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I'm looking for an example when there is no supremum of an intersection of 2 sets, also no supremum of an union (not the same example).

Each set needs to have a supremum, but not their intersection. Same goes for the union.

I'd really appreciate same example for infimum.

(I do undersand what all of the mentioned above are, I just struggle with finding a suitable example.)

Andrés E. Caicedo
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curleecurlees
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  • What is the universe of these sets? Because $\sup$ and $\inf$ are defined on sets containing ordered elements. – Rushabh Mehta Jan 21 '20 at 01:16
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    If the universe is the reals, then the union example you want isn't possible. – Rushabh Mehta Jan 21 '20 at 01:19
  • What about naturals or integers? I do not know what the universe is supposed to be, the problem was presented to me like this. – curleecurlees Jan 21 '20 at 01:21
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    Same problem with any subset of the reals. – Rushabh Mehta Jan 21 '20 at 01:23
  • But are you talking about the union or also about interjection? I was thinking what if it was for example an interjection between sets [1,2,3] and [7,8] ? Then their interjection is empty so it cannot have a supremum. Does that make sense? – curleecurlees Jan 21 '20 at 01:27
  • And as to the union I couldn't come up with anything, but my teacher believes it exists. – curleecurlees Jan 21 '20 at 01:27
  • I only mentioned union :P. I can prove union doesn't exist. If $A\cup B$ has no supremum while $\sup A=a$, $\sup B=b$, note that there must exist some $x\in A\cup B$ such that $x>a,b$, since otherwise, $A\cup B$ would be bounded from above, and by the supremum property, it has a supremum. But, $x\in A$ or $x\in B$, violating the supremum of $A$ or $B$. – Rushabh Mehta Jan 21 '20 at 01:28
  • You're right.. Does infinity have supremum? – curleecurlees Jan 21 '20 at 01:39
  • No, it's not a real. – Rushabh Mehta Jan 21 '20 at 01:40

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