A binary relation is a relation which relates pairs of objects.
Thus, the relation of triangle similarity is a binary relation over the set of triangles but the relation which says is a factorization of over the positive integers is not a binary relation because it takes 3 arguments.
Formal Definition and Notation
Formally, we say that a relation on sets and is a subset of (the Cartesian product of and ). We often write instead of . If (the case of most common interest), then we say that is a relation on .
Thus, in the example of above, we may let be the set of ordered pairs of triangles in the Euclidean plane which are similar to each other. We could also define a relation on the power set of a set , so that , or , if and only if and are subsets of and is a subset of . This is a common example of an order relation.
More generally, we say that a relation is a mathematical sentence in which two letters, and , are of particular interest. This more general definition is useful because it admits relations whose "domain" is a class of sets too large to constitute a set. For instance, the relation defined as applies to all sets, not just sets contained in some larger set.
Domain and Range
The domain of a binary relation over and , written , is defined to be the set . It is thus the set of the first components of the ordered pairs in .
The range of a binary relation over and , written , is defined to be the set . It is thus the set of the second components of the ordered pairs in .
Reflexivity, Symmetry and Transitivity
A binary relation over is defined to be reflexive if .
A binary relation over is defined to be symmetric if .
A binary relation over is defined to be anti-symmetric if .
A binary relation over is defined to be transitive if .
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