Set operations in material set theories are defined in terms
of the membership predicate ∈, e.g.

Typically sets are represented as lists of unique elements, e.g. {1,2,3},
which together with the set operations define an algebra of lists. But sets
can also be represented as bit vectors (indicator functions) where each bit
corresponds to an element of a universal set. Set operations are then carried
out by the logic operators $\land$, $\lor$, and $\lnot$. Here is an example
in J.

J has a nice function called under that applies a transformation to its
operands before computing the given algebraic operations, and then applies the
transformation’s inverse to the result. We can use it to show the equivalence
of the two algebras.

The algebras are boolean and because of their one-to-one mapping (if all sets are
restricted to be subsets of $U$) they are said to be isomorphic.

I'm a software engineer currently living in Melbourne, working remotely for the
Psychiatry Neuroimaging Laboratory
in Boston and
a startup in Canada. My interests are data analysis pipelines and inference,
and I'm unduly obsessed with understanding design principles behind concise,
uncomplicated software systems.