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Abstract: What makes sets, or more precisely, the category {\bf Set} important inMathematics are the well known {\it two} specific ways in which arbitrarymappings $f : X \longrightarrow Y$ between any two sets $X, Y$ can {\it fail}to be bijections. Namely, they can fail to be injective, and-or to besurjective. As for bijective mappings they are rather trivial, since with somerelabeling of their domains or ranges, they simply become permutations, or evenidentity mappings. \\ To the above, one may add the {\it third} property ofsets, namely that, between any two nonvoid sets there exist mappings. \\ Thesethree properties turn out to be at the root of much of the interest which thecategory {\bf Set} has in Mathematics. Specifically, these properties create acertain {\it dynamics}, or for that matter, lack of it, on the level of thecategory {\bf Set} and of some of its subcategories.



Author: Elemer E Rosinger

Source: https://arxiv.org/



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