Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context. ;
almost all: A shorthand term for "all except for a
set of
measure zero", when there is a
measure to speak of, with the phrases
almost surely and
almost everywhere having related meanings. For example, "almost all
real numbers are
transcendental" because the
algebraic real numbers form a
countable subset of the real numbers with measure zero. One can also speak of "almost all"
integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage, such as "almost all prime numbers are
odd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with
generic, below. ;
arbitrarily large: Notions which arise mostly in the context of
limits, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicate
P is satisfied by arbitrarily large values, can be expressed in more formal notation by . See also
frequently. The statement that quantity
f(
x) depending on
x "can be made" arbitrarily large, corresponds to . ;
arbitrary: A shorthand for the
universal quantifier. An arbitrary choice is one which is made unrestrictedly, or alternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set. Also much in general-language use among mathematicians: "Of course, this problem can be arbitrarily complicated". ;
eventually:In the context of limits, this is shorthand meaning
for sufficiently large arguments; the relevant argument(s) are implicit in the context. As an example, the function log(log(
x))
eventually becomes larger than 100"; in this context, "eventually" means "for
sufficiently large x." ;
factor through: A term in
category theory referring to composition of
morphisms. If for three
objects A,
B, and
C a map f \colon A \to C can be written as a composition f = h \circ g with g \colon A \to B and h \colon B \to C, then
f is said to
factor through any (and all) of B, g, and h. ; finite: When said of the value of a variable assuming values from the non-negative
extended reals \R_{\geq 0}\cup\{\infty\}, the meaning is usually "not infinite". For example, if the
variance of a
random variable is said to be finite, this implies it is a non-negative real number, possibly zero. In some contexts though, for example in "a small but finite amplitude", zero and infinitesimals are meant to be excluded. When said of the value of a variable assuming values from the extended natural numbers \N\cup\{\infty\}, the meaning is simply "not infinite". When said of a set or a mathematical whose main component is a set, it means that the
cardinality of the set is less than
\aleph_0. ; frequently: In the context of limits, this is shorthand for
arbitrarily large arguments and its relatives; as with
eventually, the intended variant is implicit. As an example, the
sequence (-1)^n is frequently in the
interval (1/2, 3/2), because there are arbitrarily large
n for which the value of the sequence is in the interval. ; formal, formally: Qualifies anything that is sufficiently precise to be translated straightforwardly in a
formal system. For example. a
formal proof, a
formal definition. ;
generic: This term has similar connotations as
almost all but is used particularly for concepts outside the purview of
measure theory. A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its
complement satisfies some (context-dependent) notion of smallness. For example, a property which holds on a
dense Gδ (
intersection of countably many
open sets) is said to hold generically. In
algebraic geometry, one says that a property of points on an
algebraic variety that holds on a dense
Zariski open set is true generically; however, it is usually not said that a property which holds merely on a dense set (which is not Zariski open) is generic in this situation. ; in general: In a descriptive context, this phrase introduces a simple characterization of a broad class of , with an eye towards identifying a unifying principle. This term introduces an "elegant" description which holds for "
arbitrary" objects. Exceptions to this description may be mentioned explicitly, as "
pathological" cases. ;
left-hand side, right-hand side (LHS, RHS): Most often, these refer simply to the left-hand or the right-hand side of an
equation; for example, x = y + 1 has x on the LHS and y + 1 on the RHS. Occasionally, these are used in the sense of
lvalue and rvalue: an RHS is primitive, and an LHS is derivative. ; nice: A mathematical is colloquially called
nice or
sufficiently nice if it satisfies hypotheses or properties, sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informal antonym for
pathological. For example, one might conjecture that a
differential operator ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interesting
topological invariant should be computable "for nice
spaces X." ; Mathematical object|: Anything that can be assigned to a
variable and for which
equality with another object can be considered. The term was coined when variables began to be used for
sets and
mathematical structures. ; onto: A function (which in mathematics is generally defined as mapping the elements of one set
A to elements of another
B) is called "
A onto
B" (instead of "
A to
B" or "
A into
B") only if it is
surjective; it may even be said that "
f is onto" (i. e. surjective). Not translatable (without circumlocutions) to some languages other than English. ; proper: If, for some notion of substructure, are substructures of themselves (that is, the relationship is
reflexive), then the qualification
proper requires the objects to be different. For example, a
proper subset of a set
S is a subset of
S that is different from
S, and a
proper divisor of a number
n is a divisor of
n that is different from
n. This
overloaded word is also non-jargon for a
proper morphism. ; : A characteristic that a mathematical object may have or not; for example "being positive". Properties are often expressed with
formulas and are used for specifying
sets and subsets, typically with
set-builder notation. ; regular : A function is called
regular if it satisfies satisfactory continuity and differentiability properties, which are often context-dependent. These properties might include possessing a specified number of
derivatives, with the function and its derivatives exhibiting some
nice property (see
nice above), such as
Hölder continuity. Informally, this term is sometimes used synonymously with
smooth, below. These imprecise uses of the word
regular are not to be confused with the notion of a
regular topological space, which is rigorously defined. ; resp.: (Respectively) A convention to shorten parallel expositions. "
A (resp.
B) [has some relationship to]
X (resp.
Y)" means that
A [has some relationship to]
X and also that
B [has (the same) relationship to]
Y. For example,
squares (resp. triangles) have 4 sides (resp. 3 sides); or
compact (resp.
Lindelöf) spaces are ones where every open
cover has a finite (resp. countable) open subcover. ; sharp: Often, a mathematical theorem will establish constraints on the behavior of some ; for example, a function will be shown to have an
upper or lower bound. The constraint is
sharp (sometimes
optimal) if it cannot be made more restrictive without failing in some cases. For example, for
arbitrary non-negative real numbers
x, the
exponential function ex, where
e = 2.7182818..., gives an upper bound on the values of the
quadratic function x2. This is not sharp; the gap between the functions is everywhere at least 1. Among the exponential functions of the form α
x, setting α =
e2/
e = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α3 = 8 2. In applied fields the word "tight" is often used with the same meaning. ;
smooth:
Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to
analyticity, and still others which are more complicated. Each such usage attempts to invoke the physically intuitive notion of smoothness. ; strong, stronger: A theorem is said to be
strong if it deduces restrictive results from general hypotheses. One celebrated example is
Donaldson's theorem, which puts tight restraints on what would otherwise appear to be a large class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not only should such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. A theorem, result, or condition is further called
stronger than another one if a proof of the second can be easily obtained from the first but not conversely. An example is the sequence of theorems:
Fermat's little theorem,
Euler's theorem,
Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see
sharp above) is a stronger result than a non-sharp one. Finally, the adjective
strong or the adverb
strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a
strong antichain is an
antichain satisfying certain additional conditions, and likewise a
strongly regular graph is a
regular graph meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as "antichain"). ;
sufficiently large, suitably small, sufficiently close: In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as that predicate
P holds for sufficiently large values, can be expressed in more formal notation by ∃
x : ∀
y ≥
x :
P(
y). See also
eventually. ; upstairs, downstairs: A descriptive term referring to notation in which two are written one above the other; the upper one is
upstairs and the lower,
downstairs. For example, in a
fiber bundle, the total space is often said to be
upstairs, with the base space
downstairs. In a
fraction, the
numerator is occasionally referred to as
upstairs and the
denominator downstairs, as in "bringing a term upstairs". ;
up to, modulo, mod out by: An extension to mathematical discourse of the notions of
modular arithmetic. A statement is true
up to a condition if the establishment of that condition is the only impediment to the truth of the statement. Also used when working with members of
equivalence classes, especially in
category theory, where the
equivalence relation is (categorical) isomorphism; for example, "The tensor product in a weak
monoidal category is associative and unital up to a
natural isomorphism." ; vanish: To assume the value 0. For example, "The function sin(
x) vanishes for those values of
x that are integer multiples of
π." This can also apply to limits: see
Vanish at infinity. ; weak, weaker: The converse of
strong. ; well-defined: Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of some ; the result of the definition must then be independent of this choice. ==Proof terminology==