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".
Read more about this topic: Triviality (mathematics)
Famous quotes containing the words trivial and, trivial and/or solutions:
“For my part, I would rather look toward Rutland than Jerusalem. Rutland,—modern town,—land of ruts,—trivial and worn,—not too sacred,—with no holy sepulchre, but profane green fields and dusty roads, and opportunity to live as holy a life as you can, where the sacredness, if there is any, is all in yourself and not in the place.”
—Henry David Thoreau (1817–1862)
“All true histories contain instruction; though, in some, the treasure may be hard to find, and when found, so trivial in quantity that the dry, shrivelled kernel scarcely compensates for the trouble of cracking the nut.”
—Anne Brontë (1820–1849)
“Science fiction writers foresee the inevitable, and although problems and catastrophes may be inevitable, solutions are not.”
—Isaac Asimov (1920–1992)