Triviality (mathematics) - Trivial and Nontrivial Solutions

Trivial and Nontrivial Solutions

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

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) = ex, the exponential function.

Similarly, mathematicians often describe Fermat's Last Theorem as asserting that there are no nontrivial integer solutions to the equation when n is greater than 2. Clearly, there are some solutions to the equation. For example, is a solution for any n, but such solutions are all obvious and uninteresting, and hence "trivial".