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I am given two subsets $A,B$ of $\mathbb{R}$ which are not empty and are bounded above.

Now according to a lemma, both of these sets have supremums. My issue is part of the question deals with sup$(A\cup B )$. What exacty does that mean?

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    $\begingroup$ Are you asking what the symbol $\cup$ means? It is set union. $\endgroup$
    – Eff
    Jun 20, 2016 at 17:50
  • $\begingroup$ Sorry I should have been more clear. It's obviously not the sum supA and supB. What exactly does the unity of two supremums mean beyond the literal meaning? $\endgroup$
    – RonaldB
    Jun 20, 2016 at 17:53
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    $\begingroup$ The most likely circumstance is that $A,B$ are subsets of a totally ordered set. The word "group" often refers not to sets per se, but to a structure studied in abstract algebra, e.g. a "group" of permutations, etc. $\endgroup$
    – hardmath
    Jun 20, 2016 at 17:54
  • $\begingroup$ Well, both $A$ and $B$ are sets, and $A\cup B$ is a set. So what is the supremum of $A\cup B$, i.e. $\sup (A\cup B)$? $\endgroup$
    – Eff
    Jun 20, 2016 at 17:55
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    $\begingroup$ Not the "unity(?) of two supremums" but the supremum of the union of two sets. Parentheses make a difference. $\endgroup$
    – hardmath
    Jun 20, 2016 at 17:56

1 Answer 1

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First, remind yourself of what sup($A$) means. sup($A$) is the least upper bound of $A$. That is, it's the number $s$ such that every element of $A$ is less than or equal to $s$. In addition, $s$ is less than or equal to any other upper bound on $A$.

Your question has to do with taking the sup of a union. It means exactly the same thing as it did before, only your input, your set, has changed. Recall what $A\cup B$ means. $A\cup B$ is the collection of all elements that are in either $A$ or $B$. So, when looking for sup($A\cup B$), we seek the smallest upper bound for the set $A\cup B$. Is it sup($A$)? Is it sup($B$)? Is it a linear combination of the two? Is it, perhaps, some other function of sup($A$) and sup($B$)? Perhaps you can do some investigating now.

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  • $\begingroup$ That's what I thought. Thanks for such a concise answer ! $\endgroup$
    – RonaldB
    Jun 20, 2016 at 18:10
  • $\begingroup$ You are very welcome :) $\endgroup$ Jun 20, 2016 at 19:52

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