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Express the notion of a minimum of a set of number (where numbers are defined via sets). That is, define a relation Min(S,x) using logic and set-theoretic operations such that it is true whenever x is the minimum element in S.

  • I understand the definition of numbers in terms of sets. I did that in part a of this same question.
  • I understand what a function or relation is

    But I have no idea what this is asking, nor how to answer it.


Edit: My definition of a number is as follows:

0 = {}
$n+1 = n\cup \{n\}$

So :

  • 1 = {{}}
  • 2 = {{}, {{}}}
  • 3 = {{}, {{}}, {{}, {{}}}}
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    (I only just saw this: if you want it brought to my attention, you have to put `@Brian` somewhere in the comment.) Yes, now you’ve got it. To say that $x$ is the minimum element of $S$ you must say first that $x\in S$, which you’ve done, and then that $x$ is minimal. That means that $x$ is less than or equal to everything in $S$. Thus, for each $y\in S$, you want either x, which means $x\in y$, or $x=y$. In some of your earlier attempts you were forgetting to allow for the $x=y$ possibility.2012-11-05

2 Answers 2

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I'd like to thank Brian M. Scott above in the comments for helping me understand the problem and coax me to an answer.

As I understand it, the logic is as follows:

For a number to be the minimum number in a set of numbers, it must

  1. Be a member of the set, and
  2. Be a member of every other member of that set

For the first part, we have $Min(S,x)\iff x\in S$

For the second part, we have

$Min(S,x)\iff \forall y\in S\{x\subseteq y\}$

I've chosen $x\subseteq y$ instead of $x\in y\vee x=y$ because I like having less text, but both are equivalent. The reason why $x\subseteq y$ works is because x can legitimately be a subset of itself, but can only also be a subset of any natural number greater than it. So it'll return false only if there is a natural number less than x in the set S.

Putting 1 and 2 together now, we have:

$Min(S,x)\iff x\in S\wedge \forall y\in S\{x\subseteq y\}$

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Hint:

You can notice either of the following observations:

  1. $n; and
  2. $n\leq m\iff n\subseteq m$.

Therefore if $S$ is a set of natural numbers, its minimum is the element which is either the member of everyone else; or a subset of everyone (including itself, of course).