LN5 (Special Binary Relations)

LN5 (Special Binary Relations) - Special Binary Relations...

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Special Binary Relations
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Special relations Definition . A binary relation on a set is reflexive if for all , . Example . Let . Consider the following binary relations on : 1. 2. 3. Among the three given relation, only and are reflexive. Note that for any set , a binary relation on is reflexive if and only if .
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Definition . A binary relation on a set is irreflexive if for all . Note that is irreflexive if and only if . Example . Let . The following relation, Is not reflexive and not irreflexive. The relation Is not reflexive and it is irreflexive. Note that reflexive and irreflexive are not negations of each other. If a relation is not reflexive, it does not follow that it is irreflexive.
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Exercises 1. Show that for any set , the binary relation is reflexive while the relation is irreflexive. 2. Show that for any set , the binary relations and are reflexive but is irreflexive. Definition . A binary relation on a set is called symmetric if it satisfies the following property:
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