“union”的概念、定义、翻译、参考文献-科学参考

union

Physics

See sets.

Mathematics

The union of sets A and B is the set consisting of all objects that belong to A or B (or both), and it is denoted by A ∪ B (read as ‘A union B’). Thus the term ‘union’ is used for both the resulting set and the operation, a binary operation on the set of all subsets of a given set. The following properties hold:
1. (i) For all A, A ∪ A = A and A ∪ Ø = A.
2. (ii) For all A and B, A ∪ B = B ∪ A; that is, ∪ is commutative.
3. (iii) For all A, B, and C, (A ∪ B) ∪ C = A ∪ (B ∪ C); that is, ∪ is associative.
  
  Given a family of sets A_i with index set I, the union
  
  $⋃_{i ε I} A_{i}$
  
  consists of those elements x such that x is an element of A_i for at least one i in I.
Compare disjoint union. For the union of two events, see event.

Statistics

The union of two events (see sample space) A and B is the event ‘either A or B occurs’, denoted by A ⋃ B. It should be noted that the ‘or’ is inclusive and therefore the union includes the case when both events occur. The exclusive union of A and B is the event ‘either A occurs or B occurs but not both’, i.e. (A ∩ B′) ⋃ (A′ ∩ B), where A′ and B′ are the complementary events to A and B, respectively. For any events A, B, C, $union$ where S is the sample space and φ is the empty set. The union of the n events A₁, A₂,…, A_n is the event ‘at least one of A₁, A₂,…, A_n occurs’. It is denoted by A₁ ⋃ A₂ ⋃⋯⋃ A_n. See also intersection; Venn diagram.

Chemical Engineering

A ring-like device used to couple or link together pipes and tubes.

Computer

1. (of two sets) The set that results from combining the elements of two sets S and T, say, usually expressed as
$S \cup T$
∪ is regarded as an operation on sets, the union operation, which is commutative and associative. Symbolically
$S \cup T = {x | x \in S or x \in T}$
The union of S and the empty set is S. See also set algebra.
2. (of two graphs, G₁ and G₂) The graph that includes all the vertices and edges of G₁ and G₂, i.e. that contains the union of the two sets of vertices and of the two sets of edges as its vertices and edges.

Philosophy

The union of two sets is the set of things that are members of either. Formally, x ∈ S ∪ R iff x ∈ S ∨ x ∈ R. However, the union of a single set, ∪ A, is the union of all its subsets, that is x ∈ ∪ A iff (∃S) (S ∈ A & x ∈ S). This connects with the first notion via the fact that ∪ {S, R}=S ∪ R.

Economics

See customs union; payments union; trade union.

单词	union
释义	union Physics See sets. Mathematics The union of sets A and B is the set consisting of all objects that belong to A or B (or both), and it is denoted by A ∪ B (read as ‘A union B’). Thus the term ‘union’ is used for both the resulting set and the operation, a binary operation on the set of all subsets of a given set. The following properties hold: (i) For all A, A ∪ A = A and A ∪ Ø = A. (ii) For all A and B, A ∪ B = B ∪ A; that is, ∪ is commutative. (iii) For all A, B, and C, (A ∪ B) ∪ C = A ∪ (B ∪ C); that is, ∪ is associative. Given a family of sets A_i with index set I, the union $⋃_{i ε I} A_{i}$ consists of those elements x such that x is an element of A_i for at least one i in I. Compare disjoint union. For the union of two events, see event. Statistics The union of two events (see sample space) A and B is the event ‘either A or B occurs’, denoted by A ⋃ B. It should be noted that the ‘or’ is inclusive and therefore the union includes the case when both events occur. The exclusive union of A and B is the event ‘either A occurs or B occurs but not both’, i.e. (A ∩ B′) ⋃ (A′ ∩ B), where A′ and B′ are the complementary events to A and B, respectively. For any events A, B, C, $union$ where S is the sample space and φ is the empty set. The union of the n events A₁, A₂,…, A_n is the event ‘at least one of A₁, A₂,…, A_n occurs’. It is denoted by A₁ ⋃ A₂ ⋃⋯⋃ A_n. See also intersection; Venn diagram. Chemical Engineering A ring-like device used to couple or link together pipes and tubes. Computer 1. (of two sets) The set that results from combining the elements of two sets S and T, say, usually expressed as $S \cup T$ ∪ is regarded as an operation on sets, the union operation, which is commutative and associative. Symbolically $S \cup T = {x \| x \in S or x \in T}$ The union of S and the empty set is S. See also set algebra. 2. (of two graphs, G₁ and G₂) The graph that includes all the vertices and edges of G₁ and G₂, i.e. that contains the union of the two sets of vertices and of the two sets of edges as its vertices and edges. Philosophy The union of two sets is the set of things that are members of either. Formally, x ∈ S ∪ R iff x ∈ S ∨ x ∈ R. However, the union of a single set, ∪ A, is the union of all its subsets, that is x ∈ ∪ A iff (∃S) (S ∈ A & x ∈ S). This connects with the first notion via the fact that ∪ {S, R}=S ∪ R. Economics See customs union; payments union; trade union.
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