 The notions of the subset relation and the intersection and union of sets are defined in PM exactly as they are now (albeit with different terminology). The complement of a set and the universal class \(\rV\) are not allowed in set theory, and rejected as “proper classes”. In PM, as they only are a set of entities of a given type \(\tau\), they form a set of the next higher type, \((\tau)/1\). The complement of a set of a given type is the set of all entities (of that type) that are not in the set. Each empty set will be the complement of the universal set (of a given type \(\tau\) ) and so there will be the empty set of type \(\tau\).