I have this question and can't quite grasp it..I'll write down what it says then go through what I've tried.
Let $R$ be a reflexive and transitive relation in $X$. Let $S$ be a relation in $X$ such that $(x,y) \in S \iff (x,y) \in R \land (y,x) \in R$.
We need to prove $S$ is an equivalence relation, i.e it is reflexive, symmetric, and transitive...I know the definition of all of these (Reflexive is $\forall x(x,x) \in R$ , symmetric is $(x,y) \in R \implies (y,x) \in R$ and transitive is if $(x,y) \in R \land (y,z) \in R \implies (x,z) \in R$
Okay...right now I'm trying to prove S is reflexive but I can't think of any ordered pair that would give me $(x,y) \in R \land (y, x) \in R$ they would have to be the same variable but R isn't antisymmetric so I'm not sure I can say that. I think I might need some pair of ordered pairs to make this work but once again not sure what..
I'm pretty new to this stuff so please be kind =x I don't want hand holding but I'm just not sure how to approach this problem. I know my definitions but I'm not sure how they'll apply..thanks for the time and help.
You’re outthinking yourself: antisymmetry has nothing to do with it. $R$ is reflexive, so if $x\in X$, then $\langle x,x\rangle\in R$. The reversed pair is identical — reversing $\langle x,x\rangle$ gives you $\langle x,x\rangle$ — so it’s in $R$ as well, and therefore by definition $\langle x,x\rangle\in S$. This shows that $S$ is reflexive.
It’s clear from the definition that $S$ is symmetric: that’s built into it. Thus, all that’s left is to show that $S$ is transitive. Suppose, then, that $\langle x,y\rangle\in S$ and $\langle y,z\rangle\in S$; you need to show that $\langle x,z\rangle\in S$, which means that you need to show that $\langle x,z\rangle\in R$ and $\langle z,x\rangle\in R$. See if you can do that on your own, but feel free to leave a comment if you get stuck.