<p>Yes, the relation <span class="math-container">$B$</span> over <span class="math-container">$X$</span> is not complete since <span class="math-container">$3 \in X$</span> but <span class="math-container">$(3,3) \notin B$</span>. </p>

<p>The variables <span class="math-container">$x$</span> and <span class="math-container">$y$</span> in the completeness definition "<span class="math-container">$\forall x,y \in X, xBy \lor yBx$</span>" are treated as placeholders and thus are not necessarily distinct. In fact, completeness implies reflexivity when you let <span class="math-container">$y = x$</span> in the definition.</p>

CC BY-SA 4.0
2018-10-02T14:39:32.563Z
2018-10-02T14:39:32.563Z
Kenneth Rios
<p>I have a set <span class="math-container">$X = \{1,2,3\}$</span> and a binary relation <span class="math-container">$B = \{(1,1),(1,2),(1,3),(2,3),(3,1)\}$</span>. </p>

<p>I am trying to understand if this relation is complete.</p>

<p>The completeness definition I am using is if for each <span class="math-container">$x,y$</span> in <span class="math-container">$X$</span>, either <span class="math-container">$xBy$</span> or <span class="math-container">$yBx$</span>.</p>

<p>At first I thought <span class="math-container">$B$</span> was complete but both <span class="math-container">$(2,2)$</span> and <span class="math-container">$(3,3)$</span> are not in <span class="math-container">$B$</span>.
In the definition, if I set <span class="math-container">$x=2,y=2$</span> then I think we should have <span class="math-container">$(2,2)$</span> in <span class="math-container">$B$</span>.
On the other hand, it seems meaningless to me because isn't <span class="math-container">$2B2$</span> related with reflexivity? So for completeness do we actually need <span class="math-container">$(2,2) \in B$</span>?</p>

CC BY-SA 4.0
2018-10-02T13:59:36.013Z
You did indeed find a violation of completeness. Actually, completeness implies reflexivity.
Michael Greinecker
<microeconomics><preferences>
Completeness from an example
Ravendi
# Completeness from an example

(Asked by: username_0 on 2018-10-02)

I have a set $X = \{1,2,3\}$ and a binary relation $B = \{(1,1),(1,2),(1,3),(2,3),(3,1)\}$.

I am trying to understand if this relation is complete.

The completeness definition I am using is if for each $x,y$ in $X$, either $xBy$ or $yBx$.

At first I thought $B$ was complete but both $(2,2)$ and $(3,3)$ are not in $B$.
In the definition, if I set $x=2,y=2$ then I think we should have $(2,2)$ in $B$.
On the other hand, it seems meaningless to me because isn't $2B2$ related with reflexivity? So for completeness do we actually need $(2,2) \in B$?

**Tags:** microeconomics, preferences

> **Comments:**
> **username_1**: You did indeed find a violation of completeness. Actually, completeness implies reflexivity.  

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## 1 answers

### Answer by username_2 on 2018-10-02. Score: 5 | Accepted answer

Yes, the relation $B$ over $X$ is not complete since $3 \in X$ but $(3,3) \notin B$.

The variables $x$ and $y$ in the completeness definition "$\forall x,y \in X, xBy \lor yBx$" are treated as placeholders and thus are not necessarily distinct. In fact, completeness implies reflexivity when you let $y = x$ in the definition.


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