Connected for all x, y ∈ X, if x ≠ y then xRy or yRx. For example, "is ancestor of" is a transitive relation, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric. Transitive for all x, y, z ∈ X, if xRy and yRz then xRz. For example, > is an asymmetric relation, but ≥ is not.Īgain, the previous 3 alternatives are far from being exhaustive as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric nor antisymmetric, let alone asymmetric. A relation is asymmetric if and only if it is both antisymmetric and irreflexive. Asymmetric for all x, y ∈ X, if xRy then not yRx.
For example, ≥ is an antisymmetric relation so is >, but vacuously (the condition in the definition is always false). Antisymmetric for all x, y ∈ X, if xRy and yRx then x = y. For example, "is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x. Symmetric for all x, y ∈ X, if xRy then yRx. The previous 2 alternatives are not exhaustive e.g., the red binary relation y = x 2 given in the section § Special types of binary relations is neither irreflexive, nor reflexive, since it contains the pair (0, 0), but not (2, 2), respectively. For example, > is an irreflexive relation, but ≥ is not. Irreflexive (or strict) for all x ∈ X, not xRx.
For example, ≥ is a reflexive relation but > is not. Some important properties that a relation R over a set X may have are: In the boolean maxtrix representing R div, the element in line x, column y is " ".
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