Math Learning Log

I’m revisiting and expanding my math foundations, starting with core notation and concepts I want to be fluent with.

Strategy

• Work through each section until the symbols feel natural.
• Use examples in my notes (not just memorizing meanings).
• Check items off once I can both recognize and use them in context.

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Basic arithmetic & relations

• = – equals
• ≠ – not equal
• ≈ – approximately equal
• ≡ – identically equal / congruent (depends on context)
• < – less than
• > – greater than
• ≤ – less than or equal to
• ≥ – greater than or equal to
• + – plus
• − – minus
• ± – plus or minus
• × – times / multiply
• · – dot multiply (also used for dot product)
• ÷ or / – divide
• ^ – power / exponent (e.g. x^2)
• √ – square root
• ∛ – cube root

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Algebra & functions

• x, y, z – generic variables
• a, b, c, k – generic constants
• f(x), g(x) – functions of x
• f: ℝ → ℝ – “maps to” arrow (function from reals to reals)
• ∘ – function composition (f ∘ g)
• f⁻¹ – inverse function of f
• ( ... ) – grouping / parentheses
• [ ... ] – grouping / intervals / indexing
• { ... } – set / collection of things
• : or | – “such that” (in set notation or conditions)

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Sets & number systems

• ∈ – “is an element of” (x ∈ A)
• ∉ – “is not an element of”
• ⊂ – proper subset
• ⊆ – subset (possibly equal)
• ⊃ – proper superset
• ⊇ – superset (possibly equal)
• ∅ – empty set
• ∪ – union (A ∪ B)
• ∩ – intersection (A ∩ B)
• ℕ – natural numbers
• ℤ – integers
• ℚ – rational numbers
• ℝ – real numbers
• ℂ – complex numbers

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Intervals

• (a, b) – open interval (a < x < b)
• [a, b] – closed interval (a ≤ x ≤ b)
• (a, b] – half-open interval
• [a, b) – half-open interval

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Exponents & logs

• a^b – a to the power b
• e – Euler’s number (~2.718)
• ln(x) – natural logarithm (base e)
• log(x) – logarithm (base depends on context)
• exp(x) – e^x (exponential function notation)

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Limits & calculus core

• lim – limit
• x → a – “x tends to a”
• Δx – small change in x (discrete)
• d/dx – derivative with respect to x
• dy/dx – derivative of y with respect to x
• df/dx – derivative of f with respect to x
• ∂/∂x – partial derivative with respect to x
• ∫ – integral (indefinite/definite depending on context)
• ∫_a^b – integral from a to b
• dx – “with respect to x” (integration/derivative variable)

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Summation & products

• Σ – summation sign
• ∑_{i=1}^n – sum from i = 1 to n
• Π – product sign
• ∏_{i=1}^n – product from i = 1 to n

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Logic & quantifiers

• ¬ – not (negation)
• ∧ – and
• ∨ – or
• ⇒ – implies
• ⇔ – if and only if (iff, equivalence)
• ∀ – “for all”
• ∃ – “there exists”
• ⊤ – true
• ⊥ – false / contradiction

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Probability & statistics

• P(A) – probability of event A
• P(A | B) – probability of A given B
• E[X] – expected value
• Var(X) – variance
• Cov(X,Y) – covariance
• μ – population mean
• σ – standard deviation
• σ² – variance
• X ~ N(μ, σ²) – “X is distributed as normal with mean μ and variance σ²”

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Linear algebra & vectors

• ⃗v or v – vector
• ‖v‖ – norm / length of vector v
• A – matrix (often uppercase)
• Aᵀ – transpose of A
• det(A) or |A| – determinant of matrix A
• I – identity matrix
• 0 – zero vector or zero matrix (context-dependent)

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Asymptotics / CS

• O(n) – big-O (upper bound on growth rate)
• Θ(n) – theta (tight bound)
• Ω(n) – omega (lower bound)
• o(n) – little-o (strictly smaller order)
• ≪ – “much less than” (informal)

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Other common squiggles

• ∝ – proportional to
• ∞ – infinity
• argmax – value where a function reaches its maximum
• argmin – value where a function reaches its minimum
• ⊕ – direct sum / XOR (context-dependent)
• ⊗ – tensor / Kronecker product (context-dependent)
• ∴ – therefore
• ∵ – because

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Notes

I’ll use this page to:

• Add quick examples for symbols as I practice
• Write down any confusions and how I resolved them
• Track my comfort level over time (by checking items off)