Relations
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General Binary Relations

This section lists general constructors of relations.

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Properties and Attributes of Binary Relations
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Binary Relations on Points

We have special construction methods when the underlying X of our relation is the set of integers \{ 1, \ldots, n \}.

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Closure Operations and Other Constructors
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Equivalence Relations
equivalence relation
An equivalence relation E over the set X is a relation on X which is reflexive, symmetric, and transitive. A partition P is a set of subsets of X such that for all R, S \in P, R \cap S is the empty set and \cup P = X. An equivalence relation induces a partition such that if (x,y) \in E then x, y are in the same element of P.

Like all binary relations in &GAP; equivalence relations are regarded as general endomorphic mappings (and the operations, properties and attributes of general mappings are available). However, partitions provide an efficient way of representing equivalence relations. Moreover, only the non-singleton classes or blocks are listed allowing for small equivalence relations to be represented on infinite sets. Hence the main attribute of equivalence relations is which provides the partition induced by the given equivalence.

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Attributes of and Operations on Equivalence Relations
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Equivalence Classes
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