Properties of binary relations

binary relation over a set in which every element is related to itself

type of binary relation

binary relation such that if A is related to B and is different from it then B is not related to A

partial order where all elements can be compared

function whose actual domain of definition may be smaller than its input set

thumb|x R y defined by xinteger division|//4≤y//4 is a preorder on the [[natural numbers. It corresponds to the equivalence relation x E y defined by x//4=y//4. The set of equivalence classes is partially ordered, and thus can be shown as a Hasse diagram (depicted).]]

binary relation such that if A is related to B then B is not related to A

binary endorelation such that for every pair of elements either first is related with second or second is related with first

partial order where every two distinct comparable elements have another element between them

law (every real number is either positive, negative, or zero)

generalization of total orderings allowing ties, axiomatized as strict weak orders, total preorders, or ordered partitions

relation ∼ such that, for every a, b, c, if a∼b and a∼c, then b∼c

In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. That is, we can find three values a, b, and c where the transitive condition does not hold.