Aczel's theory includes another important useful feature: solving equations in the universe of Hypersets.
Let 
be the universe of hypersets with atoms from a
given set A and let 
be the universe of hypersets
with atoms from another given set 
such that 
and
X is defined as 
. The elements of X can be considered as
indeterminates ranging over the universe 
. The
sets which can contain atoms from X in their construction are called
X-sets. A system of equations is a set of equations
for each 
. For example, choosing 
and 
(thus 
), consider the
system of equations
A solution to a system of equations is a family of pure sets
(sets which can have only sets but no atoms as elements), one
for each 
, such that for each 
, 
.
Here, 
is a substitution operation (defined below) and
is the pure set obtained from a by substituting 
for
each occurrence of an atom x in the construction of a.
The Substitution Lemma states that for each family of pure
sets 
, there exists a unique operation 
which assigns a
pure set 
to each X-set a, viz.,
The Solution Lemma can now be stated (Barwise & Moss 1991).
If 
is an X-set, then the system of equations 
has a unique solution, i.e., a unique family of pure sets
such that for each 
, 
.
This lemma can be stated somewhat differently. Letting X again be
the set of indeterminates, g a function from X to 
, and h
a function from X to A, there exists a unique function f for all
such that
Obviously, 
is the set of indeterminates and 
is the set
of atoms in each X-set 
of an equation 
. In the
above example, 
, 
, 
, and
, 
, 
, and
one can compute the solution
The Solution Lemma is an elegant result, but not every system of equations has a solution. First of all, the equations have to be in the form suitable for the Solution Lemma. For example, a pair equations such as
cannot be solved since it requires the solution to be stated in terms
of the indeterminate z. (These are analogous to the Diophantine
equations.) As another example, the equation 
cannot be
solved because Cantor has proved (in ZFC
) that there is no set
which contains its own power set---no matter what axioms are added to
ZFC
.
As another example due to (Barwise & Etchemendy 1987), it may be verified that the system of equations
has a unique solution in the universe of Hypersets depicted in Figure 10 with x = a, y = b, and z = c.
Figure 10: The solution to a system of equations (adapted from (Barwise & Etchemendy 1987))
This technique of solving equations in the universe of hypersets can be very useful in modeling information which can be cast in the form of equations (Pakkan 1993), e.g., situation theory (Barwise & Perry 1983), databases, etc. since it allows us to assert the existence of some graphs (the solutions of the equations) without having to depict them with graphs. We now give an example from databases.