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 May 24th, 2019, 10:27 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory Symmetry between all null and all unbounded sets Is there a greatest symmetry between the set of all null sets and the set of all unbounded sets?
 May 26th, 2019, 12:53 PM #2 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 464 Thanks: 29 Math Focus: Number theory I was trying to ask what sets are closed under both addition and multiplication. The null set, the set whose only member is zero, and the set of transfinite numbers come to mind. Is this correct? Can you offer any other "unconventional" set(s)?
May 26th, 2019, 01:48 PM   #3
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The set of even numbers fit in this category since for any two even numbers $a,b$ and their halves $c,d \in \mathbb{Z}$

$a+b = 2(c+d)$ even, and

$a \times b = 4(c \times d)$ also even.

Not super unconventional, but a property I find interesting.

Quote:
 Originally Posted by Loren Is this correct?
I believe so, $0+0$ or $0 \times 0$ will of course be closed and given two transfinites, $p,q$, $p+q$ and $p \times q$ should be transfinite as well.

Last edited by Greens; May 26th, 2019 at 01:52 PM. Reason: Grammar

May 26th, 2019, 07:45 PM   #4
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Quote:
 Originally Posted by Loren I was trying to ask what sets are closed under both addition and multiplication.
Any subring of the real numbers (or complex numbers if you prefer). Examples would be the rational numbers, the computable numbers, in fact any of the many subfields of the reals such as the rationals adjoined with $\sqrt 2$, etc.

Then there are the integers, and as mentioned the even integers.

I'm sure there are many more.

https://en.wikipedia.org/wiki/Subring

Last edited by Maschke; May 26th, 2019 at 08:15 PM.

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