lec1017 - Function vs. Relation Composition I looked over...

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Function vs. Relation Composition I looked over the book and my notes and found a discrepancy between the two. In particular, I had told you that when you compose a 2 relations, for example R A × B, S B × C, that this composition would be written, just as if R and S were functions: S ° R = { (a,c) | a A c C ( 5 b | (a,b) R (b,c) S) } But, composition with relations, is written the other way around, so that it is “intuitive” given the graph of the relation. Thus, the definition we have for relation composition with the 2 relations above is as follows: R ° S = { (a,c) | a A c C ( 5 b | (a,b) R (b,c) S) } BUT, if we were composing two functions f : A B and g : B C, the definition remains as I showed you: g f = {(a, c) | a A c C ( 5 b B | (a, b) f (b, c) g)}. With that in mind, and the idea that I should separate my presentation of relations and functions, I have reorganized the notes from my past three lectures. I apologize for the inconvenience, but I feel that this organization will clarify the material presented over the past three lectures.
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Last lecture, I asked you all to find a counter-example to the following (note that I have written this correctly, for relations R, S and T defined over the sets A, B and C as described in the last lecture): (R S) (R T) R (S T), where R: A B and S,T: B C. In creating this counter-example, one thing to realize is that in
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lec1017 - Function vs. Relation Composition I looked over...

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