I am interested in understanding the correct grammar for the use of the term "interval" vs. "intervals" (plural).
For example: which of these is correctly stated?
- A polynomial $p(x)>0$ on the interval $(-2, 0)\cup(2, \infty)$
- A polynomial $p(x)>0$ on the intervals $(-2, 0)\cup(2, \infty)$
I understand that the union of two intervals is not nessarily an interval (can be proven by counterexample because the intersection of two intervals may be an empty set).
That being the case, it seems that when defining the solution set to an inequality using interval notation the use of "intervals" is more appropriate than "interval." But, I have seen both statements. Which is correct?