well-ordered relation (english) - meaning, definition, translation, pronunciation

Part of Speech

"Well-ordered relation" functions as a noun phrase in English.

Phonetic Transcription

/wɛl ˈɔrdərd rɪˈleɪʃən/

Meaning and Usage

A "well-ordered relation" is a concept used in mathematics, particularly in order theory. It refers to a binary relation on a set that is both total and well-founded. This means that any two elements are comparable (total) and that every non-empty subset has a least element (well-founded).

• Frequency of Use: This term is primarily used in academic or technical contexts, especially in mathematics and formal logic. It is more common in written form than in spoken discourse due to its technical nature.

Example Sentences

1. A well-ordered relation ensures that for any two elements in the set, one can determine which is greater.

2. Translation: Una relación bien ordenada asegura que para cualquier par de elementos en el conjunto, se puede determinar cuál es mayor.

3. In set theory, a well-ordered relation is essential for defining ordinal numbers.

4. Translation: En teoría de conjuntos, una relación bien ordenada es esencial para definir los números ordinales.

5. The mathematician proved that every well-ordered relation has a unique least element.

6. Translation: El matemático probó que toda relación bien ordenada tiene un único elemento mínimo.

Idiomatic Expressions

While "well-ordered relation" is a specialized term without common idiomatic expressions, it can relate to several concepts in mathematics and philosophy concerning order, structure, and comparison.

• "Better the devil you know": This phrase can be related when discussing well-ordered relations, as one may prefer a familiar order or structure over an unfamiliar one.

• Translation: Es mejor el diablo que conoces.

• "Put in order": This idiom reflects the importance of organization and comparison, akin to establishing a well-ordered relation in a set.

• Translation: Poner en orden.

• "Order of the day": Refers to what is most relevant or is a priority, relating to the importance of establishing a systematic order similar to well-ordered relations.

• Translation: Orden del día.

Example Sentences with Related Idiomatic Expressions

1. In mathematics, it's better the devil you know than introducing a more complex well-ordered relation.

2. Translation: En matemáticas, es mejor el diablo que conoces que introducir una relación bien ordenada más compleja.

3. To solve the problem, we need to put in order our definition of a well-ordered relation.

4. Translation: Para resolver el problema, necesitamos poner en orden nuestra definición de una relación bien ordenada.

5. Establishing a well-ordered relation should be the order of the day in our research discussions.

6. Translation: Establecer una relación bien ordenada debería ser el orden del día en nuestras discusiones de investigación.

Etymology

The term "well-ordered relation" can be broken down into: - "Well" from Old English "wel," meaning in a good manner. - "Ordered" from the Latin "ordinatus," meaning arranged or organized. - "Relation" comes from the Latin "relatio," meaning to bring back, connection, or reference.

Synonyms and Antonyms

• Synonyms: Total order, linear order, ordered set.
• Antonyms: Disorder, chaotic relation, non-comparable relation.