If you keep removing elements from a nonempty set, will you eventually get the null set? Some context below in a problem which I made myself and have been trying to solve:(adsbygoogle = window.adsbygoogle || []).push({});

A set S having property P is smallest with respect to property P if S is an improper subset of any set having property P.

A set S having property P is minimal with respect to property P if any proper subset of S fails to have property P.

I am trying to prove/disprove the following: Let M be the only minimal set having property P. Then M is smallest with respect to property P, i.e. any set having property P is an improper superset of M.

My proof:

If M is the only set having property P, then the conjecture is trivially true. Otherwise suppose, on the contrary, that there is some set M' having property P, but which fails to be a superset of M.

Then since M' is not minimal, there exists a proper subset of M', namely M'', having property P. But M'' is also not a minimal set, so there exists a proper subset of M'', namely M''', having property P. This process can continue indefinitely, since M is not a subset of M'. By repeatedly removing elements from M, we should obtain that the null set also has property P.This contradicts the fact that M is minimal with respect to property P. Thus M' must be the smallest set having property P.

It is the part in bold whose rigor I do not completely trust. What can I do to improve the rigor, or what is the flaw in my reasoning? This is the first "difficult" problem I have ever made so I want to make sure I solve it correctly. All help is appreciated. Thanks!

BiP

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Does constantly removing elements from a set produce the null set?

**Physics Forums | Science Articles, Homework Help, Discussion**