If the term contains no free variables, then the set consists of either zero members or one member, depending on the truth value of the embedded sentence. The empty relation is true of the empty set but false of all other objects. In KIF, the functions union , intersection , difference , and complement are defined as follows. Two sets are disjoint if and only if there is no object that is a member of both sets. Sets are pairwise-disjoint if and only if every set is disjoint from every other set.
Sets are mutually-disjoint if and only if there is no object that is a member of all of the sets.
For example, the sets and and are mutually disjoint but not pairwise disjoint. We close this section with two axioms that allow us to conclude that sets of various sorts do, in fact, exist. This axiom is not absolutely essential for set theory. However, it makes many proofs a lot easier, and so it is commonly included among the axioms of set theory. The second axiom is the axiom of choice .