SET THEORY
Basic Set Operations
| Symbol | Name | Read As | Example | Meaning |
|---|
| ∈ | Element of | "is an element of" | 3 ∈ {1,2,3} | 3 is in the set |
| ∉ | Not element of | "is not an element of" | 5 ∉ {1,2,3} | 5 is not in the set |
| ⊆ | Subset | "is a subset of" | {1,2} ⊆ {1,2,3} | Every element of first is in second (allows equality) |
| ⊊ | Proper subset | "is a proper subset of" | {1,2} ⊊ {1,2,3} | Subset but strictly not equal |
| ⊇ | Superset | "is a superset of" | {1,2,3} ⊇ {1,2} | Contains all elements of second set (allows equality) |
| ⊋ | Proper superset | "is a proper superset of" | {1,2,3} ⊋ {1,2} | Superset but strictly not equal |
| ∪ | Union | "union" | A ∪ B | Everything in A or B or both |
| ∩ | Intersection | "intersection" | A ∩ B | Only what's in both A and B |
| ∖ or - | Set difference | "minus" or "set difference" | A ∖ B | Elements in A but not in B |
| ∅ or {} | Empty set | "the empty set" | A ∩ B = ∅ | Set with no elements |
| 𝒫(A) or P(A) | Power set | "power set of A" | 𝒫({1,2}) | Set of all subsets of A |
| ∣ or : | Such that | "such that" | {x ∣ x > 0} | Defines set membership condition |
∣A∣ or #A | Cardinality | "cardinality of A" | ∣{1,2,3}∣ = 3 | Number of elements in set A |
| × | Cartesian product | "cross" | A × B | All ordered pairs (a,b) where a ∈ A, b ∈ B |
| Aᶜ or A' or Ā | Complement | "A complement" | Aᶜ | Everything not in A (within universal set) |
| ⊎ | Disjoint union | "disjoint union" | A ⊎ B | Union where A and B are assumed disjoint, or tagged to force disjointness |
| △ | Symmetric difference | "symmetric difference" | A △ B | (A ∖ B) ∪ (B ∖ A) |
Note on subset notation: Convention varies by text. In this guide, ⊆ allows equality while ⊊ requires strict inequality. Some texts use ⊂ for subset (possibly equal) and ⊊ for proper subset.
Common Number Sets
| Symbol | Name | Read As | Meaning |
|---|
| ℕ | Natural numbers | "the naturals" | {0, 1, 2, 3, ...} or {1, 2, 3, ...} (convention varies) |
| ℕ₀ | Naturals with zero | "N-zero" or "N-naught" | {0, 1, 2, 3, ...} (explicit zero inclusion) |
| ℕ* or ℕ⁺ | Positive naturals | "N-star" or "N-plus" | {1, 2, 3, ...} (explicit zero exclusion) |
| ℤ | Integers | "the integers" | {..., -2, -1, 0, 1, 2, ...} |
| ℚ | Rational numbers | "the rationals" | All fractions p/q where p,q are integers, q ≠ 0 |
| ℝ | Real numbers | "the reals" | All numbers on number line |
| ℂ | Complex numbers | "the complex numbers" | Numbers of form a + bi where i² = -1 |
| ℤ⁺ | Positive integers | "positive integers" | {1, 2, 3, ...} |
| ℝ⁺ | Positive reals | "positive reals" | All positive real numbers |
| ℝ⁻ | Negative reals | "negative reals" | All negative real numbers |
| ℚ⁺ | Positive rationals | "positive rationals" | Positive fractions |
| ℙ | Prime numbers | "the primes" | {2, 3, 5, 7, 11, ...} |
LOGIC AND BOOLEAN ALGEBRA
Logical Operators
| Symbol | Name | Read As | Truth Condition | Example |
|---|
| ¬ or ~ or ! | Negation/NOT | "not" | Opposite of P | ¬P is true when P is false |
| ∧ or & or · | Conjunction/AND | "and" | Both true | P ∧ Q true only when both P and Q are true |
| ∨ or ∣ or + | Disjunction/OR | "or" | At least one true | P ∨ Q true when P or Q or both are true |
| ⊕ or ⊻ | Exclusive OR/XOR | "xor" or "exclusive or" | Exactly one true | P ⊕ Q true when exactly one is true |
| → or ⇒ or ⊃ | Implication | "implies" or "if...then" | False only when P true and Q false | P → Q |
| ↔ or ⇔ or ≡ | Biconditional/IFF | "if and only if" or "iff" | Both same value | P ↔ Q true when both match |
| ⊤ or T or 1 | Tautology/True | "true" or "top" | Always true | Universal truth |
| ⊥ or F or 0 | Contradiction/False | "false" or "bottom" | Always false | Universal falsehood (in logic context) |
| ⊨ | Semantic entailment | "models" or "entails" | Logical consequence | A ⊨ B means B is true whenever A is |
| ⊢ | Syntactic entailment | "proves" or "derives" | Provable from | A ⊢ B means B is provable from A |
| ⊬ | Does not prove | "does not prove" | Not provable | A ⊬ B means B not provable from A |
| ⊭ | Does not model | "does not model" | Not a consequence | A ⊭ B means B not true when A is |
Note on ⊥: This symbol is heavily overloaded. See "Context-Dependent Symbols" section below for all meanings.
QUANTIFIERS
| Symbol | Name | Read As | Example | Meaning |
|---|
| ∀ | Universal quantifier | "for all" or "for every" | ∀x ∈ S, P(x) | Statement is true for every x in S |
| ∃ | Existential quantifier | "there exists" | ∃x ∈ S, P(x) | Statement is true for at least one x in S |
| ∃! | Unique existence | "there exists a unique" | ∃!x, P(x) | Exactly one x makes P(x) true |
| ∄ | Does not exist | "there does not exist" | ∄x ∈ S, P(x) | No x in S makes P(x) true |
RELATIONS AND FUNCTIONS
Relations
| Symbol | Name | Read As | Example | Meaning |
|---|
| R ⊆ A×B | Relation | "R is a relation from A to B" | R ⊆ ℕ×ℕ | R is a set of ordered pairs |
| aRb | Infix notation | "a is related to b by R" | 3R5 | (a,b) ∈ R |
| R⁻¹ | Inverse relation | "R inverse" | If aRb then bR⁻¹a | Reverse all pairs |
| R ∘ S | Composition | "R composed with S" | (a,c) ∈ R∘S | Apply S first, then R: (a,c) ∈ R∘S iff ∃b: (a,b) ∈ S and (b,c) ∈ R |
| dom(R) | Domain | "domain of R" | dom(R) | Set of all first elements |
| ran(R) or img(R) | Range/Image | "range of R" | ran(R) | Set of all second elements |
| cod(R) | Codomain | "codomain of R" | cod(R) | Target set of relation |
Functions
| Symbol | Name | Read As | Example | Meaning |
|---|
| f: A → B | Function | "f from A to B" | f: ℕ → ℝ | Function mapping A to B |
| f(x) | Function application | "f of x" | f(3) = 9 | Value of f at x |
| f ∘ g | Function composition | "f composed with g" | (f∘g)(x) = f(g(x)) | Apply g first, then f |
| f⁻¹ | Inverse function | "f inverse" | f⁻¹(y) = x | Reverses the function (requires f bijective) |
| f⁻¹(B) | Preimage | "f inverse of B" | f⁻¹(B) | Set of inputs mapping to B (defined for any function) |
| ↦ | Maps to | "maps to" | x ↦ x² | Shows mapping |
| ↪ | Injection/Embedding | "injects into" or "embeds in" | f: A ↪ B | One-to-one function (injective) |
| λ | Lambda | "lambda" | λx.x+1 | Anonymous function definition |
| idₐ or id | Identity function | "identity on A" | id(x) = x | Maps each element to itself |
| ⊥ | Undefined/Bottom | "undefined" or "bottom" | f(x) = ⊥ | Function undefined at x (partial function context) |
| argmax | Argument maximum | "argmax" | argmax f(x) | Value of x that maximizes f |
| argmin | Argument minimum | "argmin" | argmin f(x) | Value of x that minimizes f |
COMPARISON AND ORDERING
| Symbol | Name | Read As | Meaning |
|---|
| = | Equals | "equals" | Exactly the same value |
| ≠ or ≢ | Not equal | "not equal to" | Different values |
| < | Less than | "less than" | Strictly smaller |
| > | Greater than | "greater than" | Strictly larger |
| ≤ or ⩽ | Less than or equal | "less than or equal to" | Smaller or same |
| ≥ or ⩾ | Greater than or equal | "greater than or equal to" | Larger or same |
| ≪ | Much less than | "much less than" | Significantly smaller |
| ≫ | Much greater than | "much greater than" | Significantly larger |
| ≈ | Approximately equal | "approximately equal to" | Close to |
| ≡ | Equivalent to | "equivalent to" or "identical to" | Context-dependent: logical equivalence, congruence (mod n), or identity |
| ≢ | Not equivalent | "not equivalent to" | Not the same structure |
| ∝ | Proportional to | "proportional to" | Constant multiple relationship |
| := or ≔ | Definition | "is defined as" | Defining a new term |
| ≜ | Equal by definition | "equal by definition" | Defined to be equal |
| ≝ | Defined as | "defined as" | Definition symbol |
| ≺ | Precedes | "precedes" | Comes before in ordering |
| ≻ | Succeeds | "succeeds" | Comes after in ordering |
| ⪯ | Precedes or equals | "precedes or equals" | Comes before or equal |
| ⪰ | Succeeds or equals | "succeeds or equals" | Comes after or equal |
ARITHMETIC AND ALGEBRA
Basic Operations
| Symbol | Name | Read As | Example |
|---|
| + | Plus/Addition | "plus" | 2 + 3 = 5 |
| - | Minus/Subtraction | "minus" | 5 - 2 = 3 |
| × or · or * | Times/Multiplication | "times" | 2 × 3 = 6 |
| ÷ or / | Division | "divided by" | 6 ÷ 2 = 3 |
| ± | Plus-minus | "plus or minus" | x = ±3 |
| ∓ | Minus-plus | "minus or plus" | Opposite of ± |
Exponents and Roots
| Symbol | Name | Read As | Example |
|---|
| xⁿ | Exponentiation | "x to the n" or "x to the power of n" | 2³ = 8 |
| √x or x^(1/2) | Square root | "square root of x" | √9 = 3 |
| ∛x or x^(1/3) | Cube root | "cube root of x" | ∛8 = 2 |
| ⁿ√x or x^(1/n) | nth root | "nth root of x" | ⁴√16 = 2 |
| ∣x∣ | Absolute value | "absolute value of x" | ∣-5∣ = 5 (for scalars) |
| ⌊x⌋ | Floor | "floor of x" | ⌊3.7⌋ = 3 |
| ⌈x⌉ | Ceiling | "ceiling of x" | ⌈3.2⌉ = 4 |
| n! | Factorial | "n factorial" | 5! = 120 |
| n!! | Double factorial | "n double factorial" | 8!! = 8·6·4·2 = 384 |
Modular Arithmetic
| Symbol | Name | Read As | Example | Meaning |
|---|
| mod | Modulo | "mod" | 7 mod 3 = 1 | Remainder after division |
| ≡ (mod n) | Congruent modulo | "congruent to ... modulo n" | 7 ≡ 1 (mod 3) | Same remainder when divided by n |
| a ∣ b | Divides | "a divides b" | 3 ∣ 12 | a divides b evenly (b is a multiple of a) |
| a ∤ b | Does not divide | "a does not divide b" | 3 ∤ 10 | a doesn't divide b evenly |
| gcd(a,b) | Greatest common divisor | "gcd of a and b" | gcd(12,8) = 4 | Largest number dividing both |
| lcm(a,b) | Least common multiple | "lcm of a and b" | lcm(12,8) = 24 | Smallest number both divide |
SUMMATION AND PRODUCTS
| Symbol | Name | Read As | Example | Meaning |
|---|
| ∑ | Summation | "sum from ... to ..." | ∑(i=1 to n) i | Add up all terms |
| ∏ | Product | "product from ... to ..." | ∏(i=1 to n) i | Multiply all terms (equals n!) |
| ∫ | Integral | "integral of" | ∫f(x)dx | Area under curve (calculus) |
| ∂ | Partial derivative | "partial" | ∂f/∂x | Derivative with respect to one variable |
| ∇ | Nabla/Gradient | "del" or "gradient" | ∇f | Vector of partial derivatives |
| Δ | Delta/Change | "delta" or "change in" | Δx | Change in x |
| d/dx | Derivative | "d by dx" | d/dx f(x) | Rate of change |
| ∮ | Contour integral | "contour integral" | ∮ f(z)dz | Integral along closed path |
PROBABILITY AND STATISTICS
| Symbol | Name | Read As | Example | Meaning |
|---|
| P(A) or Pr[A] | Probability | "probability of A" | P(heads) = 0.5 | Likelihood of event A |
| P(A∣B) | Conditional probability | "probability of A given B" | P(A∣B) | Probability of A when B occurred |
| E[X] or μ | Expected value/Mean | "expected value of X" or "mu" | E[X] = 5 | Average value |
| Var(X) or σ² | Variance | "variance of X" or "sigma squared" | Var(X) | Spread of data |
| σ | Standard deviation | "sigma" | σ = 2 | Square root of variance |
| ~ | Distributed as | "is distributed as" | X ~ N(0,1) | X follows specified distribution |
| ∩ | Intersection (in probability) | "intersection" | P(A ∩ B) | Event intersection: both events occur |
| ∪ | Union (in probability) | "union" | P(A ∪ B) | Event union: at least one event occurs |
| A ⊥ B or A ⊥⊥ B | Independent | "A independent of B" | A ⊥⊥ B | Events don't affect each other |
| Cov(X,Y) | Covariance | "covariance of X and Y" | Cov(X,Y) | Joint variability |
| ρ | Correlation coefficient | "rho" | ρ(X,Y) | Normalized covariance (-1 to 1) |
| 𝟙ₐ or Iₐ | Indicator function | "indicator of A" | 𝟙ₐ(x) = 1 if x ∈ A | 1 if condition met, 0 otherwise |
Note on independence: A ⊥⊥ B is standard statistical notation; A ⊥ B is common in probability texts. Plain ⊥ alone typically means "false" or "bottom" in other contexts.
COMBINATORICS
| Symbol | Name | Read As | Formula | Meaning |
|---|
| C(n,k) or ⁿCₖ or (n choose k) | Combinations | "n choose k" | n!/(k!(n-k)!) | Ways to choose k from n (order doesn't matter) |
| P(n,k) or ⁿPₖ | Permutations | "n permute k" | n!/(n-k)! | Ways to arrange k from n (order matters) |
| n! | Factorial | "n factorial" | n×(n-1)×...×1 | Product of all positive integers up to n |
| n!! | Double factorial | "n double factorial" | n(n-2)(n-4)... | Product of every other integer down to 1 or 2 |
SEQUENCES AND SERIES
| Symbol | Name | Read As | Example | Meaning |
|---|
| aₙ | Sequence term | "a sub n" | a₃ = 7 | The nth term of sequence |
| {aₙ} | Sequence | "sequence a sub n" | {1,2,3,4,...} | Ordered list |
| lim | Limit | "limit as ... approaches ..." | lim(n→∞) aₙ | Value sequence approaches |
| → | Approaches/Converges to | "approaches" or "goes to" | n → ∞ | Gets arbitrarily close to |
| ∞ | Infinity | "infinity" | lim(n→∞) | Unbounded growth |
| ... or ⋯ | Ellipsis | "and so on" | 1,2,3,... | Pattern continues |
| ∑ₙ₌₁^∞ | Infinite sum | "sum from n equals one to infinity" | ∑ₙ₌₁^∞ 1/n² | Converges or diverges |
COMPLEXITY AND BIG-O NOTATION
| Symbol | Name | Read As | Meaning |
|---|
| O(f(n)) | Big-O | "Big-O of f of n" | Upper bound on growth rate |
| Ω(f(n)) | Big-Omega | "Big-Omega of f of n" | Lower bound on growth rate |
| Θ(f(n)) | Big-Theta | "Big-Theta of f of n" | Tight bound (both upper and lower) |
| o(f(n)) | Little-o | "Little-o of f of n" | g(n) ∈ o(f(n)) if lim(n→∞) g(n)/f(n) = 0 |
| ω(f(n)) | Little-omega | "Little-omega of f of n" | g(n) ∈ ω(f(n)) if lim(n→∞) g(n)/f(n) = ∞ |
| f(n) ~ g(n) | Asymptotically equal | "f asymptotically equals g" | lim(n→∞) f(n)/g(n) = 1 (stricter than Θ) |
Note: f(n) ~ g(n) means the ratio approaches 1, which is a stronger condition than f(n) ∈ Θ(g(n)).
GRAPH THEORY
| Symbol | Name | Read As | Meaning |
|---|
| G = (V,E) | Graph | "graph G equals the pair V comma E" | V is vertices, E is edges |
| v ∈ V | Vertex | "v is a vertex in V" | v is a node |
| e ∈ E | Edge | "e is an edge in E" | e is a connection |
| (u,v) | Directed edge | "edge from u to v" | Edge with direction (arc) |
| {u,v} | Undirected edge | "edge between u and v" | Edge without direction |
| deg(v) | Degree | "degree of v" | Number of edges connected to v |
| deg⁺(v) | Out-degree | "out-degree of v" | Outgoing edges from v (directed graphs) |
| deg⁻(v) | In-degree | "in-degree of v" | Incoming edges to v (directed graphs) |
| ∣V∣ | Order | "order of the graph" | Number of vertices |
| ∣E∣ | Size | "size of the graph" | Number of edges |
| u ~ v | Adjacent | "u is adjacent to v" | u and v are connected |
| Kₙ | Complete graph | "K sub n" | Graph where all vertices connected |
| Cₙ | Cycle graph | "C sub n" | Circular graph with n vertices |
| Pₙ | Path graph | "P sub n" | Linear graph with n vertices |
| Kₘ,ₙ | Complete bipartite | "K sub m comma n" | Two groups, all cross-edges |
| d(u,v) | Distance | "distance from u to v" | Shortest path length between u and v (∞ if no path exists) |
| δ(G) | Minimum degree | "delta of G" | Smallest vertex degree in G |
| Δ(G) | Maximum degree | "Delta of G" | Largest vertex degree in G |
| χ(G) | Chromatic number | "chi of G" | Minimum colors needed for proper coloring |
| ω(G) | Clique number | "omega of G" | Size of largest complete subgraph |
Note: ω is also used for little-omega complexity. Context distinguishes graph theory usage.
MATRICES AND LINEAR ALGEBRA
| Symbol | Name | Read As | Meaning |
|---|
| A or A | Matrix | "matrix A" | Rectangular array of numbers |
| aᵢⱼ | Matrix element | "a sub i j" | Element in row i, column j |
| Aᵀ or A' | Transpose | "A transpose" | Flip rows and columns |
| A⁻¹ | Inverse | "A inverse" | Matrix that undoes A (AA⁻¹ = I) |
| det(A) or ∣A∣ | Determinant | "determinant of A" | Scalar value from square matrix (single bars for determinant) |
| tr(A) | Trace | "trace of A" | Sum of diagonal elements |
| v or v⃗ | Vector | "vector v" | Ordered list of numbers |
| 0 | Zero vector | "zero vector" | All components are 0 |
| u · v | Dot product | "u dot v" | Sum of component products |
| u × v | Cross product | "u cross v" | Vector perpendicular to both (3D only) |
| ‖v‖ | Norm/Magnitude | "norm of v" | Length of vector |
| ⊗ | Tensor product | "tensor product" | Generalized outer product |
| rank(A) | Rank | "rank of A" | Dimension of column/row space |
| span(v₁,...,vₙ) | Span | "span of v1 through vn" | All linear combinations of vectors |
| dim(V) | Dimension | "dimension of V" | Size of basis for vector space V |
| ker(A) or null(A) | Kernel/Null space | "kernel of A" | Solutions to Ax = 0 |
| im(A) or col(A) | Image/Column space | "image of A" | All possible Ax values |
| λ | Eigenvalue | "lambda" | Scaling factor for eigenvector (Av = λv) |
| I or Iₙ | Identity matrix | "identity" or "I sub n" | Diagonal matrix of ones |
Note on ∣A∣: Can mean determinant (matrix), cardinality (set), or absolute value (scalar). Context determines meaning.
AUTOMATA AND FORMAL LANGUAGES
| Symbol | Name | Read As | Meaning |
|---|
| Σ | Alphabet | "sigma" | Finite set of symbols |
| Σ* | Kleene star | "sigma star" | All possible strings over Σ (Kleene closure, includes empty string) |
| ε or λ | Empty string | "epsilon" or "lambda" | String with no characters |
| w ∈ Σ* | String | "w is a string over sigma" | Sequence of symbols from Σ |
| ∣w∣ | String length | "length of w" | Number of characters in w |
| L | Language | "language L" | Set of strings (subset of Σ*) |
| L₁ ∪ L₂ | Language union | "L1 union L2" | Strings in either language |
| L₁ · L₂ or L₁L₂ | Concatenation | "L1 concatenated with L2" | Strings from L₁ followed by L₂ |
| L* | Kleene closure | "L star" | Zero or more concatenations of strings from L |
| L⁺ | Positive closure | "L plus" | One or more concatenations of strings from L |
| ⊢ | Derives | "derives" | One step in grammar derivation |
| ⊢* | Derives in zero or more steps | "derives star" | Derivation sequence (reflexive transitive closure) |
| ⇒ | Production / Derivation | "derives" | Grammar production step (notation varies by text) |
| δ | Transition function | "delta" | State transition in automaton |
| q₀ | Initial state | "q naught" or "q zero" | Starting state in automaton |
| F | Final states | "F" | Set of accepting states |
| ⟨q, w⟩ | Configuration | "configuration q comma w" | Current state and remaining input |
CALCULUS
| Symbol | Name | Read As | Meaning |
|---|
| lim(x→a) | Limit | "limit as x approaches a" | Value function approaches as x nears a |
| f'(x) or df/dx | Derivative | "f prime of x" or "df dx" | Rate of change of f with respect to x |
| f''(x) or d²f/dx² | Second derivative | "f double prime" or "d squared f dx squared" | Rate of change of rate of change |
| ∫ | Integral | "integral" | Area under curve / antiderivative |
| ∫ₐᵇ | Definite integral | "integral from a to b" | Area between a and b |
| ∞ | Infinity | "infinity" | Unbounded |
| dx | Differential | "dx" or "dee x" | Infinitesimal change in x |
| ∂f/∂x | Partial derivative | "partial f partial x" | Derivative with respect to x, holding other variables constant |
| ∇ | Gradient | "del" or "nabla" | Vector of partial derivatives |
| ∇² or Δ | Laplacian | "del squared" or "laplacian" | Sum of second partial derivatives (∂²/∂x² + ∂²/∂y² + ...) |
| ∇· | Divergence | "del dot" or "divergence" | Scalar field from vector field |
| ∇× | Curl | "del cross" or "curl" | Vector field rotation (3D only) |
Note: ∂ also used as boundary operator in topology (different context).
NUMBER THEORY
| Symbol | Name | Read As | Meaning |
|---|
| p ∣ n | p divides n | "p divides n" | n is a multiple of p (n = kp for some integer k) |
| p ∤ n | p does not divide n | "p does not divide n" | n is not a multiple of p |
| gcd(a,b) | Greatest common divisor | "gcd of a and b" | Largest positive integer dividing both a and b |
| lcm(a,b) | Least common multiple | "lcm of a and b" | Smallest positive integer divisible by both a and b |
| a ≡ b (mod n) | Congruent modulo n | "a congruent to b mod n" | a and b have same remainder when divided by n |
| ϕ(n) | Euler's totient | "phi of n" | Count of positive integers ≤ n coprime to n |
| π(x) | Prime counting function | "pi of x" | Number of primes ≤ x |
MISCELLANEOUS SYMBOLS
| Symbol | Name | Read As | Meaning |
|---|
| ∴ | Therefore | "therefore" | Logical conclusion follows |
| ∵ | Because | "because" | Logical reason precedes |
| □ or ∎ | QED | "Q-E-D" or "end of proof" | Proof complete |
| ▢ | End of example | "end of example" | Example complete |
| ⊕ | Direct sum / XOR | "direct sum" or "XOR" | Context-dependent: algebra or logic |
| ⊗ | Tensor product | "tensor product" | Generalized product operation |
| ∘ | Composition | "compose" or "circle" | Function composition |
| ⋆ or * | Convolution / Star | "convolution" or "star" | Context-dependent: signal processing or Kleene star |
| ≅ | Isomorphic to | "is isomorphic to" | Structure-preserving equivalence |
| ≃ | Homotopy equivalent / Isomorphic | "homotopy equivalent" or "isomorphic to" | Topology (homotopy equivalence) and abstract algebra (structural equivalence) |
SPECIAL BRACKETS AND GROUPING
| Symbol | Name | Read As | Usage |
|---|
| ( ) | Parentheses | "open paren ... close paren" | Grouping, function arguments |
| [ ] | Square brackets | "open bracket ... close bracket" | Indexing, optional parameters, closed intervals |
| { } | Curly braces | "open brace ... close brace" | Sets, code blocks |
| ⟨ ⟩ or 〈 〉 | Angle brackets | "angle bracket" | Vectors, inner products, ordered pairs |
| ⌊ ⌋ | Floor brackets | "floor of" | Round down to nearest integer |
| ⌈ ⌉ | Ceiling brackets | "ceiling of" | Round up to nearest integer |
| ∣ ∣ | Absolute value/Norm | "absolute value" or "norm" | Distance from zero (single bars: scalars; some texts use for vectors) |
| ‖ ‖ | Double bars | "norm" | Vector/matrix magnitude (preferred notation for norms) |
| ⟦ ⟧ | Semantic brackets | "semantic brackets" | Denotational semantics interpretation |
CONTEXT-DEPENDENT SYMBOLS
Many symbols have multiple meanings depending on mathematical context:
⊥ (Bottom/False/Undefined/Independence)
| Context | Meaning | Example |
|---|
| Logic | False / Contradiction | P ∧ ¬P = ⊥ |
| Type theory | Bottom type (uninhabited) | The type with no values |
| Domain theory | Least element / Undefined | f(x) = ⊥ means f is undefined at x |
| Partial orders | Minimal element | Smallest element in partial order |
| Probability | Independence (as A ⊥ B) | Events are independent |
∣ (Vertical bar)
| Context | Meaning | Example |
|---|
| Set builder notation | "Such that" | {x ∣ x > 0} |
| Number theory | "Divides" | 3 ∣ 12 |
| Probability | "Given" | P(A ∣ B) |
| Absolute value | Delimiter | ∣x∣ = 3 |
| Determinant | Delimiter | ∣A∣ = det(A) |
* (Star/Asterisk)
| Context | Meaning | Example |
|---|
| Arithmetic | Multiplication | 3 * 4 = 12 |
| Formal languages | Kleene star | Σ* |
| Signal processing | Convolution | f * g |
| Set notation | Exclusion of element | ℕ* = ℕ ∖ {0} |
· (Dot/Middle dot)
| Context | Meaning | Example |
|---|
| Arithmetic | Multiplication | 3 · 4 = 12 |
| Linear algebra | Dot product | u · v |
| Function composition | Composition | (f · g)(x) |
⊕ (Circled plus)
| Context | Meaning | Example |
|---|
| Logic | XOR | P ⊕ Q |
| Abstract algebra | Direct sum | V ⊕ W |
| Modular arithmetic | Addition mod 2 | 1 ⊕ 1 = 0 |
~ (Tilde)
| Context | Meaning | Example |
|---|
| Logic | Negation | ~P |
| Statistics | "Distributed as" | X ~ N(0,1) |
| Asymptotics | Asymptotic equivalence (f(n)/g(n) → 1) | f(n) ~ g(n) |
Note: ~ is standard for asymptotic equivalence; ≃ is used in topology/algebra, not typically for growth rates.
∂ (Partial/Del)
| Context | Meaning | Example |
|---|
| Calculus | Partial derivative | ∂f/∂x |
| Topology | Boundary operator | ∂M |
∇ (Nabla/Del)
| Context | Meaning | Example |
|---|
| Vector calculus | Gradient | ∇f |
| Vector calculus | Del operator | ∇· (divergence), ∇× (curl) |
→ (Arrow)
| Context | Meaning | Example |
|---|
| Logic | Implication | P → Q |
| Functions | Function type | f: A → B |
| Limits | Approaches | x → ∞ |
| Automata | Transition | q₁ → q₂ |
∣A∣ (Bars around symbol)
| Context | Meaning | Example |
|---|
| Sets | Cardinality | ∣{1,2,3}∣ = 3 |
| Matrices | Determinant | ∣A∣ = det(A) |
| Scalars | Absolute value | ∣-5∣ = 5 |
| Vectors | Norm | ∣v∣ |
ω (Lowercase omega)
| Context | Meaning | Example |
|---|
| Complexity theory | Little-omega | ω(n) |
| Graph theory | Clique number | ω(G) |
| Physics | Angular frequency | ω = 2πf |
Δ (Delta)
| Context | Meaning | Example |
|---|
| Arithmetic | Change / Difference | Δx = x₂ - x₁ |
| Calculus | Laplacian operator | Δf = ∇²f |
| Graph theory | Maximum degree | Δ(G) |
≡ (Triple bar)
| Context | Meaning | Example |
|---|
| Logic | Logical equivalence | P ≡ Q |
| Number theory | Congruence modulo n | a ≡ b (mod n) |
| Programming | Identity / Definitional equality | x ≡ y |
NOTES ON CONVENTIONS
- Subset notation: This guide uses ⊆ for subset (allowing equality) and ⊊ for proper subset. Some texts use ⊂ to mean ⊆. Always check your textbook's conventions.
- Natural numbers: Convention varies. ℕ may include 0 or start at 1. Use ℕ₀ for {0,1,2,...} and ℕ* or ℕ⁺ for {1,2,3,...} when clarity is needed.
- Disjoint union: ⊎ typically assumes sets are disjoint or uses tagging to force disjointness.
- Independence in probability: Write A ⊥ B or A ⊥⊥ B for independent events, not plain ⊥.
- Asymptotic equivalence: f(n) ~ g(n) means lim(n→∞) f(n)/g(n) = 1, which is stronger than f(n) ∈ Θ(g(n)).
- Parallel notation: ∥ means parallel in geometry. In concurrency theory, parallel composition is written P | Q, P || Q, or occasionally P ∥ Q depending on the formalism.
- Multiple meanings: Always check context. Symbols like |, *, ·, ~, ∂, ∇, →, ⊕, ⊥, ω, Δ, and ≡ have different meanings in different mathematical areas.
- Inverse function vs preimage: The notation f⁻¹(B) for preimage is defined for any function. The inverse function f⁻¹(x) requires f to be bijective. Context determines which is meant.