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The mathematical notation past arithmetic: set algebra, the number systems, the quantifiers and turnstiles of formal logic, proof marks and the operators of analysis. What holds the group together is that almost every member is a mark whose *shape* carries a claim a reader is never told about — a bar that turns a depth into a height, a stroke that separates derivation from truth, a doubling that turns a magnitude into a length. Several records exist mainly to state the trap: that the two subset signs are genuinely ambiguous between authors, that a definition sign cannot be false, that an unbracketed tuplet leaves its reference implied.
A sideways U opening leftwards, saying one set contains another.
A sideways U with a bar beneath, allowing the two sets to be identical.
A sideways U with a struck-through bar, excluding the case of equality.
A struck-through epsilon, denying membership of a set.
A backslash between two sets, removing one from the other.
A triangle between two sets, taking what is in one but not both.
A multiplication cross between sets, forming ordered pairs.
A script capital P applied to a set, naming all of its subsets.
An overbar on a set name, meaning everything outside it.
The first Hebrew letter with a zero subscript, counting the whole numbers.
Double-struck capitals naming the standard number systems.
A vertical bar or colon separating a variable from its condition.
An enlarged sideways U with limits, taking the union of a family.
An enlarged upside-down U with limits, intersecting a family.
A union sign with a dot or a squared form, tagging each element by origin.
A backwards E with an exclamation mark, asserting a unique solution.
A vertical stroke with a bar to its right, meaning 'proves'.
A vertical stroke with two bars, meaning 'entails' or 'models'.
A vertical bar between two propositions, meaning 'not both'.
A downward arrow between propositions, meaning 'neither'.
A plus inside a circle, true when exactly one operand is.
An upside-down T standing for a contradiction.
A T standing for a proposition true in every case.
An open square before a proposition, meaning 'necessarily'.
An open diamond before a proposition, meaning 'possibly'.
A filled or hollow square ending a proof.
A colon before an equals sign, marking a definition rather than a claim.
Three horizontal bars, asserting equality in every case.
A wave over an equals sign, saying two structures are the same.
An arrow with a tail bar, naming what a function does to an element.
A plain right arrow naming a function's domain and codomain.
A small raised ring between two functions.
A vertical bar with a subscript, cutting a function down to part of its domain.
A vertical stroke asserting one number divides another exactly.
A struck-through vertical stroke, denying exact division.
An exclamation mark after a number, multiplying it down to one.
Two exclamation marks, skipping every other factor.
A capital theta bounding a growth rate from both sides.
A capital omega bounding a growth rate from below.
A lowercase o asserting negligible growth.
Two numbers in brackets, or the letters gcd, naming their largest shared divisor.
A number in square brackets, naming a whole class of numbers.
A triangle on its side, marking a subgroup closed under conjugation.
A circled plus joining structures into one that splits cleanly.
A circled multiplication cross joining two structures.
Angle brackets around two vectors, giving a number.
Doubled vertical bars measuring the size of a vector.
Three dots standing for an omitted, obvious continuation.
Two nested less-than signs, marking a difference of scale.
Two nested greater-than signs, marking dominance of scale.
The letters lim with an arrow beneath, naming a value approached.
An integral sign with a circle, integrating around a closed path.
Two integral signs side by side, integrating over an area.
A square operator combining space and time derivatives.
A triangle or squared nabla, summing second derivatives.
An upside-down capital pi, the dual of a product.
A double-shafted arrow between functors.
A Greek epsilon or lambda standing for a string of no characters.
A raised star marking a conjugate transpose or dual.
A raised capital T flipping a matrix about its diagonal.
A caret over a letter, marking an estimate or a unit vector.
A doubled upright stroke pair, denying any statistical relation.
The letters Var, or a squared sigma, measuring spread.
A lowercase z counting standard deviations from the mean.
A capital P with two subscripts, counting ordered arrangements.