# triviality

A circumstance is considered to be trivial ( Latin trivialis 'ordinary' ) that is considered obvious, easily visible or comprehensible for everyone.

Whether a circumstance is trivial or not, however, cannot necessarily be generalized: For example, the knowledge “that liquid water becomes solid at some point when temperatures drop” is certainly not trivial for members of isolated peoples in the tropics, as freezing occurs under these climatic conditions and cultural conditions cannot be observed. The assessment of whether something is trivial always depends on one's own knowledge and cultural background.

In addition, trivial in the sense of "everyday" or "insignificant" can also express an evaluation or a taste judgment, for example when evaluating literature that is only intended for entertainment.

## etymology

The adjective trivial goes back to the Latin trivialis via the French trivial . This is derived from the Latin trivium  “crossing of three ways” (literally “three way”, to tri  “three” and via “way”). Because the people met at such crossroads, the word trivialis developed the meanings "ordinary" and "accessible to everyone, generally known".

At the universities of the Middle Ages , the basic course consisted of three subjects: grammar , rhetoric , dialectic and was called trivium . This designation goes back to the original meaning "three ways". At the same time, the trivium, as a simple study, was opposed to the more demanding quadrivium (study of the four subjects arithmetic , geometry , music theory (harmony) and astronomy / astrology ). This connection also influenced the meaning of trivial , because the disciplines of the trivium were the trivial branch of the subject canon (see also trivial school ).

## Technical usage

In some technical languages , what is easily understood is referred to as trivial.

### Complexity (Theoretical Computer Science)

Trivial problems are related to the Turing reduction in the complexity class P mentioned. In this it is the two problems to which the others of class P cannot be Turing reduced. The problem is “always accept” and its complement “always reject”. With the Turing reduction, all instances of the original problem are mapped to instances of the target problem. Yes-instances are mapped to yes-instances and no-instances to no-instances. However, the trivial problems only have one of the two instance types, so that the instances of the other cannot be mapped.

A nontrivial property of a nonempty set is one that some, but not all, of the elements of the set possess. According to Rice's theorem , every nontrivial semantic (as opposed to "syntactic", i.e. directly readable from the character string of the program text) property of a program is undecidable. The problem of finding out whether a given program has this property cannot be solved algorithmically. ${\ displaystyle E}$${\ displaystyle E}$

### Software engineering

Methods or other services with the task of searching for or finding something can be described as trivial or non-trivial .

• Trivial are those that return a variable or some other simple value that they can access.
• Those that first have to find the value they are looking for are non-trivial. Usually a certain property is referenced. This is then searched for in a collection or a list-like construct according to various criteria and then returned.

### mathematics

Mathematical objects, statements or properties are called trivial if they can be specified particularly easily, i.e. H. result from a definition or a sentence without any action.

• The trivial divisors of a natural number are and themselves. You can state them without knowing anything more (such as the prime factorization ) about . All other factors are called nontrivial or real factors.${\ displaystyle n}$${\ displaystyle 1}$${\ displaystyle n}$${\ displaystyle n}$
• The trivial solution of a homogeneous system of linear equations is the zero solution . You can state it without knowing anything about it. All other solutions are called nontrivial solutions.${\ displaystyle Ax = 0}$${\ displaystyle x = 0}$${\ displaystyle A}$
• Trivial subsets of a set are the empty set and the set itself. All other subsets are called real subsets .
• A trivial group is a group that consists of only one element and the only possible trivial operation .${\ displaystyle (\ {e \}, *)}$${\ displaystyle e}$${\ displaystyle e * e = e}$
• A trivial ring is a ring that consists only of the zero element and the two operations and .${\ displaystyle (\ {0 \}, +, \, \ cdot \,)}$${\ displaystyle 0}$${\ displaystyle 0 + 0 = 0}$${\ displaystyle 0 \ cdot 0 = 0}$

## General language meanings

In addition to the meaning “simple, easily comprehensible”, trivial has other meanings in the general language, for example in the sense of “insignificant, uninteresting” or “without particular artistic value”. As Trivia mixed information is known, generally have no particular value or are irrelevant to a particular topic.

Triviality in this sense overlaps with related valuation terms. While triviality basically relates to easily understandable information, banality also has easy social access conditions (e.g. high circulation numbers, low price). With kitsch , the reflex-like emotional accessibility is in the foreground. These phenomena can also occur in combination. So true popular literature as a little complex, is everywhere available at low cost (eg. As booklets literature) and is often perceived as kitsch.

## Common names

So-called trivial names are designations for things that do not correspond to any official systematics, i.e. those that go beyond the German-speaking area, as they were usually defined in the assigned scientific subject areas. Examples of such subject areas are biology, chemistry, medicine and pharmacy, but also areas or focal points of the aforementioned scientific subject areas such as their technical branches.

## Web links

Wiktionary: Triviality  - explanations of meanings, word origins, synonyms, translations
Wiktionary: trivial  - explanations of meanings, word origins, synonyms, translations

## Individual evidence

1. a b Cf. trivial at Duden online.
2. See Trivium at Duden online.
3. Albrecht Beutelspacher : That is o. B. d. A. trivial! Vieweg, Wiesbaden 2004, ISBN 3-528-66442-8 , p. 41.
4. Julia Genz: Discourses of Valuation. Banality, triviality and kitsch. Wilhelm Fink, Munich 2011, ISBN 978-3-7705-5055-5 , pp. 62, 89 f.