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)
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—Ralph Waldo Emerson (1803–1882)
“Football strategy does not originate in a scrimmage: it is useless to expect solutions in a political compaign.”
—Walter Lippmann (1889–1974)