Trivial (mathematics)

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In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure. For non-mathematicians, they are sometimes more difficult to visualize or understand than other, more complicated objects.

Examples include:

• empty set: the set containing no members
• trivial group: the mathematical group containing only the identity element
• trivial ring: a ring defined on a singleton set.

Trivial also refers to 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 solution. For example, consider the differential equation

y' = y

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

y = 0, the zero function

while a nontrivial solution is

y (x) = e^x, the exponential function.

Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial solutions to the equation aⁿ + bⁿ = cⁿ when 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, as is a = 1, b = 0, c = 1. But such solutions are all obvious and uninteresting, and hence "trivial".

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" that shows that the theorem is true for a particular initial value such as n=0 or n = 1 and then an inductive step that shows that if the theorem is true for a certain value of n, it is also true for the value n+1. The base case is often trivial and is identified as such, although there are cases 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 also Vacuous truth.)

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. 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. Someone experienced in calculus, for example, would consider the theorem that

to be trivial. To a beginning student of calculus, though, this may not be obvious at all.

Note that triviality also depends on context. A proof in functional analysis would probably, given a number, trivially assume the existence of a larger number. When proving basic results about the natural numbers in elementary number theory though, the proof may very well hinge on the remark that any natural number has a successor (which should then in itself be proved or taken as an axiom, see Peano's axioms).