<p>In the wiki you will see it says: &quot;As described in the terminology section below, the terminology for these properties is not uniform.&quot;</p>
<p>The definition of <em>strongly connected</em> in the wiki is the same as the definition of <em>completeness</em> at your first link. Both require that:</p>
<p><span class="math-container">$$\text{for all $x,y\in X$ either $(x,y)\in\mathcal{R}$ or $(y,x)\in\mathcal{R}$ }$$</span></p>
<p>Note that the above must hold when <span class="math-container">$x=y$</span> as well as when <span class="math-container">$x\neq y$</span>.</p>
<p>The definition of <em>connected</em> / <em>complete</em> / <em>total</em> in the wiki is a weaker condition:</p>
<p><span class="math-container">$$\text{for all $x,y\in X$ with $x\neq y$ either $(x,y)\in\mathcal{R}$ or $(y,x)\in\mathcal{R}$ }$$</span></p>
<hr />
<p>For your relation, since, for example, <span class="math-container">$(1,1)\notin\mathcal{R}$</span>, the relation is <em>not</em> complete (according to the definition in your first link) and not strongly conected (according to the definition in the wiki). However, your relation <span class="math-container">$\mathcal{R}$</span> <em>is</em> connected (according to the definition in the wiki).</p>
<p>The relation <span class="math-container">$\mathcal{S}=\{(1,1),(2,2),(3,3),(1,3),(2,1),(3,2)\}$</span> is strongly connected (according to the wiki definition) and therefore connected (according to the wiki definition) and complete (according to the definition at your first link).</p>
<hr />
<p>If a relation is complete by the definition at your first link, then it is reflexive.</p>

CC BY-SA 4.0
2023-08-16T14:19:51.350Z
Thank you for your reply. My first link *$\forall x,y\in X$, either $(x,y)\in\mathcal{R}$ or  $(y,x)\in\mathcal{R}$ or (both)* is same as the connected relation **(not strongly connected)** in the wiki. I believe confusion comes in how to interpret $x=y$ in the definition. When you read this definiton, why would perceiving $x=y$ means this requirements need to be hold for $x=y$, whereas not the case that as long as $x=y$ we may ignore that  part. Just talk about first link definiton, and wiki connected relation definition. **not strongly connected**
LJNG
CC BY-SA 4.0
2023-08-16T14:38:11.497Z
No, the definition of connected in the wiki specifies $x\neq y$, while the definition of complete in your first link does not (so the definitions are not the same). The definition of complete in your first link corresponds to the definition of strongly connected in the wiki. I am not sure what you mean by "how to interpret $x=y$". It is not a matter of interpretation; the two elements $x$ and $y$ are the same or they are not.
smcc
CC BY-SA 4.0
2023-08-16T15:03:49.373Z
Thank you for your reply. The wiki definition goes: *A relation $\mathcal{R}$ on a set $X$ is called connected when for all $x,y\in X$ if $x\neq y$ then $x\mathcal{R} y$ or $y\mathcal{R} x$; **equivalently**, when for all $x,y\in X$, $x\mathcal{R} y$ or $y\mathcal{R} x$ or **$x=y$**.(THere is what I meant how to interpret $x=y$)* This equivalent definition is the same as the definition in my first link: *A preference relation on $X$ is complete if for every pair $x$ and $y$ from $X$ , either $x \succsim y$ or $ y\succsim x$ **(or both)***
LJNG
CC BY-SA 4.0
2023-08-16T15:17:12.827Z
No, that definition of connected is not the same as the definition of complete in your first link. The definition of connected allows for it not to be the case that $x\mathcal{R} x$, whereas the definition of complete requires that $x\succsim x$.
smcc
CC BY-SA 4.0
2023-08-16T15:21:10.067Z
Thank you for your reply, may you please enlighten me the subtle difference. this preference relation $\succsim$ look the same as $\mathcal{R}$ to me, just notation differences.
LJNG
CC BY-SA 4.0
2023-08-16T15:29:41.673Z
The definition of complete requires $x\succsim x$, I think probably $x\succsim x$ this is not a consequence of above fromal definition, rather a common sense embeded in that every good is weakly prefer to itself ? Could you logically derive $x\succsim x$ is a consequence of that complete definition ?  Much appreciated
LJNG
CC BY-SA 4.0
2023-08-16T15:31:15.017Z
The formal definitions of completeness (at the first link) and strong connectedness both require that $x$ is related to itself (for all $x\in X$). That is the crucial difference to connectedness. Strong connectedness implies connectedness, but not conversely. It is common sense that every consumption bundle is weakly preferred to itself. That is why strong connectedness is assumed rather than just connectedness. Also, indifference ($x\sim y$) can be defined as $x\succsim y$ and $y\succsim x$, so if it were not the case that $x\succsim x$, then it would not be the case that $x\sim x$.
smcc
CC BY-SA 4.0
2023-08-16T15:45:46.677Z
Thank you so much, I think I get it. The complete definition is actually said $\forall x,y\in X$, $x\mathcal{R}y$ or $y\mathcal{R}x$ or $(x\mathcal{R}y)\land (y\mathcal{R}x)$, this is the subtle different from that $x=y$ in the connected definition. $x\sim x$ is the consequence of $(x\mathcal{R}y)\land (y\mathcal{R}x)$, wheras $x=y$ in the connected definition allowing such case not in the relation still making the relation being connected.
LJNG
CC BY-SA 4.0
2023-08-16T15:50:12.910Z
In mathematics, "A or B" includes the possibility that "A and B".  There is no need to write  "A or B or (A and B)".
smcc
CC BY-SA 4.0
2023-08-16T15:55:42.877Z
Yes, it is. just infrom myself that $(x\mathcal{R}y)\land (y\mathcal{R}x)$ as the interpretation for **(or both)** in the complete preference definition is different from that $x=y$ in the connected relation. Anyway, thank you so much
LJNG
CC BY-SA 4.0
2023-08-16T14:09:03.100Z
2023-08-16T14:49:31.800Z
2023-08-16T14:49:31.800Z
smcc
<p>Is the relation <span class="math-container">$\mathcal{R}=\{(1,2),(2,3),(1,3)\}$</span> on <span class="math-container">$X=\{1,2,3\}$</span> complete?</p>
<p>By looking at the completeness definition in preference: <a href="https://assets.press.princeton.edu/chapters/s9890.pdf" rel="nofollow noreferrer">Definition 1.1(c)</a>, this is same as the connected relation in the maths definition <a href="https://www.wikiwand.com/en/Connected_relation" rel="nofollow noreferrer">wiki-connected relation</a>. If I am right, for any distinct pair of elements in <span class="math-container">$X$</span>, they are comparable, otherwise they are the same elements <strong>[They are the same might just leave the chance that <span class="math-container">$(1,1)\notin\mathcal{R}$</span>?]</strong>.</p>
<p>In my example, I think this satisfied complete preference(connected relation), even though in the context of daily language that any good should be weakly prefer to itself. Please correct me if this is not connected relation, from definitional or logical disprove.</p>
<p>This question pondered by a question asking to show that complete binary relation on<span class="math-container">$X$</span> is also reflexive.</p>

<microeconomics><decision-theory><choice-theory>
Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?
LJNG
# Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?

(Asked by: username_0 on 2023-08-16)

Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?

By looking at the completeness definition in preference: [Definition 1.1(c)](https://assets.press.princeton.edu/chapters/s9890.pdf), this is same as the connected relation in the maths definition [wiki-connected relation](https://www.wikiwand.com/en/Connected_relation). If I am right, for any distinct pair of elements in $X$, they are comparable, otherwise they are the same elements **[They are the same might just leave the chance that $(1,1)\notin\mathcal{R}$?]**.

In my example, I think this satisfied complete preference(connected relation), even though in the context of daily language that any good should be weakly prefer to itself. Please correct me if this is not connected relation, from definitional or logical disprove.

This question pondered by a question asking to show that complete binary relation on$X$ is also reflexive.

**Tags:** microeconomics, decision-theory, choice-theory


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## 1 answers

### Answer by username_1 on 2023-08-16. Score: 0 | Accepted answer

In the wiki you will see it says: "As described in the terminology section below, the terminology for these properties is not uniform."

The definition of *strongly connected* in the wiki is the same as the definition of *completeness* at your first link. Both require that:

$$\text{for all $x,y\in X$ either $(x,y)\in\mathcal{R}$ or $(y,x)\in\mathcal{R}$ }$$

Note that the above must hold when $x=y$ as well as when $x\neq y$.

The definition of *connected* / *complete* / *total* in the wiki is a weaker condition:

$$\text{for all $x,y\in X$ with $x\neq y$ either $(x,y)\in\mathcal{R}$ or $(y,x)\in\mathcal{R}$ }$$

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For your relation, since, for example, $(1,1)\notin\mathcal{R}$, the relation is *not* complete (according to the definition in your first link) and not strongly conected (according to the definition in the wiki). However, your relation $\mathcal{R}$ *is* connected (according to the definition in the wiki).

The relation $\mathcal{S}=\{(1,1),(2,2),(3,3),(1,3),(2,1),(3,2)\}$ is strongly connected (according to the wiki definition) and therefore connected (according to the wiki definition) and complete (according to the definition at your first link).

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If a relation is complete by the definition at your first link, then it is reflexive.

> **Comments:**
> **username_0**: Thank you for your reply. My first link *$\forall x,y\in X$, either $(x,y)\in\mathcal{R}$ or  $(y,x)\in\mathcal{R}$ or (both)* is same as the connected relation **(not strongly connected)** in the wiki. I believe confusion comes in how to interpret $x=y$ in the definition. When you read this definiton, why would perceiving $x=y$ means this requirements need to be hold for $x=y$, whereas not the case that as long as $x=y$ we may ignore that  part. Just talk about first link definiton, and wiki connected relation definition. **not strongly connected**  
> **username_1**: No, the definition of connected in the wiki specifies $x\neq y$, while the definition of complete in your first link does not (so the definitions are not the same). The definition of complete in your first link corresponds to the definition of strongly connected in the wiki. I am not sure what you mean by "how to interpret $x=y$". It is not a matter of interpretation; the two elements $x$ and $y$ are the same or they are not.  
> **username_0**: Thank you for your reply. The wiki definition goes: *A relation $\mathcal{R}$ on a set $X$ is called connected when for all $x,y\in X$ if $x\neq y$ then $x\mathcal{R} y$ or $y\mathcal{R} x$; **equivalently**, when for all $x,y\in X$, $x\mathcal{R} y$ or $y\mathcal{R} x$ or **$x=y$**.(THere is what I meant how to interpret $x=y$)* This equivalent definition is the same as the definition in my first link: *A preference relation on $X$ is complete if for every pair $x$ and $y$ from $X$ , either $x \succsim y$ or $ y\succsim x$ **(or both)***  
> **username_1**: No, that definition of connected is not the same as the definition of complete in your first link. The definition of connected allows for it not to be the case that $x\mathcal{R} x$, whereas the definition of complete requires that $x\succsim x$.  
> **username_0**: Thank you for your reply, may you please enlighten me the subtle difference. this preference relation $\succsim$ look the same as $\mathcal{R}$ to me, just notation differences.  
> **username_0**: The definition of complete requires $x\succsim x$, I think probably $x\succsim x$ this is not a consequence of above fromal definition, rather a common sense embeded in that every good is weakly prefer to itself ? Could you logically derive $x\succsim x$ is a consequence of that complete definition ?  Much appreciated  
> **username_1**: The formal definitions of completeness (at the first link) and strong connectedness both require that $x$ is related to itself (for all $x\in X$). That is the crucial difference to connectedness. Strong connectedness implies connectedness, but not conversely. It is common sense that every consumption bundle is weakly preferred to itself. That is why strong connectedness is assumed rather than just connectedness. Also, indifference ($x\sim y$) can be defined as $x\succsim y$ and $y\succsim x$, so if it were not the case that $x\succsim x$, then it would not be the case that $x\sim x$.  
> **username_0**: Thank you so much, I think I get it. The complete definition is actually said $\forall x,y\in X$, $x\mathcal{R}y$ or $y\mathcal{R}x$ or $(x\mathcal{R}y)\land (y\mathcal{R}x)$, this is the subtle different from that $x=y$ in the connected definition. $x\sim x$ is the consequence of $(x\mathcal{R}y)\land (y\mathcal{R}x)$, wheras $x=y$ in the connected definition allowing such case not in the relation still making the relation being connected.  
> **username_1**: In mathematics, "A or B" includes the possibility that "A and B".  There is no need to write  "A or B or (A and B)".  
> **username_0**: Yes, it is. just infrom myself that $(x\mathcal{R}y)\land (y\mathcal{R}x)$ as the interpretation for **(or both)** in the complete preference definition is different from that $x=y$ in the connected relation. Anyway, thank you so much  

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