Statements and Mathematical Jargon

What is a Statement?

A statement is a sentence that is either true or false, but not both. This is the fundamental unit of mathematical discourse.

Examples of Statements

• $e^{i\pi} + 1 = 0$
• “17 is a prime number”
• “There are infinitely many primes”

Non-examples

• “This statement is false” (paradox)
• “Is 7 prime?” (question, not a statement)
• “Let $x = 5$” (command, not a statement)

Predicates

A predicate is a statement whose truth depends on the value of one or more free variables.

Examples

• $P(n) : \equiv$ ”$n$ is even”
• $Q(x, y) : \equiv$ "$x^2 + y^2 = 1$"
• “The function $f$ is differentiable”

A predicate becomes a statement once we substitute specific values for its free variables.

The Hierarchy of Mathematical Results

Mathematicians distinguish different types of true statements:

Axiom

A statement assumed true without proof. Axioms form the foundation of a mathematical theory.

Example: The axioms of Zermelo-Fraenkel set theory (ZF).

Definition

A name given to a concept, establishing terminology.

Example: “A prime number is a natural number greater than 1 divisible only by 1 and itself.”

Theorem

A major proved result of significant importance.

Example: Fermat’s Last Theorem, the Fundamental Theorem of Arithmetic.

Proposition

A proved result of lesser importance than a theorem.

Lemma

A result proved chiefly to support the proof of another result. Lemmas are “helper theorems.”

Example: The “lifting the exponent lemma” used in number theory proofs.

Corollary

A statement that follows easily from a previously proved theorem or proposition.

Example: “There are infinitely many primes congruent to 3 mod 4” is a corollary of Dirichlet’s theorem.

Conjecture

A statement believed to be true but not yet proved.

Example: The Goldbach Conjecture (every even integer greater than 2 is the sum of two primes).

Why Precision Matters

Consider the statement: “The product of two odd integers is odd.”

This hides significant logical structure. More precisely:

$\text{For all integers } m, n: \text{ if } m \text{ and } n \text{ are odd, then } m \cdot n \text{ is odd.}$

Even more formally:

$\forall m, n \in \mathbb{Z}. \left( \text{Odd}(m) \land \text{Odd}(n) \right) \Rightarrow \text{Odd}(m \cdot n)$

Tripos Credit

Marks are awarded for making such logical structure explicit, not just for getting the right answer. A solution that correctly proves the result but fails to exhibit awareness of the quantifiers and implication structure may lose marks.

Variables and Scope

Free Variables

A variable is free in a statement if the statement’s truth depends on its value.

In "$x > 5$", the variable $x$ is free.

Bound Variables

A variable is bound if it is quantified by $\forall$ or $\exists$.

In "$\forall x \in \mathbb{R}. x^2 \geq 0$", the variable $x$ is bound by the universal quantifier.

Common Notation

Logical Symbols

Symbol	Meaning
$\land$	and (conjunction)
$\lor$	or (disjunction)
$\neg$	not (negation)
$\Rightarrow$	implies
$\Leftrightarrow$	if and only if
$\forall$	for all
$\exists$	there exists

Set Notation

Symbol	Meaning
$\in$	element of
$\subseteq$	subset
$\cup$	union
$\cap$	intersection
$\setminus$	set difference
$\mathbb{N}$	natural numbers $\{0, 1, 2, \ldots\}$
$\mathbb{Z}$	integers $\{\ldots, -1, 0, 1, \ldots\}$
$\mathbb{Q}$	rational numbers
$\mathbb{R}$	real numbers

Number Notation

• $a \mid b$ means ”$a$ divides $b$”
• $a \equiv b \pmod{n}$ means ”$a$ is congruent to $b$ modulo $n$”
• $\gcd(a, b)$ means “greatest common divisor of $a$ and $b$“

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Summary

• A statement is a sentence that is either true or false
• A predicate is a statement with free variables
• Mathematical results form a hierarchy: axiom, definition, theorem, proposition, lemma, corollary, conjecture
• Precision in stating logical structure is essential for rigorous proof and earns Tripos credit