Intervals

If we have a set of real numbers, we can define a subset of this set by specifying a range of values. This subset is called an interval. To define an interval, we need two real numbers, $a$ and $b$, with $a < b$.

We can then define the following types of intervals:

• Open interval: We include all the numbers between $a$ and $b$, but not $a$ and $b$ themselves. This is denoted as $(a, b) = \{x \in \mathbb{R} \mid a < x < b\}$.
• Closed interval: We include all the numbers between $a$ and $b$, as well as $a$ and $b$ themselves. This is denoted as $[a, b] = \{x \in \mathbb{R} \mid a \leq x \leq b\}$.
• Half-open interval: We include one of the bounds but not the other. This is denoted as $[a, b) = \{x \in \mathbb{R} \mid a \leq x < b\}$ or $(a, b] = \{x \in \mathbb{R} \mid a < x \leq b\}$.
• Unbounded/Infinite interval: On one or both sides, the interval extends indefinitely. This is denoted as $(-\infty, b] = \{x \in \mathbb{R} \mid x \leq b\}$ if we want all numbers less or equal than $b$, or $(a, \infty) = \{x \in \mathbb{R} \mid x > a\}$ if we want all numbers greater than $a$. We can also have intervals that are unbounded on both sides, such as $(-\infty, \infty) = \mathbb{R}$.