# What is an open interval?

Aug 31, 2016

See explanation...

#### Explanation:

If $a$ and $b$ are Real numbers with $a < b$ then $\left(a , b\right)$ is used to denote the numbers which lie strictly between $a$ and $b$. This is called "the open interval $a$, $b$".

In symbols we could write: $\left(a , b\right) = \left\{x \in \mathbb{R} : a < x < b\right\}$

This reads $\left(a , b\right)$ is the set of elements $x$ in the set of Real numbers ($\mathbb{R}$ for short) such that $a < x$ and $x < b$.

When we want to talk about all the Real numbers we may write:

$\left(- \infty , + \infty\right)$

The symbols $- \infty$ (minus infinity) and $+ \infty$ (plus infinity) are not really numbers. You can picture them as being at the extreme left and right ends of the Real number line. Any Real number $x$ satisfies:

$- \infty < x < + \infty$

If we want to talk about any number greater than $5$, we can write:

$x \in \left(5 , + \infty\right) \text{ }$ ($x$ is in the open interval "$5$ to plus infinity")

If we want to talk about any number less than $5$, we can write:

$x \in \left(- \infty , 5\right) \text{ }$ ($x$ is in the open interval "minus infinity to $5$")

If $z \ne 5$ then either $z < 5$ so $z \in \left(- \infty , 5\right)$ or $z > 5$ so $x \in \left(5 , \infty\right)$.

The amalgamation of these two sets is called the union and is denoted:

$\left(- \infty , 5\right) \cup \left(5 , + \infty\right)$