What is the complement of this set

May 2010
63
0
(interval, actually)

\(\displaystyle [a, \infty )\)

I see the complement contains at least the interval \(\displaystyle (-\infty ,a)\). Anything else? I want to say no.

Thanks!
 
Feb 2007
666
199
USA
The complement of \(\displaystyle [a,\infty)\) is the set \(\displaystyle X\) such that:

\(\displaystyle X = \mathbb{R}\setminus [a,\infty)\).

In other words, it is the set of all real numbers, \(\displaystyle x\), such that \(\displaystyle x\in\mathbb{R}\) but not in \(\displaystyle [a,\infty)\). So then you think about which points of \(\displaystyle \mathbb{R}\) aren't in \(\displaystyle [a,\infty)\). You can see immediately that \(\displaystyle a\) won't be in \(\displaystyle X\), and since every \(\displaystyle y\in\mathbb{R}\) that is greater than \(\displaystyle a\) is also in \(\displaystyle [a,\infty)\), none of those will be in \(\displaystyle X\) either. Now we can see that \(\displaystyle X = \{x \in \mathbb{R} : x < a\}\) is the complement of \(\displaystyle [a,\infty)\). The interval \(\displaystyle (-\infty, a)\) is indeed equal to \(\displaystyle X\). Thus, your guess is correct.