CMPUT272 Lecture Notes - Lecture 20: Coimage, Cross Product, Binary Relation
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Definitions wrt functons / bender and williamson notations. B^a: set of all functions from a to b f b^a f: a -> b. So _4_^_3_ = is the set of functions from {1,2,3} -> {1,2,3,4} B^|a| (cross product of b to the cardinality a) Image(f) = elements of b that get mapped to, {f(a)|a a} . For all b b, the inverse image of b is: f^-1(b) The set of elements from a where f(a) = b f^-1(b) = {a | a a f(a) = b} f^-1(3) = {2,3}, f^-1(1) = {1}, f^-1(4) = . The coimage of f: a -> b is the set of non-empty inverse images of elements of b. Coimage(f) = {f^-1(b)|b b} coimage(f) = {{2,3}, {1}} Each is a subset of a^2 = a a . To visualize/represent r: incidence matrix a1 a2 a3 u an 1 if (u,n) r else 0 a1 a2 a3 w an .