Mathematical proofs a transition to advanced mathematics pdf


This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. The noun triviality usually mathematical proofs a transition to advanced mathematics pdf to a simple technical aspect of some proof or definition.

The origin of the term in mathematical language comes from the medieval trivium curriculum. The antonym nontrivial is commonly used by engineers and mathematicians to indicate a statement or theorem that is not obvious or easy to prove.

Trivial can also be used to describe 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 solutions. This solution is considered obvious and is called the “trivial” solution. Clearly, there are some solutions to the equation.

Trivial may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored. The base case is often trivial and is identified as such, although there are situations where the base case is difficult but the inductive step is trivial. Similarly, one might want to prove that some property is possessed by all the members of a certain set. 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 is known to be true.