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 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 the nontrivial solution is
- y = ex, 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 usually have two parts: a part that shows that if the theorem is true for a certain value of n, it is also true for the value n+1, and a so-called "base case" that shows that the theorem is true for the particular value n=0. The base case is often trivial and is identified as such. 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 up the subjectivity of judgements about triviality. Someone experienced in calculus, for example, would consider the theorem that
to be trivial. To a beginning student of calculus, though, it might be quite difficult.
