Triviality (mathematics)

In 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., groups, topological spac e).[1][2][3] 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 curriculum, which distinguishes from the more difficult quadrivium curriculum.[2][4] 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.[1][3]

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: the set containing no or null members
Trivial group: the mathematical group containing only the identity element
Trivial ring: a ring defined on a singleton set

"Trivial" can also be used to describe solutions to an equation 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

y'=y

where y=y(x) is a function whose derivative is y'. The trivial solution is

\( {\displaystyle y(x)=0} \), the zero function

while a nontrivial solution is

\( {\displaystyle y(x)=e^{x}} \), the exponential function.

The differential equation \( f''(x)=-\lambda f(x) \) with boundary conditions f(0)=f(L)=0 is important in math and physics, as it could be used to describe a particle in a box in quantum mechanics, or a standing wave 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 (sinusoids), which are called "nontrivial" solutions.[5]

Similarly, mathematicians often describe Fermat's Last Theorem 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 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, since there are none (see vacuous truth for more).

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 true.[2]

Another joke concerns 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. These jokes point out the subjectivity of judgments about triviality. The joke also applies when the first mathematician says the theorem is trivial, but is unable to prove it himself. Often, as a joke, the theorem is then referred to as "intuitively obvious". Someone experienced in calculus, for example, would consider the following statement trivial:

\( \int _{0}^{1}x^{2}\,dx={\frac {1}{3}} \)

However, to someone with no knowledge of integral calculus, this is not obvious at all.

Triviality also depends on context. A proof in functional analysis 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, 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 (for more, see Peano's axioms).

Trivial proofs

In some texts, a trivial proof refers to a statement involving a material implication P→Q, where the consequent, Q, is always true.[6] Here, the proof follows immediately by virtue of the definition of material implication, as the implication is true regardless of the truth value of the antecedent P.[6]

A related concept is a vacuous truth, where the antecedent P in the material implication P→Q is always false.[6] Here, the implication is always true regardless of the truth value of the consequent Q—again by virtue of the definition of material implication.[6]
Examples

In number theory, it is often important to find factors 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".[7]
The homogeneous matrix equation \( {\displaystyle A\mathbf {x} =\mathbf {0} } \) , where A is a fixed matrix, \( \mathbf {x} \) is an unknown vector, and \( \mathbf {0} \) is the zero vector, has an obvious solution \( \mathbf {x} =\mathbf {0} \) . This is called the "trivial solution". If it has other solutions \( {\displaystyle \mathbf {x} \neq \mathbf {0} } \), then they would be called "nontrivial"[8]
In group theory, 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, the trivial graph is a graph which has only 1 vertex and no edge.
Database theory has a concept called functional dependency, written \( X\to Y \) . The dependence \( X\to Y \) is true if Y is a subset 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 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). Accordingly, the negative even numbers are called the trivial zeros of the function, while any other zeros are considered to be non-trivial.