lec0309

lec0309 - Definitions for Binary Relations over A x A A...

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Definitions for Binary Relations over A x A A majority of the binary relations we will be dealing with are a subset of the Cartesian product of a particular set with itself. If we have R A x A, then we have the following definitions: 1) R is reflexive if 2200 a A, (a,a) R. 2) R is irreflexive if if 2200 a A, (a,a) R. 3) R is symmetric if 2200 a A, aRb bRa 4) R is anti-symmetric if aRb bRa a=b. 5) R is transitive if aRb bRc aRc. Consider the following relation R defined over {a , b, c}: R = { (a,b), (a,c), (b,a), (b,c), (c,c) } R is not reflexive since (b,b) R R is not irreflexive since (c,c) R R is not symmetric since we have (a,c) R, but (c,a) R. R is not anti-symmetric since (a,b) R and (b,a) R. R is not transitive since (b,a) R, (a,b) R, but (b,b) R.
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Now, I will show you some examples of more meaningful relations that actual have some of these properties. Consider a relation R over the set {beans, rice, chicken} that is defined as foods that go well together. The relation could be: R = { (beans, beans), (rice, rice), (chicken, chicken), (beans, rice), (rice, chicken), (rice, beans), (chicken, rice) } This relation is reflexive since for each element a, (a,a) R. Essentially, we can mix anything with itself and it’ll still be edible. This relation is also symmetric. The reason for this is that if we can mix one food first with a second food, then we can ALSO mix the second food with the first. Symbolically, for each pair
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lec0309 - Definitions for Binary Relations over A x A A...

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