[1] Let X be a non

empty set, then A = {(x, [[mu].sub.A] (x)) : x [member of] X} is called a fuzzy set on X, where [[mu].sub.A](x) [member of] [0, 1] is the degree of membership function of each x [member of] X to the set A.

It is never precisely clear whether she was born there and emigrated to Mexico, but the

empty set, the nothingness, at the core of her story finds its roots there.

(a) ([x.sub.1], [x.sub.1]) is the

empty set [empty set], which is the open internal set of [x.sub.1];

For x [member of] X, let [P.sub.A] (x) denote the set of best approximations to x in A (possibly

empty set).

In this proof, the B constructs are prefixed by "b_", and "b_empty", "b_BIG", "b_in", "b_eq", and "b_drest" respectively represent the

empty set [empty set], the set BIG (which is an infinite set, mostly only used to build natural numbers in the foundational theory), the membership operator "[member of]", the (extensional) equality "=", and the domain restriction construct "[??]".

They even sneaked on to the

empty set and sat in Cowell's judge's seat.

Then the iteration [T.sup.[n+1]] = Z x SET([T.sup.[n]]) can be seen to converge to the class of rooted labeled trees, starting with [T.sup.[0]] the

empty set. For any complex [alpha] with [absolute value of [alpha]] < [e.sup.-1], the corresponding numerical iteration [t.sup.[0]] = 0, [t.sup.[n + 1]] = [alpha]exp([t.sup.[n]]) converges to the value T([alpha]) = -W (-[alpha]) of the generating series of rooted labeled trees (W is the Lambert W function).

Let us consider a non-

empty set E = {[e.sub.1], [e.sub.2], [e.sub.3]}.

A set that contains no elements is called the

empty set. The set of existing tooth fairies or gods can be described as the

empty set.

Then, there is [empty set] = {}, an

empty set that contains no elements at all; and there is U, a universal set that contains all elements.