Doubt about equivalence relation

Question

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?

Answer 1

I believe your questions are:

Q1. Why is $\succ$ not symmetric?

Q2. Why is $\sim$ (a) symmetric; (b) reflexive; and (c) transitive?

Answers:

A1. Suppose $x \succ y$. Then by Definition 4 (refer to basic setup and definitions below), $x \succsim y$ and $y \not\succsim x$. Therefore, by the same definition, $y \not\succ x$. Thus, $\succ$ is not symmetric.

A2. Let $x,y,z\in X$.

A2(a)$\hspace{10pt}$ $x\sim y$ $\hspace{10pt}$ $\iff$ $\hspace{10pt}$$x\succsim y$ and $y\succsim x$ (Definition 3) $\hspace{10pt}$ $\iff$ $\hspace{10pt}$$y\sim x$.

A2(b) By Definition 2, $x \succsim x$. And so, by Definition 3, $x \sim x$.

A2(c) Suppose $x\sim y$ and $y\sim z$. (We will show that $x\sim z$.)

By Definition 2, we have $x\succsim y$ and $y\succsim z$. And now by the transitivity of $\succsim$, we have $x\overset{1}{\succsim} z$.

Similarly, we also have $y\succsim x$ and $z\succsim y$. And so again by the transitivity of $\succsim$, we have $z\overset{2}{\succsim} x$.

Given $\overset{1}{\succsim}$ and $\overset{2}{\succsim}$, by Definition 3, we have $x\sim z$.

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Basic setup and definitions.

Definition 1. A (binary) relation $R$ (on a set $X$) is:

• Symmetric if for all $x,y\in X$, $xRy \iff yRx$;

• Reflexive if for all $x\in X$, $xRx$;

• Transitive if for all $x,y,z\in X$, $xRy$ and $yRz$ implies $xRz$.

Let $X$ be our set of alternatives.

Definition 2. Let $\succsim$ be a reflexive and transitive relation on $X$.

We interpret $\succsim$ as the weakly-preferred-to relation.

Definition 3. $x \sim y$ if $x \succsim y$ and $y \succsim x$.

We interpret $\sim$ as the indifference relation.

Definition 4. $x \succ y$ if $x \succsim y$ and $y \not\succsim x$.

We interpret $\succ$ as the strictly-preferred-to relation.

Answer 2

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 $yRx$.