A Boolean algebra is a set with two binary operations, 
and 
, that are commutative, associative and each distributes over the
other, plus a unary operation 
. Also required are identity
elements 
and U for the binary operations that satisfy 
, 
, 
and 
for
all elements A in the set.
One interpretation of Boolean algebra is the collection of subsets of a
fixed set X. We take , 
, , and U to be
set union, set intersection, complementation, the empty set and the set X
respectively. Equality here means the usual equality of sets.
Another interpretation is the calculus of propositions in symbolic logic.
Here we take , , , and U to be
disjunction, conjunction, negation, a fixed contradiction and a fixed
tautology respectively. In this setting equality means logical equivalence.
It is not surprising then that we find analogous properties and rules
appearing in these two areas. For example, the axiom of the distributive
properties says that for sets we have 
while 
is a
familiar equivalence in logic.
From the axioms above one can prove DeMorgan's Laws (in some axiom sets this is included as an axiom). The following table contains just a few rules that hold in a Boolean algebra, written in both set and logic notation. Rows 3 and 4 are DeMorgan's Laws. Note that the two versions of these rules are identical in structure, differing only in the choice of symbols.
