## Section6.1Subsets

It is often helpful to break down large sets into smaller, more manageable sets. We introduce relations that allow us to formulate statements about the containment of the elements of one set in another set. The subset relation allows us to compare sets beyond only equality.

We give an introduction to subsets in the video in Figure 6.1.1. It is followed by a more detailed discussion.

### Definition6.1.2.

A set $$A$$ is a subset of a set $$B$$ if each element in $$A$$ is also an element in $$B\text{.}$$ If $$A$$ is a subset of $$B\text{,}$$ we write $$A\subseteq B\text{.}$$ If there is at least one element in $$A$$ that is not an element in $$B\text{,}$$ then $$A$$ is not a subset of $$B\text{.}$$ If $$A$$ is not a subset of $$B\text{,}$$ we write $$A\not\subseteq B\text{.}$$

We read $$A\subseteq B$$ as “$$A$$ is a subset of $$B$$” and $$A\not\subseteq B$$ as “$$A$$ is not a subset of $$B\text{.}$$

In Checkpoint 6.1.3 reproduce Definition 6.1.2 by filling in the blanks.

Complete the following:

Let $$A$$ and $$B$$ be sets.

We say the set $$A$$ is

• select

• a superset

• a subset

• an under set

• not a superset

• not a subset

• not an under set

of the set $$B$$ and write $$A \subseteq B$$ if
• select

• each element

• there is an element

• no element

of $$A$$
• select

• is related to

• is an element of

• is equal to

• is not related to

• is not an element of

• is not equal to

the set $$B\text{.}$$

We say the set $$A$$ is

• select

• a superset

• a subset

• an under set

• not a superset

• not a subset

• not an under set

of the set $$B$$ and write $$A\not\subseteq B$$ if
• select

• each element

• no element

• there is an element

• there are two elements

of $$A$$ that
• select

• is related to

• is an element of

• is equal to

• is not related to

• is not an element of

• is not equal to

the set $$B\text{.}$$

$$\text{a subset}$$

$$\text{each element}$$

$$\text{is an element of}$$

$$\text{not a subset}$$

$$\text{there is an element}$$

$$\text{is not an element of}$$

We give some examples for the use of the relations $$\subseteq$$ and $$\not\subseteq\text{.}$$

1. $$\{1,2\}\subseteq \{1,2,4,9\}\text{,}$$ because $$1\in\{1,2,4,9\}$$ and $$2\in\{1,2,4,9\}\text{.}$$

2. $$\{1,2\}\subseteq \mathbb{N}\text{,}$$ because $$1\in\mathbb{N}$$ and $$2\in\mathbb{N}\text{.}$$

3. $$\{1,2\}\not\subseteq \{1,3,4,9\}\text{,}$$ because $$2\not\in \{1,3,4,9\}\text{.}$$

4. $$\{2\}\subseteq \{2,3\}\text{,}$$ because $$2\in \{2,3\}\text{.}$$

5. $$\{2,3\}\subseteq\{2,3\}\text{,}$$ because $$2\in \{2,3\}$$ and $$3\in \{2,3\}\text{.}$$

6. $$\{\{2\},7\}\subseteq\{\{2\},\{2,3\},5,6,7\}\text{,}$$ because $$\{2\}\in \{\{2\},\{2,3\},5,6,7\}$$ and $$7\in \{\{2\},\{2,3\},5,6,7\}\text{.}$$

The relations $$\in$$ and $$\subseteq$$ may seem similar, but we have to consider that $$\subseteq$$ compares two sets while $$\in$$ is used to express that an element is in a set. So we cannot write $$3 \subseteq \{1,2,3\}$$ or $$3 \not\subseteq \{1,2,3\}$$ because $$3$$ is not a set.

We give some examples for the use of the relations $$\in\text{,}$$ $$\notin\text{,}$$ $$\subseteq\text{,}$$ and $$\not\subseteq\text{.}$$

1. $$3\in\{1,2,3\}\text{,}$$ as the number 3 is in the set containing the numbers 1, 2, and 3.

2. $$\{3\}\subseteq\{1,2,3\}\text{,}$$ as each element, namely the number 3, of the set $$\{3\}$$ is in the set containing the numbers 1, 2, and 3.

3. $$\{3\}\not\in\{1,2,3\}\text{,}$$ as the set $$\{3\}$$ is not in the set containing the numbers 1, 2, and 3.

4. $$\{3\}\not\subseteq\{\{1\},\{2\},\{3\}\}\text{,}$$ as the number 3 is not element of the set containing the sets $$\{1\}\text{,}$$ $$\{2\}\text{,}$$ and $$\{3\}\text{.}$$

5. $$\{1,2\}\subseteq\{1,2,3,4\}\text{,}$$ as the numbers 1 and 2 are in the set containing the numbers 1 and 2 and 3 and 4.

The empty set $$\{\}$$ does not contain any elements. So when checking whether the empty set is a subset of another set, we do not have any elements to check. So it is true that each element in $$\{\}$$ is also an element of any other set. This means that the empty set is a subset of every set.

We give examples of subset relations involving the empty set.

1. $$\displaystyle \{\} \subseteq \{2,3\}$$

2. $$\displaystyle \{\} \subseteq \{\}$$

For any set $$A$$ each element in $$A$$ is also an element of $$A\text{,}$$ thus:

Furthermore, if two sets are both subsets of each other, they contain the same elements and hence are equal.

In Checkpoint 6.1.10 decide whether sets are subsets of other sets, and if not, select a reason why.

For the given sets $$C$$ and $$D$$ determine whether the statement

\begin{equation*} C\subseteq D \end{equation*}

is true or false. If the statement is false choose the reason.

1. when $$C:=\lbrace 11\rbrace$$ and $$D: = \lbrace \rbrace$$

2. when $$C:=\lbrace 11\rbrace$$ and $$D: = \lbrace 10,11\rbrace$$

3. when $$C:=\lbrace 6, 7,8\rbrace$$ and $$D: = \lbrace 8, 7, 6 \rbrace$$

4. when $$C:=\lbrace 10, 7,6,8\rbrace$$ and $$D: = \lbrace 8, 7, 6 \rbrace$$