Notation and de nitions Sample Clauses

Notation and de nitions. 1. Hereinafter, we use the set theoreti symbols (quanti ers, propositional bundles, ∅ for the def def empty set); = for the equality by de nition; ⇔ for the equivalen e by de nition. We a ept the axiom of hoi e. A set onsisting of sets is alled a family. By P(T ) (by P′(T )), we denote the family of all (all nonempty) subsets of an arbitrary set T; the family P(T ) is also alled Boolean of the set T. If A and B are non-empty sets, then BA is the set of all fun tions from the set A to the set B (see [9℄). If f ∈ BA and C ∈ P′(A), then (f | C) ∈ BC is the restri tion of f to the set C: P def (f | C)(x) = f (x) ∀x ∈ C. We denote the image of the set C ∈ (A) under the fun tion f by f (C): def A f (C) = {f (x) : x ∈ C}. When f ∈ P(B) , f is alled a multivalued fun tion or multifun tion