# What is the complement of this set

#### director

(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!

#### Aryth

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.