Doubt about equivalence relation In class I was taking notes about equivalence relations defined as:  

Given a generic relation $R$ on $X$, $xIy$ if both $xRy$ and $yRx$

Now, I don't really understand the following proposition:  

If $R$ (the original relation) is reflexive and transitive, then $I$ (the derived relation) is reflexive, symmetric and transitive.

I understand this is an instance of $\succsim $ where the derived relations are $\sim$ and $\succ$, but the proposition holds only for the former (the strict preference obviously is not symmetric). Why is that so? 

Also, how can I prove that $R$ having the two properties leads to $I$ having the same two properties + symmetry?
If both $xRy$ and $yRx$ $\forall x,y \in X$, then the indifference relation $I$ is symmetric by definition. Not all symmetric relations are equivalence relations. However, you are given that $R$ is also reflexive and transitive. So we have that $I$ is symmetric, reflexive, and transitive. Therefore $I$, or $\sim$, is an equivalence relation.

You can verify that $\sim$ is an equivalence relation by reading off the definitions of the three properties. $R$, or $\succsim$, is a (weak) preference relation, also known as a total preorder. Total preorders are transitive, complete, and therefore reflexive.

Keep in mind that reflexivity and transitivity does not necessarily imply symmetry: $R = \{(x,x), (y,y), (x,y)\}$ given $X = \{x,y\}$ is clearly reflexive and transitive but not symmetric because $xRy$ but not $yRx$.