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

In summary, we are given a reflexive and transitive relation R on a set A, and we define another relation S such that Sxy iff (Rxy and Ryx). We want to prove that S is an equivalence relation on A. To do this, we need to show that S is reflexive, symmetric, and transitive. We can prove S is reflexive by using the fact that R is reflexive and using the definition of S. To prove S is symmetric, we use the definition of S and the fact that R is reflexive. Finally, to prove S is transitive, we use the definition of S and the fact that R is transitive.
  • #1
JoeRocket
3
0

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!
 
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  • #2
JoeRocket said:

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?
 

1. What is a set relation problem?

A set relation problem is a type of mathematical problem that involves comparing and analyzing sets of data or objects. It often involves identifying common elements, intersections, or differences between two or more sets.

2. How do I solve a set relation problem?

To solve a set relation problem, you first need to identify the sets involved and determine the type of relationship or comparison being made. From there, you can use various methods such as Venn diagrams, tables, or equations to analyze and solve the problem.

3. What are some common types of set relation problems?

Some common types of set relation problems include finding the union, intersection, or difference between sets, as well as identifying subsets or supersets. These types of problems can also involve other mathematical concepts such as probability or logic.

4. What are some strategies for solving set relation problems?

One strategy for solving set relation problems is to create a visual representation, such as a Venn diagram, to help visualize the relationship between the sets. Another strategy is to use logical reasoning and break the problem down into smaller, more manageable parts. Practice and familiarity with various methods can also help in solving set relation problems.

5. How can I check if my solution to a set relation problem is correct?

You can check if your solution to a set relation problem is correct by verifying that all the conditions and relationships between the sets have been correctly identified and applied. You can also use methods such as substitution or trial and error to check your solution. Additionally, you can compare your solution to a provided answer or ask for feedback from a teacher or peer.

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