COT3100Relations02 - Composition of Relations In math...

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Composition of Relations In math class, given two functions f(x) and g(x), you probably had to figure out the composition of the functions, which is denoted either by f(g(x)) OR f ° g(x). Basically, the way this worked is that you “plugged in” your original x into one function, THEN you used the “answer” that you got from that function to “plug in” to the second function. And the order in which you did it mattered. The same will be true of the composition of two relations. Here is the formal definition of the composition of two relations R and S, where R A x B, S B x C: R ° S = { (a,c) | a A c C ( 5 b | (a,b) R (b,c) S) } (Notice the difference in order here. When we compose a relation, we write the relations in the order we apply them, not the opposite order, as is done with functions.) Basically, when you compose the relations R and S, you get a third relation which relates elements from the set A to the set C, as long as the “answer” from relation R can be the input for relation S. We can use a directed graph again. Consider this example: A = { ABC, NBC, CBS, FOX, HBO} B = { NYPD Blue, Simpsons, Letterman, ER, X-Files, Dennis Miller Show, Monday Night Football} C = { Dennis Miller, Marge, Rick Schroeder, Gillian Anderson, Noah Wyle} R = {(ABC, NYPD Blue), (NBC, ER), (CBS,Letterman), (HBO, Dennis Miller Show), (FOX, X-Files) } S = { (MNF, Dennis Miller), (Simpsons, Marge), (ER, Noah Wyle), (Party of 5, Neve Campbell)
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(D. Miller Show,Dennis Miller),(NYPD, Rick Schroeder) } Theorems about Relation Composition If R A x B, S B x C and T C x D, then we have the following: (R ° S) ° T = R ° (S ° T) Essentially, when doing multiple relation composition, associativity is preserved. First of all, we see that both sides define a relation over the set A x D. Next, we have to prove that both define the same relation over that set. Formally, if we break down the definition, we have: (R S ) T = {(a, d)| a A and d D, and for some c C, (a, c) R S and cTd}, Since (a, c) R S means aRb and bSc for some b B, by definition of R S, the relation (R S ) T consists of pairs (a, d) A × D such that for some b B and some c C, aRb, bSc and cTd. If we break down the definition of R ° (S ° T) in a similar manner, we will get the exact same thing. Similarly, using the directed graph of the situation will lead to the same conclusion.
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Let R A × B, S B × C, and T B × C denote 3 binary relations. Then we have the following:
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This document was uploaded on 09/16/2011.

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COT3100Relations02 - Composition of Relations In math...

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