# What is meant by $sup(A\cup B )$

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?

• Are you asking what the symbol $\cup$ means? It is set union.
– Eff
Jun 20, 2016 at 17:50
• 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? Jun 20, 2016 at 17:53
• 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. Jun 20, 2016 at 17:54
• 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. $\sup (A\cup B)$?
– Eff
Jun 20, 2016 at 17:55
• Not the "unity(?) of two supremums" but the supremum of the union of two sets. Parentheses make a difference. Jun 20, 2016 at 17:56

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.