# Naive Set Theory: Relations

## Intervals

An interval is a subset of an ordered set that includes all of the elements of between two given elements. In this context, "between" can mean to either include or exclude the elements, and the distinction is so important that each case gets its own designation.

Let $A$ be a set equipped with an order relation $<$, and let $a$ and $b$ be elements of $A$ where $a<b$. The following are the four kinds of intervals on $A$:

• $(a, b) = \{ x : a < x < b \}$ is an open interval.

• $(a, b] = \{ x : a < x \leq b \}$ is a left open interval or a right closed interval.

• $[a, b) = \{ x : a \leq x < b \}$ is a right open interval or a left closed interval.

• $[a, b] = \{ x : a \leq x \leq b \}$ is a closed interval.

Open intervals do not include their endpoints and are denoted with parentheses, just like ordered pairs. In contrast, closed intervals do include their endpoints and are denoted with square brackets. Intervals that include only one endpoint are said to be half open intervals or half closed intervals, and they are denoted using the appropriate combination of parentheses and square brackets.

If an open interval $(a,b)$ is empty, then $a$ is called the immediate predecessor of $b$, and $b$ is called the immediate successor of $a$.

Rays are like intervals, but they include all elements below or above a certain point. A ray is a subset of an ordered set that contains all of the elements greater than (or equal to) or less than (or equal to) a particular interval. Like intervals, they come in four kinds:

• $(-\infty, a) = \{ x : x < a \}$ is is a right open ray.

• $(a, \infty) = \{ x : x > a \}$ is a left open ray.

• $(-\infty, a] = \{ x : x \leq a \}$ is a right closed ray.

• $[a, \infty) = \{ x : x \geq a\}$ is a left closed ray.

Intervals and rays are related to bounds, which are defined in the next section. Intervals are always defined by their bounds, which here are the elements $a$ and $b$ between which their own elements lie. By way of a hamburger analogy, bounds are the buns, and intervals are the patty. However, bounds need not apply to contiguous intervals; bounds can exist for many kinds of sets (such as the set $\{1,3,\pi,17\}$). Some problems involve determining whether elements of interest are within some given bounds, while others involve determining the bounds from the elements inside and outside of a given set. Said differently, some chefs seek patties, while others seek buns.

## Problems

1. Let $(2, 7)$ denote an interval on the integers, $\mathbb{Z}$. List the elements of $(2, 7)$, describe the sets of its upper and lower bounds, and identify its supremum and infimum.

The elements of $(2, 7)$ are $3, 4, 5$, and $6$. Its set of upper bounds is $[6, \infty)$, its set of lower bounds is $(-\infty, 3]$. Its supremum is $6$ and its infimum is $3$.

2. Let $[-3, 12)$ be an interval on the real line, $\mathbb{R}$. Describe its sets of upper and lower bounds, and identify its supremum and infimum.

The set of upper bounds for $[-3, 12)$ is $[12, \infty)$, and the set of lower bounds is $(-\infty, -3]$. The supremum is $12$ and the infimum is $-3$.