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)
“The seashore is a sort of neutral ground, a most advantageous point from which to contemplate this world. It is even a trivial place. The waves forever rolling to the land are too far-traveled and untamable to be familiar. Creeping along the endless beach amid the sun-squall and the foam, it occurs to us that we, too, are the product of sea-slime.”
—Henry David Thoreau (1817–1862)
“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)