Cf: Animated Logical Graphs • 29 https://inquiryintoinquiry.com/2019/08/11/animated-logical-graphs-29/
Re: Animated Logical Graphs • 21 https://inquiryintoinquiry.com/2019/07/12/animated-logical-graphs-21/ Re: Ontolog Forum • Joseph Simpson https://groups.google.com/g/ontolog-forum/c/wF03K5KG1vQ/m/ssNdCXssEgAJ <QUOTE JS:> I tend to view equivalence and distinction as relationships as opposed to operations. I do not know if this makes any significant difference in this context. </QUOTE> Dear Joe, I invoked the general concepts of equivalence and distinction at this point in order to keep the wider backdrop of ideas in mind but since we’ve been focusing on boolean functions to coordinate the semantics of propositional calculi we can get a sense of the links between operations and relations by looking at their relationship in a boolean frame of reference. Let B = {0, 1} and k be a positive integer. Then B^k is the set of k-tuples of elements of B. • A “k-variable boolean function” f is a mapping from B^k to B. A function of that type is typically notated as f : B^k → B. • A “k-place boolean relation” L is a subset of B^k. A relation of that type is notated as L ⊆ B^k. The correspondence between boolean functions and boolean relations may be articulated as follows. • Any k-place relation L, as a subset of B^k, has a corresponding “indicator function” (or “characteristic function”) f_L : B^k → B defined by the rule that f_L (x) = 1 if x is in L and f_L (x) = 0 if x is not in L. • Any k-variable function f : B^k → B is the indicator function of a k-place relation L_f consisting of all the x in B^k where f(x) = 1. The set L_f is called the “fiber” of 1 or the “pre-image” of 1 in B^k and is commonly notated as f⁻¹(1). Resources ========= • Logic Syllabus https://oeis.org/wiki/Logic_Syllabus • Relation Theory https://oeis.org/wiki/Relation_theory • Boolean Function https://oeis.org/wiki/Boolean_function • Boolean-Valued Function https://oeis.org/wiki/Boolean-valued_function Regards, Jon
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