Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.10 http://inquiryintoinquiry.com/2014/05/07/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-10/ Posted : May 7, 2014 at 10:36 pm Author : Jon Awbrey
Peircers, In the case of a dyadic relation F ⊆ X × Y that has the qualifications of a function f : X → Y, there are a number of further differentia that arise: • f is surjective ⇔ f is total at Y • f is injective ⇔ f is tubular at Y. • f is bijective ⇔ f is 1-regular at Y. For example, the function f : X → Y depicted below is neither total nor tubular at its codomain Y, so it can enjoy none of the properties of being surjective, injective, or bijective. Figure 40. Function f : X → Y ☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-40.jpg An easy way to extract a surjective function from any function is to reset its codomain to its range. For example, the range of the function f above is Y' = {0, 2, 5, 6, 7, 8, 9}. Thus, if we form a new function g : X → Y' that looks just like f on the domain X but is assigned the codomain Y', then g is surjective, and is described as a mapping ''onto'' Y'. Figure 41. Surjective Function g : X → Y' ☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-41.jpg The function h : Y' → Y is injective. Figure 42. Injective Function h : Y' → Y ☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-42.jpg The function m : X → Y is bijective. Figure 43. Bijective Function m : X → Y ☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-43.jpg Regards, Jon -- academia: http://independent.academia.edu/JonAwbrey my word press blog: http://inquiryintoinquiry.com/ inquiry list: http://stderr.org/pipermail/inquiry/ isw: http://intersci.ss.uci.edu/wiki/index.php/JLA oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey facebook page: https://www.facebook.com/JonnyCache
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