Noun
/ˈæks.i.əm əv pɛr/
The "axiom of pair" is a fundamental concept in set theory, specifically within the context of Zermelo-Fraenkel set theory (ZF), which is one of the most commonly used foundations for mathematics. This axiom states that for any two sets, there exists a set that contains exactly these two sets as its elements. It's a central concept because it allows for the construction of pairs and, subsequently, more complex structures in set theory.
This term is used primarily in mathematical and logical discussions, especially in advanced studies of set theory. It is more frequently found in written contexts, such as textbooks, academic papers, and research articles, rather than in casual oral speech.
The axiom of pair states that for any two sets, there exists a set containing those sets.
(El axioma de pares establece que para cualquier par de conjuntos, existe un conjunto que contiene esos conjuntos.)
Using the axiom of pair, mathematicians can construct ordered pairs crucial for defining functions.
(Usando el axioma de pares, los matemáticos pueden construir pares ordenados que son cruciales para definir funciones.)
The axiom of pair is one of the cornerstones of Zermelo-Fraenkel set theory.
(El axioma de pares es una de las piedras angulares de la teoría de conjuntos de Zermelo-Fraenkel.)
While "axiom of pair" does not typically appear in idiomatic expressions, in the context of mathematical discussions, the concepts related to it may lead to more idiomatic phrases. Here are some related expressions using the concept of pairs:
"Two sides of the same coin" (referring to two closely related aspects)
(Dos lados de la misma moneda, refiriéndose a dos aspectos estrechamente relacionados.)
"Birds of a feather flock together" (indicating similarity or connection between pairs)
(Aves de una misma pluma se agrupan, indicando similitudes o conexiones entre pares.)
"It takes two to tango" (suggesting that both parties are responsible in a situation, often implying pairs in relationships)
(Se necesitan dos para bailar un tango, sugiriendo que ambas partes son responsables en una situación, a menudo implicando pares en relaciones.)
The term "axiom" originates from the Greek word "axioma," which means "that which is deemed worthy or fit." The term "pair" comes from the Latin "paria," which refers to two that are considered as one unit.
The "axiom of pair" is a crucial concept in mathematical discourse, especially for those delving into formal set theory. Understanding its implications not only enriches mathematical comprehension but also improves logical reasoning skills.