<p>Mas-Collel, Winston and Green's Microeconomic Theory (3rd editions) offers two definitions of the upper contour set:</p>

<ol>
<li><a href="https://i.sstatic.net/Io86C.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Io86C.jpg" alt="enter image description here"></a></li>
<li><a href="https://i.sstatic.net/9e199.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9e199.jpg" alt="enter image description here"></a></li>
</ol>

<p>How can the equivalence between the two definitions be proved?</p>

CC BY-SA 4.0
2019-02-26T19:57:42.450Z
But Def M.C.3 is not a definition of the upper contour set; rather, it's a definition of quasiconcavity of a function.
Herr K.
CC BY-SA 4.0
2019-02-26T20:00:21.553Z
Or are you asking about the equivalence between $\{x\in A:f(x)\ge t\}$ and $\{y\in X:y\succsim x\}$?
Herr K.
CC BY-SA 4.0
2019-02-26T21:29:56.213Z
I have updated the question. The definition of upper contour set in M.C.3 is now highlighted.
Incognito
CC BY-SA 4.0
2019-02-26T21:33:50.363Z
Have you given the question some thoughts? This is not a particularly difficult proof.
Herr K.
<microeconomics><mathematical-economics>
Equivalence of definitions of upper contour sets
Incognito
# Equivalence of definitions of upper contour sets

(Asked by: username_0 on 2019-02-26)

Mas-Collel, Winston and Green's Microeconomic Theory (3rd editions) offers two definitions of the upper contour set:

1. [![enter image description here](https://i.sstatic.net/Io86C.jpg)](https://i.sstatic.net/Io86C.jpg)
2. [![enter image description here](https://i.sstatic.net/9e199.jpg)](https://i.sstatic.net/9e199.jpg)

How can the equivalence between the two definitions be proved?

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: But Def M.C.3 is not a definition of the upper contour set; rather, it's a definition of quasiconcavity of a function.  
> **username_1**: Or are you asking about the equivalence between $\{x\in A:f(x)\ge t\}$ and $\{y\in X:y\succsim x\}$?  
> **username_0**: I have updated the question. The definition of upper contour set in M.C.3 is now highlighted.  
> **username_1**: Have you given the question some thoughts? This is not a particularly difficult proof.  

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## 0 answers

