<p>A <em>function</em> or <em>mapping</em> in mathematics requires that there are <strong>two sets</strong> on which the function is defined.</p>
<p>The so-called <strong>classical  definition of function</strong> (ascribed to Dirichlet)  is the following:</p>
<blockquote>
<p><strong>Definition.</strong> Consider two sets <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, whose elements may be any objects whatsoever. A <em>function</em> <span class="math-container">$f$</span> is a <strong>rule</strong> that associates to each element of <span class="math-container">$A$</span> a <strong>unique</strong> element of <span class="math-container">$B$</span>.</p>
</blockquote>
<p>The set <span class="math-container">$A$</span> is called the <em>domain</em> of <span class="math-container">$f$</span>, and the set of the values of <span class="math-container">$f$</span> is called the <em>range</em> of <span class="math-container">$f$</span>.
The concept of rule is left undefinied.</p>
<p>In your example you have a set, <span class="math-container">$X^n\subset R^n$</span>, as range, and a relation <span class="math-container">$\succeq_i$</span> as domain.</p>
<p>A <em>relation</em> in set theory is <strong>defined as a set</strong>, a set of ordered pairs:</p>
<blockquote>
<p><strong>Definition.</strong> A <strong>relation</strong> on a set <span class="math-container">$A$</span> is a subset <span class="math-container">$C$</span> of the cartesian product <span class="math-container">$A\times A$</span>.</p>
</blockquote>
<p>It is, in some way, a concept more general than that of function (see for instance Munkres, <em>Topology</em>).</p>
<p>So,  in principle, from a formal point of view,  you can define a function from <span class="math-container">$\succeq_i$</span> to <span class="math-container">$X^n$</span>, that is a function that assigns an element of <span class="math-container">$X^n$</span> to each pair that forms the set <span class="math-container">$\succeq_i$</span>.</p>
<p>But what does it mean from an economic point of view? I think, nothing. This way you associate to each pair that forms the <span class="math-container">$\succeq_i$</span> set, a bundle of consumption of the consumer <span class="math-container">$i$</span>. It seems to me meanigless.</p>
<p>Instead, you can consider the set of <strong>all possible</strong> preferences <span class="math-container">$\succeq_i$</span>, that is a set of sets, and assign to each set of preferences a commodity bundle <span class="math-container">$X_i$</span>. That is, you have to take as domain not   <span class="math-container">$\succeq_i$</span>, but <span class="math-container">$\succeq_i$</span>, <span class="math-container">$i=1,...,n,...$</span>, for instance. Maybe it makes sense. But I wonder if it can be useful, I don't know.</p>
<p>As for your second question, <span class="math-container">$C_i(X^n, \succeq_i)$</span> cannot be viewed as a vector space. Vector space needs two operations, scalar multiplication and addition of vectors, <span class="math-container">$\succeq_i$</span> is not any of them.</p>
<p>Anyway, it is not clear what you mean by  <span class="math-container">$C_i(X^n, \succeq_i)$</span>: a set? a function?</p>

CC BY-SA 4.0
2022-07-17T09:30:30.887Z
How can you answer a question which you yourself admit you don't understand? You are asking the right questions at the end!
BrsG
CC BY-SA 4.0
2022-07-17T10:56:58.327Z
I understand very well the question. To answer is sufficient to know the definition of vector space. In both cases, that is if the notation indicates a vector space or a function, it cannot represent a vector space, it is obviuosly wrong to say that can be a vector space, in any case. But I asked my question because also the notation is wrong if one wants to denote a vector space. Maybe is correct in case it denotes a function. And I asked what was the idea in writing this strange notation, in order to understand the reason of the mistake.
BakerStreet
CC BY-SA 4.0
2022-07-12T15:15:30.673Z
2022-07-15T16:45:14.567Z
2022-07-15T16:45:14.567Z
BakerStreet
<p>I want to make a function that states consumer <span class="math-container">$i's$</span> consumption <span class="math-container">$C_i$</span>  in a shared bundle <span class="math-container">$X^n$</span> <span class="math-container">$\subset$</span> <span class="math-container">$\mathbb R^n$</span> depends on their preferences. That is, something like <span class="math-container">$C_i(\succeq_i)$</span>.</p>
<p>Since <span class="math-container">$\succeq_i$</span> is a binary relator that does not take any value, would I denote this as a vector space <span class="math-container">$C_i(X^n,\succeq_i)$</span>?</p>

CC BY-SA 4.0
2022-05-21T08:04:26.110Z
It is not really clear what you mean with "a function that states consumer $i$'s consumption in a shared bundle". Does $X^n$ vary? What is the output of $C_i$ (a set or an element of $\mathbb{R}^n$ or an element of $X^n$?) What restrictions do you impose on $\succeq_i$?
tdm
CC BY-SA 4.0
2022-05-21T11:17:01.217Z
Vector spaces need vector operations like vector addition and scalar multiplication. These make little sense here.
Michael Greinecker
CC BY-SA 4.0
2022-06-07T19:03:57.357Z
It seems you mean $C_i \in X^n$, i.e. the compsumption will be a set of "consumed" quantities. Isn't it?
Davius
<mathematical-economics><preferences>
Is this a function or vector space?
Eli J
# Is this a function or vector space?

(Asked by: username_0 on 2022-05-21)

I want to make a function that states consumer $i's$ consumption $C\_i$ in a shared bundle $X^n$ $\subset$ $\mathbb R^n$ depends on their preferences. That is, something like $C\_i(\succeq\_i)$.

Since $\succeq\_i$ is a binary relator that does not take any value, would I denote this as a vector space $C\_i(X^n,\succeq\_i)$?

**Tags:** mathematical-economics, preferences

> **Comments:**
> **username_1**: It is not really clear what you mean with "a function that states consumer $i$'s consumption in a shared bundle". Does $X^n$ vary? What is the output of $C_i$ (a set or an element of $\mathbb{R}^n$ or an element of $X^n$?) What restrictions do you impose on $\succeq_i$?  
> **username_2**: Vector spaces need vector operations like vector addition and scalar multiplication. These make little sense here.  
> **username_3**: It seems you mean $C_i \in X^n$, i.e. the compsumption will be a set of "consumed" quantities. Isn't it?  

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## 1 answers

### Answer by username_4 on 2022-07-12. Score: 2

A *function* or *mapping* in mathematics requires that there are **two sets** on which the function is defined.

The so-called **classical definition of function** (ascribed to Dirichlet) is the following:

> **Definition.** Consider two sets $A$ and $B$, whose elements may be any objects whatsoever. A *function* $f$ is a **rule** that associates to each element of $A$ a **unique** element of $B$.

The set $A$ is called the *domain* of $f$, and the set of the values of $f$ is called the *range* of $f$.
The concept of rule is left undefinied.

In your example you have a set, $X^n\subset R^n$, as range, and a relation $\succeq\_i$ as domain.

A *relation* in set theory is **defined as a set**, a set of ordered pairs:

> **Definition.** A **relation** on a set $A$ is a subset $C$ of the cartesian product $A\times A$.

It is, in some way, a concept more general than that of function (see for instance Munkres, *Topology*).

So, in principle, from a formal point of view, you can define a function from $\succeq\_i$ to $X^n$, that is a function that assigns an element of $X^n$ to each pair that forms the set $\succeq\_i$.

But what does it mean from an economic point of view? I think, nothing. This way you associate to each pair that forms the $\succeq\_i$ set, a bundle of consumption of the consumer $i$. It seems to me meanigless.

Instead, you can consider the set of **all possible** preferences $\succeq\_i$, that is a set of sets, and assign to each set of preferences a commodity bundle $X\_i$. That is, you have to take as domain not $\succeq\_i$, but $\succeq\_i$, $i=1,...,n,...$, for instance. Maybe it makes sense. But I wonder if it can be useful, I don't know.

As for your second question, $C\_i(X^n, \succeq\_i)$ cannot be viewed as a vector space. Vector space needs two operations, scalar multiplication and addition of vectors, $\succeq\_i$ is not any of them.

Anyway, it is not clear what you mean by $C\_i(X^n, \succeq\_i)$: a set? a function?

> **Comments:**
> **username_5**: How can you answer a question which you yourself admit you don't understand? You are asking the right questions at the end!  
> **username_4**: I understand very well the question. To answer is sufficient to know the definition of vector space. In both cases, that is if the notation indicates a vector space or a function, it cannot represent a vector space, it is obviuosly wrong to say that can be a vector space, in any case. But I asked my question because also the notation is wrong if one wants to denote a vector space. Maybe is correct in case it denotes a function. And I asked what was the idea in writing this strange notation, in order to understand the reason of the mistake.  

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