### Sets: Sets

### The notion of set

This chapter introduces the basic notions of set theory.

Sets and elements

A **set** #\blue A# is a collection of distinct objects, considered as an object in its own right. An object can be anything from a number and a letter to a combination of these (or even a set). The distinct objects in a set are called the **elements** of that set.

Two sets #\blue A# and #\green B# are **equal** (denoted by #\blue A=\green B#) if and only if they have the same elements.

If a set #\blue A# has a finite number of elements, then we call it a **finite set** and denote this number by #|\blue{A}|#. Otherwise, we call it an infinite set and write #|\blue{A}| = \infty#. The number #|\blue{A}|# (possibly infinite) is called the **size** of #\blue{A}#.

We use curly brackets to describe sets by means of their elements. Here are four ways to describe a set with a moderate number of elements (referred to the abbreviation #\text{obj}# of object). \[\begin{array}{rcl}\text{set}&=&\{\text{obj}_1, \text{obj}_2,\text{obj}_3\}\\

\text{set}&=&\{\text{obj}_1, \text{obj}_2, \ldots,\text{obj}_n\}\\

\text{set}&=&\{\text{obj}_1, \text{obj}_2, \ldots\}\\

\text{set}&=&\{\ldots,\text{obj}_{-2}, \text{obj}_{-1},\text{obj}_0,\text{obj}_1, \text{obj}_2, \ldots\}\end{array}\]* *

The first and second case are examples of finite sets, with at most #3# and #n# elements, respectively. The three dots #{\ldots}# indicate that we continue in a like manner or with the same pattern.

**Examples**

The set #\blue A# of all vowels in the English alphabet is determined by #\blue A =\{\text{a}, \text{e}, \text{i}, \text{o}, \text{u}\}#.

The elements of # \{1, 2, 3, 4\}# are #1#, #2#, #3#, and #4#. A more complicated description of the same set is # \{1, 2, 2, 3, 4, 1\}#.

Thus, # \{1, 2, 3, 4\}= \{1, 2, 2, 3, 4, 1\}#.

The size of #\blue {B} = \{ \text{apple}, \pi, 4, 7.5, \text{q}

\} # is #|\blue{B}|=5#.

The size of # \{2,4,6,8,\ldots\}# is infinite.

The set #\{3,9,27,81,\ldots,729\}# has size #6#.

The sets #\{4, 3, 2\}# and #\{2, 4, 3\}# coincide.

The set of even integers is # \{\ldots,-4,-2,0,2,4,\ldots\}#

We will be working a lot with the following sets, which are so special that they are denoted by widely accepted symbols.

Some special sets

\[\begin{array}{rcl} \mathbb{N} &=& \text{the set of all natural numbers} = \{1, 2, 3, 4, \ldots\}\\ \mathbb{Z} & = & \text{the set of all integers} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\\ \mathbb{Q} &=& \text{the set of all rational numbers} = \text{the set of all elements of the form }\frac{a}{b}\text{ for integers } a,b \text { with } b\gt 0\\ \mathbb{R} & = & \text{the set of all real numbers} = \text{all numbers on the axis of numbers } (\text{neither } \infty \text{ nor } -\infty) \\ \emptyset &=&\text{the empty set} = \{\}\end{array}\]

We also have a special symbol for expressing membership of a set.

Membership symbol

The set membership symbol #\in# is used to say that an object is **an ****element of** a set. We use the symbol #\notin# to indicate that the object is **not an element of** a set.

Let #\blue A# be a set and #a# an object.

- We write #a \in \blue A# if #a# is an element of #\blue A#.
- We write #a \notin \blue A# if #a# is not an element of #\blue A#.

**Example**

Consider the set #\blue A = \{2, 3, 4\}#.

The statement that #3# is an element of #\blue A# can be expressed as #3\in\blue A#.

The statement that #5# is not an element of #\blue A# can be expressed as #5\not\in\blue A#.

The numbers #-3#, #0#, and #14# occur twice, so we can remove one copy of each. Only four elements remain; ordering these from lowest to highest, we find #\left\{-6, -3, 0, 14\right\}#.

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