Terms of Art: The Vocabulary of Proofs

Mathematics has a technical vocabulary that school never defines; not because the terms are advanced, but because school mathematics never asks you to prove anything, and these terms are the language of proof. “Without loss of generality” appears 144 times across this site; “if and only if” 87 times; “up to isomorphism” 8 times. A reader who does not know what they mean will be lost: not because the mathematics is hard, but because the English is.

This page defines each term precisely, with examples of correct and incorrect use, so no reader is lost by vocabulary alone. The terms are ordered by how often they appear, from the most common to the most specialised.

Consider the sentence “Every group of order $p^2$ is abelian, where $p$ is prime.” Every word is English; every symbol is defined in the abstract algebra track. But to understand the sentence, you need to know that “order” means the number of elements, that "$p^2$" means $p$ times $p$, and that “abelian” means commutative. Each of those is defined in the tracks; but you need all three simultaneously to parse one sentence.

The terms on this page are worse: they are used without definition, because the author assumes you have seen them before. If you have not; because your school taught computation, not proof; you will be lost, and the loss is invisible: you will think the mathematics is hard when the English is the problem. This page exists to remove that invisibility.

The vocabulary of proof is a compression code. Each term encodes a paragraph of reasoning into two or three words. “Without loss of generality” compresses: “there are two cases, and by a symmetry of the problem, the argument for one case applies verbatim to the other.” “If and only if” compresses: “the forward implication and the reverse implication are both true.” “Up to isomorphism” compresses: “the objects may not be equal, but there is a structure-preserving bijection between them, and everything we care about is preserved by that bijection.”

Learning the vocabulary is learning to decompress: when you read “WLOG assume $x \geq 0$”, you should expand it internally to the full paragraph. Once you can do that automatically, proofs become readable in the same way that a fluent reader reads prose; not letter by letter, but phrase by phrase.

Definition. ”$P$ if and only if $Q$” (abbreviated iff, $\Leftrightarrow$, or $\iff$) means both $P \Rightarrow Q$ and $Q \Rightarrow P$.

Correct use. ”$f$ is continuous iff $f^{-1}(\text{open})$ is open”; two directions, both proved.

Common error. Proving only one direction and claiming iff. The forward direction and the reverse direction are separate proofs; each needs its own argument, and neither implies the other. The most common place this goes wrong is in “characterise” problems: “characterise the open sets” means prove both directions.

Synonym. “Exactly when” (16 uses); “the series converges exactly when $a_n \to 0$” is the same claim.

These are all the same word: “the previous step implies this step.” They are interchangeable, and the choice is stylistic; though “hence” is more common in British-influenced writing and “therefore” in American.

Why the page bothers to define these. Because they are not interchangeable with “and then”: each one is a claim that the implication is justified, not merely that two statements appear in sequence. If the implication is not justified, the word is a lie.

Definition. “For an arbitrary $x$” means “for any $x$ at all; no properties assumed beyond the stated hypotheses.”

Correct use. “Let $x$ be an arbitrary element of $G$”; you are about to prove something that works for every element, so you pick one with no special properties.

Common error. Using “arbitrary” when you mean “random.” “Arbitrary” means you have no control over which element; it is a statement of universality. “Random” means a probability distribution is involved. In a proof, “arbitrary” is a deterministic claim about all elements; “random” would change the mathematics.

Definition. ”$P$ precisely when $Q$” means $P \iff Q$; the same as “if and only if,” but emphasising that the boundary is exactly at $Q$: both directions hold and the characterisation is complete.

Correct use. ”$f$ is injective precisely when $f(a) = f(b) \Rightarrow a = b$”; the condition characterises injectivity completely.

Definition. An object is canonical if it is distinguished without any arbitrary choices; the same in every equivalent construction, for every mathematician, independent of coordinates, bases, or conventions.

Correct use. “The canonical map $V \to V^{**}$”; the map $v \mapsto (\varphi \mapsto \varphi(v))$ requires no basis or choice; it is defined for every $v$ and is the same map regardless of how you think about $V$.

Common error. Using “canonical” when you mean “natural” or “standard.” “Canonical” is a mathematical claim: the object is unique up to unique isomorphism. “Standard” is a sociological claim: most people use it. They are different claims, and confusing them leads to proofs that depend on choices that are not canonical.

Definition. $P$ is necessary for $Q$ means $Q \Rightarrow P$ (you cannot have $Q$ without $P$). $P$ is sufficient for $Q$ means $P \Rightarrow Q$. $P$ is necessary and sufficient for $Q$ means both, which is the same as “if and only if.”

Correct use. “For a matrix to be invertible, it is necessary that $\det \neq 0$ (invertible matrices have nonzero determinants) and sufficient that $\det \neq 0$ (by the adjugate formula); so a nonzero determinant is necessary and sufficient.” The two directions come from different arguments; the theorem packages them.

Why the distinction matters. “Necessary” alone is not “sufficient.” A necessary condition can hold for objects that fail the conclusion; a sufficient condition can fail for objects that satisfy it. Confusing the two is the logical error that school mathematics can hide (because school problems are usually iff), but proof-based mathematics cannot.

Definition. A map defined via representatives is well-defined if the result does not depend on which representative you chose.

Correct use. “The operation $[a] + [b] = [a + b]$ on $\mathbb{Z}/n$ is well-defined: if $[a] = [a']$ and $[b] = [b']$, then $a + b \equiv a' + b' \pmod{n}$, so the result is the same class regardless of representatives.”

Why this needs proof. When you define something on equivalence classes by referring to representatives, you must verify that choosing different representatives gives the same answer. This is a proof obligation, not a courtesy; and forgetting it is the most common error in abstract algebra. The classical counterexample: defining $[a] \times [b] = [ab]$ on $\mathbb{Z}/n$ is well-defined, but defining $[a] \times [b] = [ab] \bmod {n+1}$ would not be, because the result depends on the representative.

Definition. A property holds almost everywhere (a.e.) if the set where it fails has measure zero.

Correct use. “The function $f$ is continuous a.e.” means the set of discontinuities has Lebesgue measure zero; it can still be infinite, dense, or uncountable, as long as its measure is zero.

Common error. Confusing “almost everywhere” with “everywhere except finitely many points.” The set of measure zero can be countably infinite, or uncountable (the Cantor set has measure zero but uncountably many points). “Almost everywhere” is a measure-theoretic statement, not a cardinality one.

Definition. A statement holds up to isomorphism if it holds for every object isomorphic to the one named; the specific identity of the object is irrelevant, only its structure matters.

Correct use. “There is a unique group of order $p^2$ up to isomorphism, for $p$ prime”; wait, that is false (there are two: $\mathbb{Z}/p^2$ and $\mathbb{Z}/p \times \mathbb{Z}/p$). The correct version: “there are exactly two groups of order $p^2$ up to isomorphism.” The phrase “up to isomorphism” means: the classification counts isomorphism classes, not labelled groups - two isomorphic groups count as one.

Why “up to” matters. In category theory and representation theory, “up to” appears constantly: “unique up to unique isomorphism” is a stronger claim than “unique up to isomorphism,” and both are weaker than “unique.” Each phrase means something different, and conflating them is an error.

Definition. A degenerate case is one where the usual hypotheses collapse to something trivial; a triangle with zero area, a vector space of dimension zero, a group of order one. A nontrivial case is the opposite.

Correct use. “Assume $V$ is a nontrivial vector space” excludes $V = \{0\}$. “The degenerate case $\theta = 0$” is where the formula simplifies to something vacuous.

Why they are excluded. Degenerate cases often require separate arguments (the division-by-zero problem) or are genuinely uninteresting (a group of order one has exactly one element and one subgroup). Stating “nontrivial” in the hypothesis is how the author says “I am not dealing with the boring case.”

Definition. “Without loss of generality, assume $x \geq y$” means: the statement is symmetric in $x$ and $y$ (or there is a symmetry that maps one case to the other), so the argument for $x \geq y$ applies verbatim to $y \geq x$ by exchanging their names.

Correct use. “WLOG assume $x \geq y$”; because the statement is symmetric under $x \leftrightarrow y$.

Why it is not always valid. WLOG requires a symmetry of the problem that maps one case to the other. If the statement is not symmetric, you cannot use WLOG. Example: “WLOG assume $x \leq y$” for the claim "$x + y \geq 0$"; this is not valid if the claim is not symmetric in $x$ and $y$, and the reader will reject it. The burden of proof for the symmetry is on the writer.

Definition. Assume $P$ and $\neg Q$; derive any falsehood. Then $P \Rightarrow Q$.

Why it works. If $P \wedge \neg Q$ leads to a contradiction, then $P \wedge \neg Q$ is false. Since $\neg(P \wedge \neg Q) = P \Rightarrow Q$ by De Morgan, the implication holds.

When to use it. When the direct route has no traction; especially for statements of the form “there is no $x$ such that…” (prove the negation leads to absurdity) or “every $x$ satisfies…” (assume some $x$ does not).

Definition. Build the object explicitly. This is the strongest form of existence proof.

Contrast. A proof by contradiction of “there exists $x$ such that $P(x)$” shows that assuming no such $x$ leads to absurdity, but does not exhibit $x$. A constructive proof does. Intuitionists reject non-constructive existence proofs; classical mathematicians accept both.

Definition. If $n + 1$ objects are placed in $n$ boxes, at least one box contains at least two objects.

Why it matters. It is the simplest non-constructive existence principle: it proves that something exists (a collision, a repeated value, two people with the same birthday) without identifying which pair.

Definition. To prove $S(n)$ for all $n \geq n_0$: prove $S(n_0)$ (base case) and prove $S(k) \Rightarrow S(k+1)$ (inductive step) for all $k \geq n_0$.

Why it works. By the well-ordering principle of $\mathbb{N}$; every nonempty subset of $\mathbb{N}$ has a least element. If some $n$ fails $S(n)$, the set of failures is nonempty and has a least element $n^*$; then $S(n^* - 1)$ holds (by minimality), so $S(n^*)$ holds by the inductive step - contradiction.

Definition. Check every case. Only practical when the cases are finite and few; but when they are, it is definitive.

Correct use. “Every group of order 4 is isomorphic to $\mathbb{Z}/4$ or $\mathbb{Z}/2 \times \mathbb{Z}/2$”; proved by checking all possible Cayley tables (there are only two up to isomorphism). Modern uses include the four colour theorem (1976) and the classification of finite simple groups.

Definition. A statement of the form $\forall\varepsilon > 0\ \exists\delta > 0\ \forall x : P(x, \delta)$ means: for each $\varepsilon$ chosen first by your opponent, you may choose a $\delta$ depending on that $\varepsilon$ (and on nothing else) so that $P$ holds.

The order matters. $\forall\epsilon\ \exists\delta$ is not the same as $\exists\delta\ \forall\epsilon$: in the first, $\delta$ can depend on $\epsilon$; in the second, one $\delta$ must work for all $\epsilon$; a much stronger requirement.

Negating quantified statements. $\neg\forall\epsilon\ \exists\delta\ P \equiv \exists\epsilon\ \forall\delta\ \neg P$. To negate a string of quantifiers, flip each quantifier and negate the predicate. Getting this wrong is the most common error in analysis proofs.

• Almost everywhere; see §1.8.
• Uniformly; the bound does not depend on the point (unlike pointwise).
• Dominated; bounded by an integrable function, licensing dominated convergence.
• Compactly supported; nonzero only on a compact set.

• Up to isomorphism; see §1.9.
• Generated by; the smallest substructure containing the named elements.
• Free; satisfies no relations beyond the axioms.
• Faithful; the representation is injective (different group elements give different maps).

• Dense; every open set intersects it.
• Clopen; both open and closed (possible in disconnected spaces).
• Compact; every open cover has a finite subcover (NOT the same as “closed and bounded” outside $\mathbb{R}^n$).

• Almost surely; with probability 1 (the probability-theoretic “almost everywhere”).
• Independently; the joint law factorises; not the same as “uncorrelated.”
• Almost surely convergent; the strong law’s mode; see the ML course’s page 1 for the four modes and their differences.

• Reading “if” as “only if.” ”$f$ is continuous if it is differentiable” means differentiable ⇒ continuous, not the converse. School English uses “if” loosely; proof English uses it precisely.
• Assuming “not” distributes. $\neg\forall x\,P(x) = \exists x\,\neg P(x)$, not $\forall x\,\neg P(x)$. Getting this wrong inverts the meaning of every negated quantified statement.
• Using “obviously” or “clearly” to skip a step you cannot prove. If a step is clear, it is clear to the reader without the word. If it is not, the word does not help; it signals that the author cannot prove it either.
• Assuming every “if” is an “iff.” See §1.1. The converse must be proved separately.
• Using “unique” when you mean “unusual” or “special.” In mathematics, “unique” means exactly one. “The unique group of order 4” would be false (there are two).

1. Write the negation of: “for every $\epsilon > 0$ there exists $\delta > 0$ such that for all $x$ with $|x - x_0| < \delta$, $|f(x) - L| < \epsilon$.” (§3)
2. What does “up to isomorphism” mean in “there are two groups of order 4 up to isomorphism”? (§1.9)
3. Is “n is even” necessary, sufficient, or necessary and sufficient for ”$n^2$ is even”? Prove both directions or give a counterexample. (§1.6)
4. What does WLOG mean, and when is it valid? (§1.11)
5. What does “almost everywhere” mean, and can the exceptional set be uncountable? (§1.8)

Answers. (1) $\exists\epsilon > 0\ \forall\delta > 0\ \exists x\ (|x - x_0| < \delta \wedge |f(x) - L| \geq \epsilon)$. (2) The classification counts isomorphism classes, not labelled groups: there are exactly two isomorphism classes ($\mathbb{Z}/4$ and $\mathbb{Z}/2 \times \mathbb{Z}/2$), and every group of order 4 is isomorphic to one of them. (3) Necessary and sufficient. $n$ even $\Rightarrow n = 2k \Rightarrow n^2 = 4k^2$ is even. $n$ odd $\Rightarrow n = 2k+1 \Rightarrow n^2 = 4k^2 + 4k + 1$ is odd. So $n^2$ even forces $n$ even by contrapositive. (4) “Without loss of generality”: the statement is symmetric under the exchange being made, so proving one case suffices. Valid only when the symmetry exists. (5) The set where it fails has measure zero. Yes, the exceptional set can be uncountable (the Cantor set).

• Before You Begin; the school-to-university transition page: logic, sets, functions, proof techniques.
• Introduction to Algebraic Topology - where “up to isomorphism” and “canonical” appear in context.
• Measure Theory; where “almost everywhere” is defined on the measure-theoretic foundation.
• The Fundamental Theorems - where “uniformly” and “dense” appear in their functional-analytic context.

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