  The sets used to construct other mathematical objects are pure sets, which
means that all of their elements are sets, which are themselves pure. One set
may therefore be within another, not as an element, but as an element of an
element, or even deeper, inside several layers of sets within sets. The
introduction of the term constituent to describe a set which is within a given
set, however deep, induces an apparently novel partial order on sets, and
assigns to any given set a diagram which specifies a directed graph, or
category, herein dubbed its constituent structure, indicating which sets within
it are constituents of which others. Sets with different numbers of elements
can have exactly the same constituent structure. Consequently, constituent
structure isomorphisms between sets need not preserve the number of elements,
although they are still injective, surjective, and invertible. We consider in
detail an example of an isomorphism between a one-element set and a
five-element set, which is a surjective mapping despite the mismatch in
cardinalities. The constituent structure of a set determines the mathematical
objects for which the set is a suitable representation. Different schemes for
constructing the natural numbers, such as those of von Neumann and Zermelo,
generate sets with the same constituent structures. Objects share the
constituent structures, not the elements, of the sets used to construct or
represent them. The requirement that an object's properties be faithfully
encoded within a set's constituent structure and not its non-constituent
characteristics such as its cardinality, when made explicit, dictates a
specific and novel way of representing ordered pairs and tuples of sets as
sets, providing simple formulae for addressing and extracting sets located deep
within nested tuples.
