chap mathematical preliminaries example for the integersis an equivalence relation that partitions each element into distinct subset in other wordsfor any integer athree things are true if then aand if and cthen of coursefor distinct integers aband there are never cases where bb aor so the claims that is symmetric and transitive are vacuously true (there are never examples in the relation where these events occurbut because the requirements for symmetry and transitivity are not violatedthe relation is symmetric and transitive example if we clarify the definition of sibling to mean that person is sibling of himor herselfthen the sibling relation is an equivalence relation that partitions the set of people example we can use the modulus function (defined in the next sectionto define an equivalence relation for the set of integersuse the modulus function to define binary relation such that two numbers and are in the relation if and only if mod mod thusfor is in the relation because mod mod we see that modulus used in this way defines an equivalence relation on the integersand this relation can be used to partition the integers into equivalence classes this relation is an equivalence relation because mod mod for all if mod mod mthen mod mod mand if mod mod and mod mod mthen mod mod binary relation is called partial order if it is antisymmetric and transitive the set on which the partial order is defined is called partially ordered set or poset elements and of set are comparable under given relation if either not all authors use this definition for partial order have seen at least three significantly different definitions in the literature have selected the one that lets and <both define partial orders on the integersbecause this seems the most natural to me