<p>If both <span class="math-container">$xRy$</span> and <span class="math-container">$yRx$</span> <span class="math-container">$\forall x,y \in X$</span>, then the indifference relation <span class="math-container">$I$</span> is symmetric by definition. Not all symmetric relations are equivalence relations. However, you are given that <span class="math-container">$R$</span> is also reflexive and transitive. So we have that <span class="math-container">$I$</span> is symmetric, reflexive, and transitive. Therefore <span class="math-container">$I$</span>, or <span class="math-container">$\sim$</span>, is an equivalence relation.</p>

<p>You can verify that <span class="math-container">$\sim$</span> is an equivalence relation by reading off the definitions of the three properties. <span class="math-container">$R$</span>, or <span class="math-container">$\succsim$</span>, is a (weak) preference relation, also known as a total preorder. Total preorders are transitive, complete, and therefore reflexive.</p>

<p>Keep in mind that reflexivity and transitivity does not necessarily imply symmetry: <span class="math-container">$R = \{(x,x), (y,y), (x,y)\}$</span> given <span class="math-container">$X = \{x,y\}$</span> is clearly reflexive and transitive but not symmetric because <span class="math-container">$xRy$</span> but not <span class="math-container">$yRx$</span>.</p>

CC BY-SA 4.0
2018-09-30T00:06:04.147Z
2018-09-30T04:34:13.797Z
2018-09-30T04:34:13.797Z
Kenneth Rios
<p>In class I was taking notes about equivalence relations defined as:  </p>

<p><em>Given a generic relation <span class="math-container">$R$</span> on <span class="math-container">$X$</span>, <span class="math-container">$xIy$</span> if both <span class="math-container">$xRy$</span> and <span class="math-container">$yRx$</span></em></p>

<p>Now, I don't really understand the following proposition:  </p>

<p><em>If <span class="math-container">$R$</span></em> (the original relation) <em>is reflexive and transitive, then <span class="math-container">$I$</span></em> (the derived relation) <em>is reflexive, symmetric and transitive.</em></p>

<p>I understand this is an instance of <span class="math-container">$\succsim $</span> where the derived relations are <span class="math-container">$\sim$</span> and <span class="math-container">$\succ$</span>, but the proposition holds only for the former (the strict preference obviously is not symmetric). Why is that so? </p>

<p>Also, how can I prove that <span class="math-container">$R$</span> having the two properties leads to <span class="math-container">$I$</span> having the same two properties + symmetry? </p>

<preferences><self-study>
Doubt about equivalence relation
plastico
# Doubt about equivalence relation

(Asked by: username_0 on 2018-09-28)

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?

**Tags:** preferences, self-study

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## 1 answers

### Answer by username_1 on 2018-09-30. Score: 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 $yRx$.


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