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

Question

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?

Answer 1

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.