1-if R and S are relations on set A then prove that
a) R and S are symmetric=>RUS and R∩S are symmetric
b) R is reflexive and S is any relation=> RUS is reflexive
2-If R and S are transitive relations on set A then prove that RUS may not be a transitive relation on A
if u word limit exceeds kindly mail the solution to me at jain.naveen100@yahoo.com
a) R and S are symmetric=>RUS and R∩S are symmetric
b) R is reflexive and S is any relation=> RUS is reflexive
2-If R and S are transitive relations on set A then prove that RUS may not be a transitive relation on A
if u word limit exceeds kindly mail the solution to me at jain.naveen100@yahoo.com
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a) Suppose R and S are symmetric and let (a,b) ∈ RUS. Then (a,b) ∈ R or (a,b) ∈ S. Since these are both symmetric, (a,b) ∈ R implies that (b,a) ∈ R and if (a,b) ∈ S then (b,a) ∈ S. So in either case (b,a) ∈ RUS which is therefore symmetric.
Now suppose (a,b) ∈ R∩S. Then (a,b) ∈ R and (a,b) ∈ S. Since each is symmetric, (b, a) ∈ R and (b, a) ∈ S. Hence (b, a) ∈ R∩S which is therefore symmetric.
b) Suppose R is reflexive and S is a relation (both on A). For each a ∈ A, (a, a) ∈ R. Hence (a, a) ∈ RUS. So RUS is reflexive.
2. You can establish this by producing an example of a set A with transitive relations R and S such that RUS is not transitive. Let
A = {1, 2, 3}, R = {(1,2)}, and S = {(2,3)}.
Then R and S are transitive (vacuously), but RUS = {(1, 2), (2, 3)} which is not transitive because (1, 2) ∈ RUS and (2, 3) ∈ RUS but (1, 3) ∉ RUS.
The key to this last property is that you can have (a, b) in R but not in S and (b, c) in S but not in R. That way, R and S can be transitive while RUS is not provided neither R nor S contain (a,c).
Now suppose (a,b) ∈ R∩S. Then (a,b) ∈ R and (a,b) ∈ S. Since each is symmetric, (b, a) ∈ R and (b, a) ∈ S. Hence (b, a) ∈ R∩S which is therefore symmetric.
b) Suppose R is reflexive and S is a relation (both on A). For each a ∈ A, (a, a) ∈ R. Hence (a, a) ∈ RUS. So RUS is reflexive.
2. You can establish this by producing an example of a set A with transitive relations R and S such that RUS is not transitive. Let
A = {1, 2, 3}, R = {(1,2)}, and S = {(2,3)}.
Then R and S are transitive (vacuously), but RUS = {(1, 2), (2, 3)} which is not transitive because (1, 2) ∈ RUS and (2, 3) ∈ RUS but (1, 3) ∉ RUS.
The key to this last property is that you can have (a, b) in R but not in S and (b, c) in S but not in R. That way, R and S can be transitive while RUS is not provided neither R nor S contain (a,c).