“ordered pair”的概念、定义、翻译、参考文献-科学参考

ordered pair

Mathematics

An ordered pair consists of two objects considered in a particular order. Thus, if a ≠ b, the ordered pair (a,b) does not equal the ordered pair (b,a). See Cartesian product, pair.

Computer

A pair of objects in a given fixed order, usually represented by
$(x, y) or (x, y)$
for objects x and y. Two ordered pairs are said to be equal if and only if the first elements of each pair are equal and the second elements are equal. Typical situations in which ordered pairs are used include discussion of points in the Cartesian plane or complex numbers.
The idea can be extended to cover ordered triples, such as
$(x_{1}, x_{2}, x_{3})$
and indeed ordered n-tuples, such as
$(x_{1}, x_{2}, \dots x_{n})$

Philosophy

An object associated with a set of two objects considered in a particular order. Thus the ordered pair <a,b> is not identical with the ordered pair <b,a>, just as the queue ‘a first and b next’ is not the queue ‘b first and a next’. In set theory, ordered pairs cannot be identified with the set of their members, since {a,b}={b,a}, which is exactly what we do not want. The task is to find sets with the property that <a,b> ≠a <b,a> if a ≠b, and with the property that if <a,b>=<u,v> then a=u and b=v. One solution is that of Kazimierz Kuratowski (1896–1980), that identifies the ordered pair <a,b> with the set {{a},{a,b}}. This is the definition in general use today. See also Cartesian product.

单词	ordered pair
释义	ordered pair Mathematics An ordered pair consists of two objects considered in a particular order. Thus, if a ≠ b, the ordered pair (a,b) does not equal the ordered pair (b,a). See Cartesian product, pair. Computer A pair of objects in a given fixed order, usually represented by $(x, y) or (x, y)$ for objects x and y. Two ordered pairs are said to be equal if and only if the first elements of each pair are equal and the second elements are equal. Typical situations in which ordered pairs are used include discussion of points in the Cartesian plane or complex numbers. The idea can be extended to cover ordered triples, such as $(x_{1}, x_{2}, x_{3})$ and indeed ordered n-tuples, such as $(x_{1}, x_{2}, \dots x_{n})$ Philosophy An object associated with a set of two objects considered in a particular order. Thus the ordered pair <a,b> is not identical with the ordered pair <b,a>, just as the queue ‘a first and b next’ is not the queue ‘b first and a next’. In set theory, ordered pairs cannot be identified with the set of their members, since {a,b}={b,a}, which is exactly what we do not want. The task is to find sets with the property that <a,b> ≠a <b,a> if a ≠b, and with the property that if <a,b>=<u,v> then a=u and b=v. One solution is that of Kazimierz Kuratowski (1896–1980), that identifies the ordered pair <a,b> with the set {{a},{a,b}}. This is the definition in general use today. See also Cartesian product.
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