“set”的概念、定义、翻译、参考文献-科学参考

set

Mathematics

A well-defined collection of objects. It may be possible to define a set by listing the elements: {a, e, i, o, u} is the set consisting of the vowels of the alphabet, {1, 2,…, 100} is the set of the first 100 positive integers. The meaning of {1, 2, 3,…} is also clear: it is the set of all positive integers. It may be possible to define a set as consisting of all elements, from some universal set, that satisfy some property. Thus the set of all real numbers that are greater than 1 can be written as either {x | x ε‎ ℝ and x>1} or {x : x ε‎ ℝ and x>1}, both of which are read as ‘the set of x such that x belongs to ℝ and x is greater than 1’. The same set is sometimes written {x ε‎ ℝ | x>1}. See algebra of sets, russell’s paradox, Zermelo-Fraenkel axioms.

Computer

1. A collection of distinct objects of any sort. The objects in the set are called its members or elements. An element can occur at most once in a set and order or arrangement is unimportant. If x is a member of the set S it is customary to write
$x \in S$
If x is not a member of S this can be expressed as
$x \notin S$
and is equivalent to
$NOT (x \in S)$
i.e. ∈ and ∉ can be regarded as operators. When any element in set S is also in set T, and vice versa, the two sets are said to be identical or equal.
A finite set has a fixed finite number of members and a notation such as
${Ada, Pascal, Cobol, C}$
is possible; the members are separated by commas and here are just the names of various programming languages. When the number of elements is not finite, the set is said to be infinite and explicit enumeration of the elements is not then possible.
Infinite and finite sets can be described using a predicate or statement such as p(x) that involves x and is either true or false, thus
${x | p (x)}$
This is read as ‘the set of all elements x for which p(x) is true’, the elements being characterized by the common property p. Examples of sets described in this way are (letting R be the set of real numbers):
$\begin{array}{l} {(x, y) | x \in R, y \in R and x + y = 9} \\ {n | n is a prime number} \\ {l | l is the name of a language} \end{array}$
There is an implicit assumption here that there is some algorithm for deciding whether p(x) is true or false in any particular case.
The idea of a set is fundamental to mathematics. It forms the basis for all ideas involving functions, relations, and indeed any kind of algebraic structure. Authors differ considerably in the way they define sets. A mathematical logician will distinguish carefully between classes and sets, basically to ensure that paradoxes such as Russell’s paradox cannot occur in sets. However, the informal definition is adequate for most purposes. See also operations on sets.
2. Any data structure representing a set of elements. One example is a characteristic vector.
3. To cause the condition or state of a switch, signal, or storage location to change to the positive condition.

Philosophy

Intuitively a set is a collection of entities, called its members or elements, itself considered as a single object. The fundamental principle of the theory of sets is the principle of extensionality: sets are identical if and only if they have the same members. The union of two sets is the set A ∪ B that has as members all the things that are members of A or B (or both). The intersection A ∩ B is the set of things that are members of both. Sets are disjoint when they have no common members. The complement of a set B within a set A, A − B, is the set of elements that are in A but not in B. A set A is a subset of a set B when all the things that belong to A belong to B. This makes A itself a subset of A; a subset of A not itself identical with A is a proper subset of A. To obtain the set theoretic hierarchy, we start with a list of elements (things that are not themselves sets; in case this sounds mathematically impure, we can start simply with ∆, the null set). At the bottom level we have the set of all these elements. At the next level we add all sets of elements; at each level we have everything from the previous level, plus all sets of them. We then take the infinite union of all these sets, and continue ‘forever’. In fact, if we start with the null set at the lowest level, each ascending level becomes the power set of the set that constitutes the previous level.

单词	set
释义	set Mathematics A well-defined collection of objects. It may be possible to define a set by listing the elements: {a, e, i, o, u} is the set consisting of the vowels of the alphabet, {1, 2,…, 100} is the set of the first 100 positive integers. The meaning of {1, 2, 3,…} is also clear: it is the set of all positive integers. It may be possible to define a set as consisting of all elements, from some universal set, that satisfy some property. Thus the set of all real numbers that are greater than 1 can be written as either {x \| x ε‎ ℝ and x>1} or {x : x ε‎ ℝ and x>1}, both of which are read as ‘the set of x such that x belongs to ℝ and x is greater than 1’. The same set is sometimes written {x ε‎ ℝ \| x>1}. See algebra of sets, russell’s paradox, Zermelo-Fraenkel axioms. Computer 1. A collection of distinct objects of any sort. The objects in the set are called its members or elements. An element can occur at most once in a set and order or arrangement is unimportant. If x is a member of the set S it is customary to write $x \in S$ If x is not a member of S this can be expressed as $x \notin S$ and is equivalent to $NOT (x \in S)$ i.e. ∈ and ∉ can be regarded as operators. When any element in set S is also in set T, and vice versa, the two sets are said to be identical or equal. A finite set has a fixed finite number of members and a notation such as ${Ada, Pascal, Cobol, C}$ is possible; the members are separated by commas and here are just the names of various programming languages. When the number of elements is not finite, the set is said to be infinite and explicit enumeration of the elements is not then possible. Infinite and finite sets can be described using a predicate or statement such as p(x) that involves x and is either true or false, thus ${x \| p (x)}$ This is read as ‘the set of all elements x for which p(x) is true’, the elements being characterized by the common property p. Examples of sets described in this way are (letting R be the set of real numbers): $\begin{array}{l} {(x, y) \| x \in R, y \in R and x + y = 9} \\ {n \| n is a prime number} \\ {l \| l is the name of a language} \end{array}$ There is an implicit assumption here that there is some algorithm for deciding whether p(x) is true or false in any particular case. The idea of a set is fundamental to mathematics. It forms the basis for all ideas involving functions, relations, and indeed any kind of algebraic structure. Authors differ considerably in the way they define sets. A mathematical logician will distinguish carefully between classes and sets, basically to ensure that paradoxes such as Russell’s paradox cannot occur in sets. However, the informal definition is adequate for most purposes. See also operations on sets. 2. Any data structure representing a set of elements. One example is a characteristic vector. 3. To cause the condition or state of a switch, signal, or storage location to change to the positive condition. Philosophy Intuitively a set is a collection of entities, called its members or elements, itself considered as a single object. The fundamental principle of the theory of sets is the principle of extensionality: sets are identical if and only if they have the same members. The union of two sets is the set A ∪ B that has as members all the things that are members of A or B (or both). The intersection A ∩ B is the set of things that are members of both. Sets are disjoint when they have no common members. The complement of a set B within a set A, A − B, is the set of elements that are in A but not in B. A set A is a subset of a set B when all the things that belong to A belong to B. This makes A itself a subset of A; a subset of A not itself identical with A is a proper subset of A. To obtain the set theoretic hierarchy, we start with a list of elements (things that are not themselves sets; in case this sounds mathematically impure, we can start simply with ∆, the null set). At the bottom level we have the set of all these elements. At the next level we add all sets of elements; at each level we have everything from the previous level, plus all sets of them. We then take the infinite union of all these sets, and continue ‘forever’. In fact, if we start with the null set at the lowest level, each ascending level becomes the power set of the set that constitutes the previous level.
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