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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 
Member Joined: Oct 2018 From: USA Posts: 98 Thanks: 70 Math Focus: Algebraic Geometry 
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. 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  
Senior Member Joined: Aug 2012 Posts: 2,410 Thanks: 754  Quote:
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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