关于序偶和二元关系的基本概念(摘自Wikipedia)

序偶(Ordered pair)的基本概念：

In mathematics, an ordered pair (a, b) is a pair of mathematical objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b.(In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.)

In the ordered pair (a, b), the object a is called the first entry, and the object b the secondentry of the pair.Alternatively, the objects are called the first and second coordinates, or the left and right projections of the ordered pair.

Let (a1, b1) and (a2, b2) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is:

(a1, b1) = (a2, b2) if and only if a1 = a2, b1 =b2.

The set of all ordered pairs whose first entry is in some set A andwhose second entry is in some set B is called the Cartesian product of A and B, and written A × B. A binary relation between sets A and B is a subset of A × B.

If one wishes to employ the notation for a different purpose (such as denoting open intervals on the real number line) the ordered pair may be denoted by the variant notation .

The left and right projection of a pair p is usually denoted by π₁(p) and π₂(p), or by π_l(p) and π_r(p), respectively.

二元关系(Binary relation)的基本概念：

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A.In other words, it is a subset of the Cartesian product A² = A × A.More generally, a binary relation between two sets A and B is a subset of A × B.The terms correspondence, dyadic relation and 2-placerelation are synonyms forbinary relation.

A binary relation R is usually defined as an ordered triple (X, Y, G) where X and Y are arbitrary sets (or classes), and G is a subset ofthe Cartesian product X × Y.The sets X and Y are called the domain (or the set of departure) and codomain (or the set of destination),respectively, of the relation, and G is called its graph.The statement (x,y) ∈ G is read "x is R-related to y", and is denoted by xRy or R(x,y).The latter notation corresponds to viewing R as the characteristic function on X × Y for the set of pairs of G.

An example is the "divides"relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with no integer that is not amultiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4,0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13.

Binary relations are used in many branchesof mathematics to model concepts like "is greater than", "is equal to", and "divides" in arithmetic,"is congruent to" in geometry,"is adjacent to" in graph theory,"is orthogonal to" in linear algebraandmany more. The concept of function isdefined as a special kind of binary relation. Binary relations are also heavily used in computer science.

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关于序偶和二元关系的基本概念(摘自Wikipedia)

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