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)
“Men of genius are not quick judges of character. Deep thinking and high imagining blunt that trivial instinct by which you and I size people up.”
—Max Beerbohm (1872–1956)
“The anorexic prefigures this culture in rather a poetic fashion by trying to keep it at bay. He refuses lack. He says: I lack nothing, therefore I shall not eat. With the overweight person, it is the opposite: he refuses fullness, repletion. He says, I lack everything, so I will eat anything at all. The anorexic staves off lack by emptiness, the overweight person staves off fullness by excess. Both are homeopathic final solutions, solutions by extermination.”
—Jean Baudrillard (b. 1929)