### Sets: Sets

### Intervals

We have discussed *the notion of set* and of *subsets*. In calculus, the focus is on 'functions' between sets of real numbers. *Later*, we will discuss the concept of functions. Here, we will focus on sets of real numbers. A set of real numbers that contains all real numbers lying between any two numbers of the set is called a bounded interval.

Consider the bounded interval of all real numbers between #1# and #5#. It is intuitive that the numbers #\pi# and #4# are in this interval. But, what about the numbers #1# or #5#; are they also included? In order to provide a precise specification, we distinguish four kinds of bounded interval.

Bounded interval

Let #\blue c#, #\green d# be two real numbers with #\blue c \le \green d#. We define the following four types of bounded intervals.

Name interval |
Interval notation |
Set-builder notation |
Example |

open | #\ivoo{\blue c}{\green d}# | #\{x\in\mathbb{R}\mid \blue c\lt x\lt \green d\}# | #\ivoo{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \lt x\lt \green 8\}# |

closed | #\ivcc{\blue c}{\green d}# | #\{x\in\mathbb{R}\mid \blue c\le x\le \green d\}# | #\ivcc{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \le x\le \green 8\}# |

open-closed | #\ivoc{\blue c}{\green d}# | #\{x\in\mathbb{R}\mid \blue c\lt x\le \green d\}# | #\ivoc{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \lt x\le \green 8\}# |

closed-open | #\ivco{\blue c}{\green d}# | #\{x\in\mathbb{R}\mid \blue c\le x\lt \green d\}# | #\ivco{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \le x\lt \green 8\}# |

The point #\blue c# is called the **left boundary **or **left end point** and the point #\green d# the **right boundary** or **right end point **of the interval. The points #\blue c# and #\green d# are the **end points** or **boundary points** of the interval.

Points #x\in\mathbb{R}# with #\blue c\lt x\lt \green d# are called **interior points** of the interval.

The** length **of the interval is #\left|\blue c- \green d\right|#.

There also are unbounded intervals. In these case there is at most one real end point. If there is no left end point, then we designate #-\infty# as the left end point, which is not a real number. Similarly, if there is no right end point, we use the notation #\infty# to replace the missing the right end point.

Unbounded intervalLet #\blue c#, #\green d# be two real numbers. We define the following unbounded interval

Name interval |
Interval notation |
Set-builder notation |
Example |

left-open | #\ivoo{\blue c}{\infty}# | #\{x\in\mathbb{R}\mid x \gt \blue c \}# | #\ivoo{\blue 3}{\infty} = \{x\in\mathbb{R}\mid x\gt \blue 3\}# |

left-closed | #\ivco{\blue c}{\infty}# | #\{x\in\mathbb{R}\mid x \ge \blue c \}# | #\ivco{\blue 3}{\infty} = \{x\in\mathbb{R}\mid x\ge \blue 3\}# |

right-closed | #\ivoc{-\infty}{\green d}# | #\{x\in\mathbb{R}\mid x \le \green d\}# | #\ivoc{-\infty}{\green 8} = \{x\in\mathbb{R}\mid x \le \green 8\}# |

right-open | #\ivoo{-\infty}{\green d}# | #\{x\in\mathbb{R}\mid x\lt \green d\}# | #\ivoo{-\infty}{\green 8} = \{x\in\mathbb{R}\mid x \lt \green 8\}# |

The interval notation of #\{x\in\mathbb{R}\mid 5 \lt x \lt 12\}# is #\ivoo{5}{12}#.

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