By Aloisius Ho-Yin Louie

Mapping Origins.- From issues to Sets.- rules of Set-Valued Mappings.- Censusing Independence.- Set-Valued Mappings Redux.- The good judgment of Entailment.- The Imminence Mapping.- Imminence of Life.- Imminence in Models.- Connections.- Symbiosis.- Pathophysiology.- Relational Virology.- Therapeutics

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**Example text**

If g D f f is injective and g is surjective. 17 Bijection If a mapping f : X o Y is both one-to-one and onto, that is, both injective and surjective, then f is called bijective (or is a bijection) and that the mapping f establishes a one-to-one correspondence between the sets X and Y. 2) and assigns to each mapping f : X o Y the mapping (17) 3f : 3X o 3Y that sends each A X to its image f A Y . One readily verifies that this definition satisfies the functorial requirements 3 g D f 3 g D 3 f (the mapping that sends a subset A of the domain of f to the subset g f A of the codomain of g ) and 31X 13X (the identity morphism gets sent to the identity morphism), so 3 is a covariant functor from Set to Set.

The source (material cause) and the value (final cause) of a set-valued mapping are thus different in kind from each other, they belonging to different hierarchical levels (‘point’ versus ‘set’). 4) f : X o Y may be considered a very specialized set-valued mapping F : X –l Y such that, for each x X , the value (4) F x ^ f x ` is a singleton set. Indeed, one can make the formal definition: a set-valued mapping F : X –l Y is called single-valued if for each x X , F x is a singleton set. A ‘single-valued set-valued mapping’ F : X –l Y therefore defines a ‘standard’ mapping f : X o Y by f : x 6 the single element in F x .

Since a (single-valued) mapping is a specialized set-valued mapping through the correspondence (4), whatever properties that set-valued mappings have cannot be contradictory to (but may be weaker than) their analogs for mappings. 19. i now only goes one way. 11 Theorem equivalent: Let F : X –l Y be a set-valued mapping. The following are i. F is injective. ii. For all A, B X , A B F A F B iii. For all A, B X , F A B F A F B . iv. For all A, B X , F B A F B F A . v. For all A X , F X A Y F A .