How would you answer to this question in an exam?
My idea is that they can be both intervals or not. In fact, the intersection of 2 sets can give as a result an interval but also an empty set and the union of two sets can give as a result an interval (if the sets are equal) or two disjoint sets.
Do you agree with me? In case, how would you write an answer to such question in an exam?
Best Answer
For a subset of the real numbers to be an interval it must satisfy that if $a,b \in I \quad a<b \implies c \in I ~ \forall a<c<b$.
So, $I_1 \cup I_2$ is not always an interval (consider $[0,1] \cup [2,3]$). But $I_1 \cap I_2$ is always an interval (even if it's empty). To show this take $x,y \in (I_1 \cap I_2)$ then, for any $x<z<y$, using that $x,y \in I_1$ and that $x,y \in I_2$ you get that $z \in I_1 \cap I_2$