorphism)

Im(φ) = { φ(g) | g ∈ G } (Image of Homomorphism)

G / Ker(φ) ≅ Im(φ) (First Isomorphism Theorem for Groups)

(H * N) / N ≅ H / (H ∩ N) (Second Isomorphism Theorem for Groups)

(G / N) / (H / N) ≅ G / H (Third Isomorphism Theorem for Groups)

Z(G) = { g ∈ G | g*x = x*g for all x ∈ G } (Center of a Group)

g ⋅ x (Group Action of G on X)

Orb(x) = { g ⋅ x | g ∈ G } (Orbit of x)

Stab(x) = { g ∈ G | g ⋅ x = x } (Stabilizer of x)

|G| = |Orb(x)| * |Stab(x)| (Orbit-Stabilizer Theorem)

|G| = |Z(G)| + Σ |G:C(a)| (Class Equation)

∃H ≤ G with |H| = p^k where |G| = p^k * m (Sylow's First Theorem)

All Sylow p-subgroups are conjugate (Sylow's Second Theorem)

n\_p ≡ 1 (mod p) and n\_p divides m (Sylow's Third Theorem)

G ≅ ℤ\_p₁^k₁ × ℤ\_p₂^k₂ × ... (Fundamental Theorem of Finite Abelian Groups)

Ring Theory

I ◁ R (Ideal Definition: (I, +) ≤ (R, +) and r*x, x*r ∈ I)

R/I = { r + I | r ∈ R } (Quotient Ring / Factor Ring)

(a + I)(b + I) = (a*b) + I (Multiplication in R/I)

φ(a + b) = φ(a) + φ(b) and φ(a * b) = φ(a) * φ(b) (Ring Homomorphism)

R / Ker(φ) ≅ Im(φ) (First Isomorphism Theorem for Rings)

I is Prime ⇔ R/I is an Integral Domain (Prime Ideal Property)

I is Maximal ⇔ R/I is a Field (Maximal Ideal Property)

R is a PID if every ideal is principal (Principal Ideal Domain)

R is a UFD if factorization is unique (Unique Factorization Domain)

R is PID ⇒ R is UFD (PID implies UFD)

gcd(a, b) = sa + tb (Bézout's Identity)

Fundamental Counting Principles

|A ∪ B| = |A| + |B| (Rule of Sum for disjoint sets)

|A × B| = |A| ⋅ |B| (Rule of Product)

Permutations & Combinations

n! = n ⋅ (n-1) ⋅ ... ⋅ 2 ⋅ 1 (Factorial)

0! = 1 (Definition of Zero Factorial)

P(n, k) = n! / (n-k)! (k-Permutations of n / Partial Permutation)

C(n, k) = n! / (k! ⋅ (n-k)!) (Combinations / Binomial Coefficient)

n^k (Permutations with Replacement / k-Tuples)

C(n+k-1, k) = C(n+k-1, n-1) (Combinations with Replacement / Stars and Bars)

(n! / (k_1! ⋅ k_2! ⋅ ... ⋅ k_m!)) (Permutations with Repetition / Multinomial Coefficient)

Binomial & Multinomial Identities

C(n, k) = C(n, n-k) (Symmetry Identity)

C(n, k) = C(n-1, k-1) + C(n-1, k) (Pascal's Identity)

(x + y)^n = Σ_{k=0 to n} C(n, k) ⋅ x^(n-k) ⋅ y^k (Binomial Theorem)

Σ_{k=0 to n} C(n, k) = 2^n (Sum of Binomial Coefficients / Row Sum of Pascal's Triangle)

Σ_{k=0 to n} (-1)^k ⋅ C(n, k) = 0 (Alternating Sum of Binomial Coefficients)

Σ_{k=0 to r} C(m, k) ⋅ C(n, r-k) = C(m+n, r) (Vandermonde's Identity)

Σ_{k=0 to n} C(n, k)^2 = C(2n, n) (Central Binomial Coefficient Identity)

Σ_{i=k to n} C(i, k) = C(n+1, k+1) (Hockey-Stick Identity / Christmas Stocking Theorem)

k ⋅ C(n, k) = n ⋅ C(n-1, k-1) (Absorption Identity)

(x_1 + ... + x_m)^n = Σ_{k_1+...+k_m=n} (n! / (k_1! ⋅ ... ⋅ k_m!)) ⋅ x_1^{k_1} ⋅ ... ⋅ x_m^{k_m} (Multinomial Theorem)

Inclusion-Exclusion, Derangements & Pigeonhole Principle

|A ∪ B| = |A| + |B| - |A ∩ B| (Inclusion-Exclusion for 2 sets)

|∪_{i=1 to n} A_i| = Σ|A_i| - Σ_{i < j }|A_i ∩ A_j| + ... + (-1)^{n-1} |A_1 ∩ ... ∩ A_n| (Principle of Inclusion-Exclusion)

D_n = n! ⋅ Σ_{k=0 to n} ((-1)^k / k!) (Derangements / Subfactorial)

D_n = (n-1) ⋅ (D_{n-1} + D_{n-2}) (Derangement Recurrence)

D_n ≈ n! / e (Derangement Approximation)

If n objects are in m boxes, at least one box contains ⌈n/m⌉ objects. (Pigeonhole Principle)

Recurrence Relations & Special Numbers

F_n = F_{n-1} + F_{n-2} with F_0 = 0, F_1 = 1 (Fibonacci Numbers - Recurrence)

F_n = (φ^n - ψ^n) / sqrt(5) where φ = (1+sqrt(5))/2, ψ = (1-sqrt(5))/2 (Binet's Formula)

gcd(F_m, F_n) = F_{gcd(m,n)} (Fibonacci GCD Property)

C_n = (1 / (n+1)) ⋅ C(2n, n) (Catalan Numbers - Closed Form)

C_n = Σ_{i=0 to n-1} C_i ⋅ C_{n-1-i} (Catalan Numbers - Recurrence)

C_n = C(2n, n) - C(2n, n-1) (Catalan Numbers - Alternative Form)

S(n, k) = S(n-1, k-1) + k ⋅ S(n-1, k) (Stirling Numbers of the Second Kind - Recurrence)

s(n, k) = s(n-1, k-1) - (n-1) ⋅ s(n-1, k) (Stirling Numbers of the First Kind - Recurrence)

x^n = Σ_{k=0 to n} S(n, k) ⋅ x(x-1)...(x-k+1) (Power to Falling Factorial)

x(x-1)...(x-n+1) = Σ_{k=0 to n} s(n, k) ⋅ x^k (Falling Factorial to Power)

B_n = Σ_{k=0 to n} S(n, k) (Bell Numbers / Relation to Stirling Numbers)

B_{n+1} = Σ_{k=0 to n} C(n, k) ⋅ B_k (Bell Numbers - Recurrence)

p(n) = number of ways to write n as a sum of positive integers (Partition Function)

p(n) = p(n-1) + p(n-2) - p(n-5) - p(n-7) + ... (Euler's Pentagonal Number Theorem)

Generating Functions

G(x) = Σ_{n=0 to ∞} a_n ⋅ x^n (Ordinary Generating Function - OGF)

E(x) = Σ_{n=0 to ∞} a_n ⋅ (x^n / n!) (Exponential Generating Function - EGF)

1 / (1 - x) = Σ_{n=0 to ∞} x^n (Geometric Series Generating Function)

Σ_{n=0 to ∞} p(n)x^n = Π_{k=1 to ∞} (1 / (1 - x^k)) (Generating Function for Partitions)

Graph Theory & Enumeration

Σ_{v ∈ V} deg(v) = 2|E| (Handshaking Lemma / Degree Sum Formula)

|V| - |E| + |F| = 2 (Euler's Formula for Planar Graphs)

A tree with n vertices has n - 1 edges. (Property of Trees)

n^(n-2) (Number of labeled trees on n vertices) (Cayley's Formula)

det(L_reduced) (Number of spanning trees of a graph) (Matrix-Tree Theorem)

|E| = n(n-1) / 2 (Number of Edges in Complete Graph K_n)

Ramsey Theory

R(r, s) ≤ C(r+s-2, r-1) (Ramsey Number Upper Bound)

Basic Probability Axioms & Rules

P(∅) = 0, P(S) = 1 (Probability Axioms)

0 ≤ P(A) ≤ 1 (Nonnegativity / Bounds)

P(Aᶜ) = 1 − P(A) (Complement Rule)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (Inclusion–Exclusion)

If A ∩ B = ∅ then P(A ∪ B) = P(A) + P(B) (Finite Additivity / Disjoint)

P(A | B) = P(A ∩ B) / P(B), P(B) > 0 (Conditional Probability)

P(A ∩ B) = P(A | B) · P(B) (Multiplication Rule)

P(A₁ ∩ … ∩ Aₙ) = Π_{i=1..n} P(Aᵢ | A₁ ∩ … ∩ A_{i−1}) (Chain Rule)

If independent: P(A ∩ B) = P(A)·P(B) and more generally Π P(Aᵢ) (Independence)

P(B) = Σ_i P(B | Aᵢ)·P(Aᵢ) over partition {Aᵢ} (Law of Total Probability)

P(A | B) = P(B | A)·P(A) / P(B) (Bayes' Theorem)

Combinatorics

n! = n·(n−1)·…·1 ; C(n,k) = n!/(k!(n−k)!) (Factorial & Binomial Coefficient)

P(n,k) = n!/(n−k)! (Permutations)

Random Variables & Distributions

F_X(x) = P(X ≤ x) (Cumulative Distribution Function, CDF)

p_X(x) = P(X = x), Σ_x p_X(x) = 1 (Probability Mass Function, PMF)

f_X(x) = dF_X(x)/dx (Probability Density Function, PDF)

P(a < X ≤ b) = F_X(b) − F_X(a) (Interval Probability via CDF)

f_{X,Y}(x,y) = joint pdf/pmf ; f_X(x) = ∫ f_{X,Y}(x,y) dy (Joint & Marginal)

f_{X|Y}(x|y) = f_{X,Y}(x,y) / f_Y(y) (Conditional PDF / PMF)

Expectation, Variance & Covariance

E[X] = Σ x p(x) or ∫ x f(x) dx (Expectation / Mean)

E[g(X)] = Σ g(x)p(x) or ∫ g(x)f(x) dx (Law of the Unconscious Statistician)

E[aX + b] = a E[X] + b ; E[X + Y] = E[X] + E[Y] (Linearity of Expectation)

Var(X) = E[(X − μ)^2] (Variance, definition)

Var(X) = E[X^2] − (E[X])^2 (Computational Variance)

Var(aX + b) = a^2 Var(X) ; Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y) (Variance Rules)

Cov(X,Y) = E[(X − μ_X)(Y − μ_Y)] = E[XY] − E[X]E[Y] (Covariance)

ρ(X,Y) = Cov(X,Y) / (σ_X σ_Y) (Pearson Correlation Coefficient)

Moment Generating Functions & Inequalities

M_X(t) = E[e^{tX}] (Moment Generating Function, MGF)

M_{X+Y}(t) = M_X(t)·M_Y(t) if X and Y independent (MGF Product Property)

Markov: P(X ≥ a) ≤ E[X]/a for X ≥ 0 (Markov's Inequality)

Chebyshev: P(|X − μ| ≥ kσ) ≤ 1/k^2 (Chebyshev's Inequality)

Limit Theorems

Weak Law: X̄_n → μ in probability as n → ∞ (Weak Law of Large Numbers)

Strong Law: X̄_n → μ almost surely as n → ∞ (Strong Law of Large Numbers)

CLT: (Σ_{i=1..n} X_i − nμ) / (σ√n) → N(0,1) (Central Limit Theorem)

Common Distributions

Bernoulli(p): P(X=1)=p, P(X=0)=1−p ; E[X]=p, Var(X)=p(1−p) (Bernoulli)

Binomial(n,p): P(X=k)=C(n,k)p^k(1−p)^{n−k} ; E=np, Var=np(1−p) (Binomial)

Poisson(λ): P(X=k)=e^{−λ} λ^k / k! ; E=Var=λ (Poisson)

Geometric(p): P(X=k)=(1−p)^{k−1} p, k=1,2,... ; E=1/p (Geometric)

Negative Binomial(r,p): P(X=k)=C(k−1,r−1) p^r (1−p)^{k−r} (Negative Binomial)

Uniform(a,b): f(x)=1/(b−a), a ≤ x ≤ b ; E=(a+b)/2, Var=(b−a)^2/12 (Continuous Uniform)

Exponential(λ): f(x)=λ e^{−λ x}, x ≥ 0 ; E=1/λ, memoryless: P(X>s+t|X>s)=P(X>t) (Exponential)

Normal(μ,σ^2): f(x)=(1/(σ√(2π))) e^{−(x−μ)^2/(2σ^2)} ; Z=(X−μ)/σ ~ N(0,1) (Normal / Gaussian)

Sum of independent normals is normal (Normal closure property)

χ^2_k distribution: sum of k independent Z_i^2 ; (n−1) s^2 / σ^2 ~ χ^2_{n−1} (Chi-Square)

t_{ν} distribution: T = (X̄ − μ) / (s/√n) ∼ t_{n−1} (Student's t)

F_{d1,d2} distribution: ratio of scaled chi-squares (F Distribution)

Sampling & Inference

Sample mean X̄ = (1/n) Σ X_i ; sample variance s^2 = (1/(n−1)) Σ (X_i − X̄)^2 (Sample Statistics)

SE(X̄) = σ / √n ; estimated SE = s / √n (Standard Error of the Mean)

Confidence Intervals: X̄ ± z_{α/2}·(σ/√n) (σ known) ; X̄ ± t_{α/2,n−1}·(s/√n) (σ unknown) (Confidence Intervals)

Likelihood L(θ) = Π_i f(x_i; θ) ; θ̂_MLE = argmax_θ L(θ) ; Fisher Information I(θ) = −E[∂^2 log f/∂θ^2] ; Var(θ̂) ≥ 1/(n I(θ)) (Likelihood, MLE, Fisher Info, Cramér–Rao)