Suppose that R and S are relations on a nonempty set A Deter

Suppose that R and S are relations on a non-empty set A. Determine if each of the following statements is true or false. Prove each true statement. For each false statement, give a counterexample using A = {1, 2, 3}. If R and S are both anti-symmetric, then R \\ S is anti-symmetric. If neither R nor S is symmetric, then R union S is not symmetric. If R and S are both equivalence relations, then so is R intersection S.

Solution

1) The statement is TRUE

Reason: Let R\\S be a symmetric set, then for each element a and b belonging to the set (a,b) and (b,a) will belong to R\\S

Since the given element belong to R\\S

Hence we can say that (a,b) and (b,a) will belong to R and (a,b) and (b,a) will not belong to S

(a,b) and (b,a) belongs to R means that R is symmetric in nature, but R is antisymmetric in nature

Hence there is a contradiction, so our assumption was wrong and R\\S is antisymmetric in nature

2) FALSE

Let us consider

R = {(1,1),(2,2),(3,3), (1,2),(1,3),(2,3)} R is not symmetric

S = {(1,1),(2,2),(3,3),(2,1),(3,1),(3,2)} S is not symmetric

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

But in this case RUS is symmetric in nature, hence the statement is FALSE

3) The statement is TRUE