Assume the If and S are relations on a set A If R and S are

Assume the If and S are relations on a set A. If R and S are reflexive, is If R S reflexive? Prove or disprove the statement. If R and S are symmetric? is R S symmetric? Prove or disprove the statement. If R and S are symmetric? is R S symmetric? Prove or disprove the statement. If R and S are transitive, is R S transitive? Prove or disprove the statement.

Solution

Let R and S be relations on a set A.

(a)

Let xA. Since R and S are reflexive, (x,x)R and S(x,x)S.

Therefore, (x,x)RS.

Therefore, RS is reflexive.

(b)

Given that R and S are symmetric, and let (a,b)RS.

Therefore (a,b)R and R is symmetric, so (b,a)R.

Similarly, (a,b)S, and S is symmetric, so (b,a)S.

So, (b,a)RS.

This implies that RS must be symmetric.

(c)

Given that R and S are symmetric, and let (a,b)RUS.

Therefore either (a,b)R or (a,b)S.

So, either (b,a)R or (b,a)S.

So, (b,a)RUS.

This implies that RUS must be symmetric.

(d)

Take R = {(1,2),(2,1),(1,1),(3,3)}

and S = {(2,1),(2,3)}

So RUS = {(1,2),(2,1),(1,1),(2,3),(3,3)}

In this case, RUS contains (1,2) and (2,3) but not (1,3), and therefore is not transitive.