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1.2.1Small and complete preimages of a set
ОглавлениеLet X, Y be sets, F : X → P(Y) a multimap.
Definition 1.2.1. The small preimage of a set D ⊂ Y is the set
Definition 1.2.2. The complete preimage of a set D ⊂ Y is the set
It is clear that .
Let A ⊂ X; D ⊂ Y; {Dj}j∈J a family of subsets of Y, J a set of indices. The next properties of small and complete preimages follow immediately from the definitions (verify!).
Lemma 1.2.3.
Lemma 1.2.4.
Let us observe the properties of small and complete preimages while passing to various set-theoretic operations on multimaps.
Definition 1.2.5. Let F0, F1 : X → P(Y) be multimaps. The multimap F0 ∪ F1 : X → P(Y),
is called the union of the multimaps F0 and F1.
Definition 1.2.6. Let F0, F1 : X → P(Y) be multimaps such that F0(x) ∩ F1(x) ≠ for all x ∈ X. The multimap F0 ∩ F1 : X → P(Y),
is called the intersection of the multimaps F0 and F1.
The following properties can be easily verified (do it!).
Lemma 1.2.7. If D ⊂ Y then
(a) ;
(b) .
Lemma 1.2.8. If D ⊂ Y then
(a) ;
(b) .
Definition 1.2.9. Let X, Y, and Z be sets, F0 : X → P(Y), F1 : Y → P(Z) multimaps. The multimap F1 ○ F0 : X → P(Z),
is called the composition of the multimaps F0 and F1.
Lemma 1.2.10. Let D ⊂ Z then
(a) ;
(b) .
Verify these relations!
Definition 1.2.11. Let X, Y0, and Y1 be sets, F0 : X → P(Y0), F1 : X → P(Y1) multimaps. The multimap F0 × F1 : X → P(Y0 × Y1),
is called the Cartesian product of the multimaps F0 and F1.
Lemma 1.2.12. Let D0 ⊂ Y0, D1 ⊂ Y1, then
(a) ;
(b) .
Verify these relations!
