Relation mathematics - Remarks

Relations are classified according to the number of sets in the cartesian product, in other words the number of terms in the expression: Unary relation or property: L(u) Binary relation: L(u, v) or u L v Ternary relation: L(u, v, w) Quaternary relation: L(u, v, w, x) Relations with more than four terms are usually referred to a ...

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Because a relation as above defines uniquely an n-ary predicate that holds for x1, ..., xn if (x1, ..., xn) is in G(R), and vice versa, the relation and the predicate are often denoted with the same symbol. So, for example, the following two statements are considered to be equivalent: Relations are classified according to the number of sets in the Cartesian product; in other words the number of terms in the expre ...

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Relation mathematics, Relation mathematics - Definition, Relation mathematics - Remarks

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There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these definitions is as follows: Definition 1. A relation L over the sets X1, …, Xk is a subset of the cartesian product of those sets, written L ⊆ X1 × … × Xk. Under this definition, then, ...

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Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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There are two definitions of k-place relations that are commonly encountered in mathematics. In order of simplicity, the first of these definitions is as follows: Definition 1. A relation L over the sets X1, …, Xk is a subset of their cartesian product, written L ⊆ X1 × … × Xk. Under this definition, then, ...

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Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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A relation over the sets X1, ..., Xn is an (n + 1)-tuple R=(X1, ..., Xn, G(R)) where G(R) is a subset of X1 × ... × Xn (the Cartesian product of these sets). If X=X1=X2=...=Xn, R is simply called a relation over X. G(R) is called the graph of R and, similar to the case of binary relations, R is often identified with its graph. An n-ary predi ...

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A more typical example of a 2-place relation in mathematics is the relation of divisibility between two positive integers n and m that is expressed in statements like "n divides m" or "n goes into m". This is a relation that comes up so often that a special symbol "|" is reserved to express it, allowing one to write "n|m" for "n divides m". To express the binary relation of divisibility in terms of sets, we have the set P of positive integers, P ...

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Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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The definition of relation given in the next Section formally captures a concept that is actually quite familiar from everyday life. For example, consider the relationship, involving three roles that people might play, expressed in a statement of the form "X suspects that Y likes Z". The facts of a concrete situation could be organized in a Table like the following: Each row of the Table records a fact or makes an assertion of the form "X suspects that Y likes Z". For instance, the first row says, in effect ...

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Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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For lines L in three-dimensional space, there is a ternary relation picking out the triples of lines that are coplanar. This does not reduce to the binary symmetric relation of coplanarity of pairs of lines. In other words, writing P(L, M, N) when the lines L, M, and N lie in a plane, and Q(L, M) for the binary relation, it is not true that Q(L, M), Q(M, N) and Q(N, L) together imply PSee also:

Relation mathematics, Relation mathematics - Informal introduction, Relation mathematics - Example: divisibility, Relation mathematics - Formal definitions, Relation mathematics - Example: coplanarity, Relation mathematics - Remarks, Relation mathematics - Bibliography

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