Create a discrete math problem in function and relations are

Create a discrete math problem in function and relations area. And also please provide a solution for the problem.

Solution

A )    Suppose R is tanasitive relation on A .

        If R is transitive, then R is the subset of A such that (a,b) is in R and (b,c) is in R,

        and, due to transitivity (a,c) is in R when (a,b) and (b,c) have the same b for all a, b, c in A.

       Now, for R composed with R , that means there is an (a,b) in R and an (b,c) in R, and R composed with R .

       By the definition of composition, the pair (a,c) is in R o R if and only if

        there exists a b such that (a,b) and (b,c) are both in R.

       But if there is such a b, then by transitivity (a,c) is in R.

       Thus if (a,c) is in R o R, then (a,c) is in R. It follows that R o R R

       That means R o R is transitive relation

       is the set of all (a,c) in A such that (a,b) and (b,c) have the same b.