Trivial (mathematics)
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g.,
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
s,
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s). The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval
trivium The trivium is the lower division of the seven liberal arts and comprises grammar, logic, and rhetoric. The trivium is implicit in ''De nuptiis Philologiae et Mercurii'' ("On the Marriage of Philology and Mercury") by Martianus Capella, but t ...
curriculum, which distinguishes from the more difficult
quadrivium From the time of Plato through the Middle Ages, the ''quadrivium'' (plural: quadrivia) was a grouping of four subjects or arts—arithmetic, geometry, music, and astronomy—that formed a second curricular stage following preparatory work in the ...
curriculum. The opposite of trivial is nontrivial, which is commonly used to indicate that an example or a solution is not simple, or that a statement or a theorem is not easy to prove. The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at all. And there can be an argument about how quickly and easily a problem should be recognized for the problem to be treated as trivial. So, triviality is not a universally agreed property in mathematics and logic.


Trivial and nontrivial solutions

In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others *
Empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
: the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
containing no or null members *
Trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
: the mathematical
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
containing only the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
*
Trivial ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
: a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
defined on a
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
''"''Trivial''"'' can also be used to describe solutions to an
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solutions. For example, consider the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
y'=y where y = y(x) is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
whose
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
is y'. The trivial solution is the
zero function 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
y(x) = 0 while a nontrivial solution is the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
y(x) = e^x . The differential equation f''(x) = -\lambda f(x) with boundary conditions f(0) = f(L) = 0 is important in mathematics and physics, as it could be used to describe a
particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypo ...
in quantum mechanics, or a
standing wave In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect ...
on a string. It always includes the solution f(x) = 0, which is considered obvious and hence is called the "trivial" solution. In some cases, there may be other solutions (
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
s), which are called "nontrivial" solutions. Similarly, mathematicians often describe
Fermat's last theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
as asserting that there are no ''nontrivial'' integer solutions to the equation a^n + b^n = c^n, where ''n'' is greater than 2. Clearly, there are some solutions to the equation. For example, a = b = c = 0 is a solution for any ''n'', but such solutions are obvious and obtainable with little effort, and hence "trivial".


In mathematical reasoning

''Trivial'' may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored. For instance, proofs by
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
have two parts: the "base case" which shows that the theorem is true for a particular initial value (such as ''n'' = 0 or ''n'' = 1), and the inductive step which shows that if the theorem is true for a certain value of ''n'', then it is also true for the value ''n'' + 1. The base case is often trivial and is identified as such, although there are situations where the base case is difficult but the inductive step is trivial. Similarly, one might want to prove that some property is possessed by all the members of a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty, the property is trivially possessed by all the members of the empty set, since there are none (see
vacuous truth In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she d ...
for more). The judgement of whether a situation under consideration is trivial or not depends on who considers it since the situation is obviously true for someone who has sufficient knowledge or experience of it while to someone who has never seen this, it may be even hard to be understood so not trivial at all. And there can be an argument about how quickly and easily a problem should be recognized for the problem to be treated as trivial. The following examples show the subjectivity and ambiguity of the triviality judgement. * A common joke in the mathematical community is to say that "trivial" is synonymous with "proved"—that is, any theorem can be considered "trivial" once it is known to be proved as true. * Two mathematicians who are discussing a theorem: the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial. But can we say that this theorem is trivial even if it takes a lot of time and effort to prove it? * When a mathematician says that a theorem is trivial, but he is unable to prove it by himself at the moment that he says it as trivial. Then is the theorem trivial? * Often, as a joke, a problem is referred to as "intuitively obvious". For example, someone experienced in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
would consider the following statement trivial:\int_0^1 x^2\, dx = \fracHowever, to someone with no knowledge of integral calculus, this is not obvious at all so is not trivial. Triviality also depends on context. A proof in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
would probably, given a number, trivially assume the existence of a larger number. However, when proving basic results about the natural numbers in
elementary number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, the proof may very well hinge on the remark that any natural number has a successor – a statement which should itself be proved or be taken as an
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
so is not trivial (for more, see
Peano's axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
).


Trivial proofs

In some texts, a ''trivial proof'' refers to a statement involving a material implication ''P''→''Q,'' where the
consequent A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
''Q'', is always true. Here, the proof follows immediately by virtue of the definition of material implication in which as the implication is true regardless of the truth value of the antecedent ''P'' if the consequent is fixed as true. A related concept is a
vacuous truth In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she d ...
, where the antecedent ''P'' in a material implication ''P''→''Q'' is false. In this case, the implication is always true regardless of the truth value of the consequent ''Q'' – again by virtue of the definition of material implication.


Examples

*In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, it is often important to find
factors Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, suc ...
of an integer number ''N''. Any number ''N'' has four obvious factors: ±1 and ±''N''. These are called "trivial factors". Any other factor, if it exists, would be called "nontrivial". *The homogeneous
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
equation A\mathbf=\mathbf, where A is a fixed matrix, \mathbf is an unknown vector, and \mathbf is the zero vector, has an obvious solution \mathbf=\mathbf. This is called the "trivial solution". Any other solutions, with \mathbf\neq\mathbf, are called "nontrivial". *In
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, there is a very simple group with just one element in it; this is often called the "trivial group". All other groups, which are more complicated, are called "nontrivial". *In
graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, the trivial graph is a graph which has only 1 vertex and no edge. *
Database theory Database theory encapsulates a broad range of topics related to the study and research of the theoretical realm of databases and database management systems. Theoretical aspects of data management include, among other areas, the foundations of q ...
has a concept called
functional dependency In relational database theory, a functional dependency is a constraint between two sets of attributes in a relation from a database. In other words, a functional dependency is a constraint between two attributes in a relation. Given a relation ' ...
, written X \to Y . The dependence X \to Y is true if ''Y'' is a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of ''X'', so this type of dependence is called "trivial". All other dependences, which are less obvious, are called "nontrivial". * It can be shown that
Riemann's zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
has zeros at the negative even numbers −2, −4, … Though the proof is comparatively easy, this result would still not normally be called trivial; however, it is in this case, for its ''other'' zeros are generally unknown and have important applications and involve open questions (such as the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
). Accordingly, the negative even numbers are called the trivial zeros of the function, while any other zeros are considered to be non-trivial.


See also

* Degeneracy *
Initial and terminal objects In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...
*
List of mathematical jargon The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in l ...
*
Pathological Pathology is the study of the causal, causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when us ...
*
Trivialism Trivialism is the logical theory that all statements (also known as propositions) are true and that all contradictions of the form "p and not p" (e.g. the ball is red and not red) are true. In accordance with this, a trivialist is a person who b ...
*
Trivial measure In mathematics, specifically in measure theory, the trivial measure on any measurable space (''X'', Σ) is the measure ''μ'' which assigns zero measure to every measurable set: ''μ''(''A'') = 0 for all ''A'' in Σ. Properties of the trivial mea ...
*
Trivial representation In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ...
*
Trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequ ...


References


External links


Trivial entry at MathWorld
{{DEFAULTSORT:Trivial (Mathematics) Mathematical terminology