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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, especially order theory, a strict weak ordering is a binary relation on a set S that is a strict partial order (a transitive relation that is irreflexive, or equivalently, that is asymmetric) in which the relation "neither a b nor b a" is transitive. The equivalence classes of this "incomparability relation" partition the elements of S, and are totally ordered by . Conversely, any total order on a partition of S gives rise to a strict weak ordering in which x y if and only if there exists sets A and B in the partition with x in A, y in B, and A B in the total order. Strict weak orders are often used in microeconomics to model preferences