# 13.1: Basics of Sets

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An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.

## Set

A set is a collection of distinct objects, called elements of the set

A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.

## Example 1

Some examples of sets defined by describing the contents:

1. The set of all even numbers
2. The set of all books written about travel to Chile

Some examples of sets defined by listing the elements of the set:

1. {1, 3, 9, 12}
2. {red, orange, yellow, green, blue, indigo, purple}

A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.

## Notation

Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.

The symbol $$\in$$ means “is an element of”.

A set that contains no elements, $$\{ \}$$, is called the empty set and is notated $$\emptyset$$

## Example 2

Let $$A=\{1,2,3,4\}$$

To notate that 2 is element of the set, we'd write $$2 \in A$$

Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.

## Subset

A subset of a set $$A$$ is another set that contains only elements from the set $$A$$, but may not contain all the elements of $$A$$.

If $$B$$ is a subset of $$A,$$ we write $$B \subseteq A$$

A proper subset is a subset that is not identical to the original set – it contains fewer elements.

If $$B$$ is a proper subset of $$A$$, we write $$B \subset A$$

## Example 3

Consider these three sets

$$A=$$ the set of all even numbers$$\quad B=\{2,4,6\} \quad C=\{2,3,4,6\}$$

Here $$B \subset A$$ since every element of $$B$$ is also an even number, so is an element of $$A$$.

More formally, we could say $$B \subset A$$ since if $$x \in B,$$ then $$x \in A$$

It is also true that $$B \subset C$$.

$$C$$ is not a subset of $$A$$, since $$C$$ contains an element, 3 , that is not contained in $$A$$

## Example 4

Suppose a set contains the plays “Much Ado About Nothing”, “MacBeth”, and “A Midsummer’s Night Dream”. What is a larger set this might be a subset of?

###### Solution

There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.

##### Try it Now 1

The set $$A=\{1,3,5\} .$$ What is a larger set this might be a subset of?

There are several answers: The set of all odd numbers less than 10. The set of all odd numbers. The set of all integers. The set of all real numbers.

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