all it is not true that This condition holds if and only if the induced relation on is reflexive, where is defined by declaring that is true if and only if is . : For all if then : For all if is true then is false. : For all if is incomparable with (meaning that neither nor is true) and if is incomparable with then is incomparable with Two elements are incomparable with respect to if and only if they are equivalent with respect to the induced relation (which by definition means that are both true), where as before, is declared to be true if and only if is false. Thus this condition holds if and only if the symmetric relation on defined by " are equivalent with respect to " is a transitive relation, meaning that whenever are -equivalent and also are -equivalent then necessarily are -equivalent. This can also be restated as: whenever and also then necessarily Properties (1), (2), and (3) are the defining properties of a strict partial order, although there is some redundancy in this list as asymmetry (3) implies irreflexivity (1), and also because irreflexivity (1) and transitivity (2) together imply asymmetry (3). The incomparability relation is always symmetric and it will be reflexive if and only if is an irreflexive relation (which is assumed by the above definition). Consequently, a strict partial order is a strict weak order if and only if its induced incomparability relation is an equivalence relation. In this case, its equivalence classes partition and moreover, the set of these equivalence classes can be strictly totally ordered by a binary relation, also denoted by that is defined for all by: for some (or equivalently, for all) representatives Conversely, any strict total order on a partition of gives rise to a strict weak ordering on defined by if and only if there exists sets in this partition such that Not every partial order obeys the transitive law for incomparability. For instance, consider the partial order in the set defined by the relationship The pairs are incomparable but and are related, so incomparability does not form an equivalence relation and this example is not a strict weak ordering. For transitivity of incomparability, each of the following conditions is necessary, and for strict partial orders also sufficient: If then for all either or both. If is incomparable with then for all ,