# lec0220 - Problems from last time. (R S) (R T) R (S T) is...

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Problems from last time. .. (R S) (R T) R (S T) is false. Here is a counter example: A = {1,2} B = {a,b} C = {x,y} R = {(1,a), (1,b)} S = {(a,x)} T = {(b,x)} Here we have S T = , so R (S T) = . But, R S = {(1,x)} and R T = {(1,x)} so (R S) (R T) = {(1,x)}, proving the statement false. Prove or disprove: If R S = R T, then S = T. This statement is also false. Simply use the counter example given in the problem above to validate this claim.

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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 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.
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 {jelly, bread, ham} that is defined as foods that go well together. The relation could be: R = { (jelly, jelly), (bread, bread), (ham, ham), (jelly, bread), (bread, jelly), (ham, bread), (bread, ham) } This relation is reflexive since for each element a, (a,a) R. Essentially, we can mix anything with itself and it’ll still be

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lec0220 - Problems from last time. (R S) (R T) R (S T) is...

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