One immediate, and quite remarkable, result of this work is that it shows how one can define the general notions of relation and function in purely set-theoretic terms. It had long been recognised that relations/functions can be conceived as sets of ordered pairs, and Kuratowski's work now shows how to define the ordered pair primitively. The ordered pair (a, b) can be considered informally as the unordered pair M = {a, b}, together with an ordering relation a &lt; b. Suppose this relation is treated now via the theory of inclusion chains. The only maximal inclusion chains in the power set of M are {∅, {a}, {a, b}} and {∅, {b}, {a, b}}. Using Kuratowski's definition of the ordering ‘&lt;’ derived from a maximal inclusion chain, these chains must then correspond to the orderings a &lt; b and b &lt; a on {a, b} respectively. If ∅ is ignored, the resulting chain {{a}, {a, b}} is thus associated with the relation a &lt; b, and so with the ordered set (pair) (a, b). It is then quite natural to define (a, b) as {{a}, {a, b}} (see Kuratowski 1921: 170–171). One can now define the product a × b of a and b as the set of all ordered pairs whose first member is in a and whose second member is in b; relations on a can now be treated as subsets of a × a, and functions from a to b as certain subsets of a × b. Thus, many of the representational problems faced by Zermelo's theory are solved at a stroke by Kuratowski's work, building as it does on Zermelo's own.