17. Irreflexive Relation
The chapter delves into various types of binary relations, including irreflexive, symmetric, asymmetric, antisymmetric, and transitive relations, outlining their definitions and characteristics. It also discusses matrices representing these relations and explores their implications in terms of directed graphs. Moreover, the chapter emphasizes the distinctions and potential overlaps between reflexivity and irreflexivity, as well as the absence of direct relationships among symmetric, asymmetric, and antisymmetric properties.
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What we have learnt
- Irreflexive relations do not contain any pairs of the form (a, a).
- Symmetric relations require that if (a, b) is in the relation, then (b, a) must also be present.
- Antisymmetric relations allow (a, b) and (b, a) only if a equals b.
- Asymmetric relations disallow (b, a) whenever (a, b) is present, and diagonal entries are zero.
- Transitive relations state that if (a, b) and (b, c) exist, then (a, c) must also exist.
Key Concepts
- -- Irreflexive Relation
- A relation is irreflexive if no element in the set is related to itself.
- -- Symmetric Relation
- A relation is symmetric if whenever (a, b) is in the relation, (b, a) must also be in the relation.
- -- Asymmetric Relation
- A relation is asymmetric if for (a, b) in the relation, (b, a) cannot be in the relation.
- -- Antisymmetric Relation
- A relation is antisymmetric if both (a, b) and (b, a) can only exist if a equals b.
- -- Transitive Relation
- A relation is transitive if whenever (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation.
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