[Math] Show that the intersection of any two intervals is an interval

elementary-set-theorylogic

So i've come across this question, with a follow up question of showing that the union of any two intervals need not be an interval.

I don't see how this could possibly be the case. The general structure of my proof would be to consider the case where:

  1. The intersection yields an empty set;
  2. The intersection yields a set with 1 element;
  3. The intersection yields a set with 2 or more elements;

and then consider each case and show that each is an interval.

But a union can only yield one of those three possibilities too. To elaborate: If any of the three possibilities were not an interval, then an intersection is not necessarily an interval, so each of them must be an interval. But each of these possibilities for the intersection are also the only possibilities for a union, meaning a union of intervals must be an interval too, which is not true.

I'm obviously wrong, but why?

Best Answer

By definition, a set $A\subset{\mathbb R}$ is an interval if $$\forall x, \ y\in A,\quad\forall t\in{\mathbb R}:\qquad x\leq t\leq y\quad\Rightarrow\quad t\in A\ .\tag{1}$$ It is then obvious (on logical grounds, no case distinctions needed) that the intersection of two intervals is an interval. Of course it is allowed to go through the motions anyway:

Let $A$ and $B$ be intervals, let $x$, $y\in A\cap B$, and assume $x\leq t\leq y$. Then $t\in A$ as well as $t\in B$, hence $t\in A\cap B$. This shows that $A\cap B$ passes the test $(1)$.

Note that the claim would not be true if we would not accept the empty set as an interval.

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