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:
“If the pages of this book contain some successful verse, the reader must excuse me the discourtesy of having usurped it first. Our nothingness differs little; it is a trivial and chance circumstance that you should be the reader of these exercises and I their author.”
—Jorge Luis Borges (1899–1986)
“In our most trivial walks, we are constantly, though unconsciously, steering like pilots by certain well-known beacons and headlands, and if we go beyond our usual course we still carry in our minds the bearing of some neighboring cape; and not till we are completely lost, or turned round,—for a man needs only to be turned round once with his eyes shut in this world to be lost,—do we appreciate the vastness and strangeness of nature.”
—Henry David Thoreau (1817–1862)
“Every man is in a state of conflict, owing to his attempt to reconcile himself and his relationship with life to his conception of harmony. This conflict makes his soul a battlefield, where the forces that wish this reconciliation fight those that do not and reject the alternative solutions they offer. Works of art are attempts to fight out this conflict in the imaginative world.”
—Rebecca West (1892–1983)