Statements

In mathematics, a statement (also known as a proposition) is a declarative sentence that is either true or false but not both. There must be no ambiguity as to whether the statement is true or false. The statement cannot be an opinion or vague sentence (for example, "chocolate tastes good" or "red is an ugly color") that does not have a set truth value. Some examples of basic statements and their truth values:

• $2$ is an even number (TRUE)
• Negative numbers are less than $0$ (TRUE)
• $3+4=6$ (FALSE)
• Quadrilaterals have five sides (FALSE)
• For all real numbers $n$, $n^{2}\geq 0$ (TRUE)
• All multiples of $3$ are even (FALSE)

Terminology

A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time.

• An axiom or postulate is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a definition, which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects. Historically, axioms were regarded as "self-evident"; today they are merely assumed to be true.
• A conjecture is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example, Goldbach's conjecture and Collatz conjecture). The term hypothesis is also used in this sense (for example, Riemann hypothesis), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example problem when people are not sure whether the statement should be believed to be true. Fermat's Last Theorem was historically called a theorem, although, for centuries, it was only a conjecture.
• A theorem is a statement that has been proven to be true based on axioms and other theorems.
• A proposition is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in propositional logic. In classical geometry the term "proposition" was used differently: in Euclid's Elements, all theorems and geometric constructions were called "propositions" regardless of their importance.
• A lemma is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a theorem, though the term "lemma" is usually kept as part of its name (e.g. Gauss's lemma, Zorn's lemma, and the fundamental lemma).
• A corollary is a proposition that follows immediately from another theorem or axiom, with little or no required proof. A corollary may also be a restatement of a theorem in a simpler form, or for a special case: for example, the theorem "all internal angles in a rectangle are right angles" has a corollary that "all internal angles in a square are right angles" - a square being a special case of a rectangle.
• A generalization of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a special case (a corollary).

Other terms may also be used for historical or customary reasons, for example:

• An identity is a theorem stating an equality between two expressions, that holds for any value within its domain (e.g. Bézout's identity and Vandermonde's identity).
• A rule is a theorem that establishes a useful formula (e.g. Bayes' rule and Cramer's rule).
• A law or principle is a theorem with wide applicability (e.g. the law of large numbers, law of cosines, Kolmogorov's zero–one law, Harnack's principle, the least-upper-bound principle, and the pigeonhole principle).

A few well-known theorems have even more idiosyncratic names, for example, the division algorithm, Euler's formula, and the Banach–Tarski paradox.

Connectives

We can use connectives to create compound statements out of two or more basic statements, or to transform statements. Here are some common connectives and how they are used (let $P$ and $Q$ be statements):

• Conjunction: "$P$ and $Q$", also denoted as "$P\land Q$". A conjunctive statement is only true if both $P$ and $Q$ are true, and false otherwise.
• Disjunction: "$P$ or $Q$", also denoted as "$P\lor Q$". A disjunctive statement is true if either $P$ or $Q$ are true, and false only when both $P$ and $Q$ are false.
• Negation: "not $P$", also denoted as "$\neg P$". A negation reverses the truth value of the original statement (that is, if $P$ is true then $\neg P$ is false, and vice versa).
• Implication: "if $P$, then $Q$" or "$P$ implies $Q$", also denoted as "$P\rightarrow Q$". An implication is true if $P$ and $Q$ are both true, as well as when $P$ is false, regardless of the truth value of $Q$ (that is, $Q$ can be true or false when $P$ is false, and the implication will still be true). An implication is only false if $P$ is true and $Q$ is false.
• Biconditional: "$P$ if and only if $Q$", also denoted as "$P\Leftrightarrow Q$". A biconditional is only true when both $P\Rightarrow Q$ and $Q\Rightarrow P$" are true statements; that is, $P\Leftrightarrow Q$ is true when $P$ and $Q$ have the same truth value, and false when $P$ and $Q$ have different truth values.

Truth table of some basic logical statements given statements $P$ and $Q$:

$P$	$Q$	$P\land Q$	$P\lor Q$	$P\Rightarrow Q$	$Q\Rightarrow P$	$P\Leftrightarrow Q$	$\neg P$	$\neg Q$
T	T	T	T	T	T	T	F	F
T	F	F	T	F	T	F	F	T
F	T	F	T	T	F	F	T	F
F	F	F	F	T	T	T	T	T

Resources

• Course Textbook, pages 4-25

Licensing

Content obtained and/or adapted from:

• Statements and Logical Operators, OpenStax under a CC BY-NC-SA license