Is S an Equivalence Relation Given R is Reflexive and Transitive?

[JoeRocket]

Homework Statement

Let R be a reflexive and transitive relation on a set A. Define another relation, S, such that, for any x, y ∈ A, Sxy iff (Rxy and Ryx).

Prove:
S is an equivalence relation on A.

Homework Equations

S is an equivalence relation if it is symmetric, reflexive and transitive.
S is reflexive if for every x, then Rxx.
S is symmetric if for every x,y - if Rxy, then Ryx.
S is transitive if for every x,y,z - if (Rxy and Ryx) then x = y.

The Attempt at a Solution

What I do not understand is how to get any information about S other than Sxy. For example if I need to prove that S is symmetric, then I need Sxx. How can I know Sxx when the only info I have about the relation S is about Sxy?

I would really appreciate some help on this problem - I have been stuck for days!

 

[HallsofIvy]

Homework Statement

Let R be a reflexive and transitive relation on a set A. Define another relation, S, such that, for any x, y ∈ A, Sxy iff (Rxy and Ryx).

Prove:
S is an equivalence relation on A.

Homework Equations

S is an equivalence relation if it is symmetric, reflexive and transitive.
S is reflexive if for every x, then Rxx.
S is symmetric if for every x,y - if Rxy, then Ryx.
S is transitive if for every x,y,z - if (Rxy and Ryx) then x = y.

The Attempt at a Solution

What I do not understand is how to get any information about S other than Sxy. For example if I need to prove that S is symmetric, then I need Sxx. How can I know Sxx when the only info I have about the relation S is about Sxy?

I would really appreciate some help on this problem - I have been stuck for days!

You do have "information about S", you have its definition. Also you do not "need Sxx" to prove S is symmetric- you need that to prove S is reflexive.

To prove S is reflexive: Let x be any member of set A. You know that R is reflexive so you know Rxx. Then it is true that "Rxx" and "Rxx" (where I have reversed the order of the "x"s!) so, from the definition of S, Sxx.

To prove S is symmetric: let x and y be members of A such that Sxy. Then, by definition of S, Rxy and Ryx. You want to prove that "Syx" which means that "Ryx and Rxy".

To prove S is transitive: let x, y, and z be members of A such that Sxy and Syz. Then, by definition of S, Rxy, Ryx, Ryz, and Rzy. Now you want to prove that "Sxz" which means you need to prove "Rxz and Rzx". Can you prove that from the fact that R is reflexive and transitive?