Proving that $\sim$ is an equivalence relation

Question

I have the following theorem I need to prove:

Let $R$ be a quasi-order relation on a set $X$, and let $\sim$ be defined by

$$x\sim y \;\;<=>\;\;(x,y)\in R\;\;\text{and}\;\;(y,x)\in R$$

Then $\sim$ is an equivalence relation that satisfies: if $x_1\sim y_1$ and $x_2 \sim y_2$, then $(x_1,x_2)\in R$ if and only if $(y_1,y_2)\in R$.

Definitions:

• Let $X$ and $Y$ be sets, and $R\subseteq X \times Y$ a relation between the elements of $X$ and $Y$, i.e. $R$ consists of some pairs $(x,y)$ with $x\in X$ and $y\in Y$.

We usually consider cases where $R\subseteq X \times X$

• Relation $R$ is reflexive if $(x,x)\in R$ for all $x \in X$

• Relation $R$ is symmetric if $(x,y)\in R$ implies $(y,x)\in R$

• Relation $R$ is transitive if $(x,y)\in R$ and $(y,z)\in R$ imply $(x,z)\in R$

• Relation $R$ is quasi-order if it is reflexive and transitive

• Relation $R$ is equivalence relation if it is reflexive, symmetric and transitive

I know I need to show that $\sim$ is reflexive, symmetric and transitive. What confuses me here most is the definition of $\sim$. How do I interpret this definition:

$$x\sim y \;\;<=>\;\;(x,y)\in R\;\;\text{and}\;\;(y,x)\in R\;\;\;?$$

Am I supposed to read this definition like this: If there is a relation $\sim$ between the elements $x$ and $y$, then $(x,y)\in R\;\;\text{and}\;\;(y,x)\in R$. I don't see $\sim$ as a set here and that confuses me here...because that's what a relation is...a set of elements $(x,y)$...right?

Hope I explained my confusion right...I fail to see $\sim$ as a set in the first place, where I could begin showing that this set is reflexive, symmetric, etc...

How to proceed with this proof?

P.S. the author of my reference uses also the notation $xRy$ to denote $(x,y)\in R$

Answer 1

$R$ is a relation on $X$, i.e. $\ R \subseteq X \times X$.

Also $\sim$ is a relation on $X$, i.e. $\ \sim \ \subseteq X \times X$, defined by :

$x \sim y$ iff $(x,y) \in R$ and $(y,x) \in R$.

We have to show that $\sim$ is :

reflexive, symmetric and transitive.

1) Reflexive :

Consider $x \in X$. We have that $(x,x) \in R$, because $R$ is reflexive; thus, trivially : $(x,x) \in R$ and $(x,x) \in R$, i.e. $x \sim x$.

2) Symmetric :

Consider $x,y \in X$ such that $x \sim y$. This means, by definition of $\sim$ : $(x,y) \in R$ and $(y,x) \in R$.

But trivially, also : $(y,x) \in R$ and $(x,y) \in R$, i.e. $y \sim x$, and thus we have that : if $x \sim y$, then $y \sim x$, that is the symmetry of $\sim$ .

And so on ...