Everything about Trivial Mathematics totally explained
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're sometimes more difficult to visualize or understand than other, more complicated objects.
Examples include:
Trivial also refers to solutions to an
equation that have a very simple structure, but for the sake of completeness can't 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 -> e
x, the
exponential function.
Similarly, mathematicians often describe
Fermat's last theorem as asserting that there are no nontrivial 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, as is
a = 1,
b = 0,
c = 1. But such solutions are all obvious and uninteresting, and hence "trivial".
Trivial may also refer to any easy
case of a proof, which for the sake of completeness can't be ignored. For instance, proofs by
mathematical induction usually have two parts: a part that shows that if the theorem is true for a certain value of
n, it's also true for the value
n+1, and a so-called "base case" that shows that the theorem is true for the particular value
n=0. The base case is often trivial and is identified as such. Similarly, one might want to prove that some property is possessed by all the members of a certain set. The main part of the proof will consider the case of a nonempty set, and examine the members in detail; in the case where the set is empty, the property is trivially possessed by all the members, since there are none. (See also
Vacuous truth.)
A common joke in the mathematical community is to say that "trivial" is synonymous with "proved" — that is, any theorem can be considered "trivial" once it's known to be true. Another joke concerns two mathematicians who are discussing a theorem; the first mathematician says that the theorem is "trivial". In response to the other's request for an explanation, he then proceeds with twenty minutes of exposition. At the end of the explanation, the second mathematician agrees that the theorem is trivial. These jokes point out the subjectivity of judgements about triviality. Someone experienced in
calculus, for example, would consider the theorem that
»
to be trivial. To a beginning student of calculus, though, this may not be obvious at all.
Note that triviality also depends on context. A proof in
functional analysis would probably, given a number, trivially assume the existence of a larger number. When proving basic results about the natural numbers in
elementary number theory though, the proof may very well hinge on the remark that any natural number has a successor (which should then in itself be proved or taken as an
axiom, see
Peano's axioms).
Further Information
Get more info on 'Trivial Mathematics'.
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