Subsets

Sets: Sets

Subsets

We discuss the notion of subset. These are sets that are part of another set.

Subset

Let $\blue A$ and $\green B$ be sets.

We say set $\blue A$ is a subset of $\green B$ if every element of $\blue A$ is also an element of $\green B$ .

We denote this by $\blue A\subseteq \green B$ or by $\green B \supseteq \blue A$ .

Examples

Let $\blue A = \{1, 4, 6\}$ and $\green B = \{1, 2, 3, 4, 5, 6\}$ .

Since $1\in\green B$ , $4\in\green B$ , and $6\in\green B$ , we have $\blue A \subseteq \green B$ .

The set $\orange C$ is not a subset of $\green B$ if at least one element of $\orange C$ is not an element of $\green B$ .

We denote this by $\orange C \not \subseteq \green B$ .

In addition, let $\orange C = \{2.5, \pi, 5\}$ .

The element $2.5$ of $C$ does not belong to $\green B$ (and similarly for $\pi$ ). Therefore, $\orange C \not \subseteq \green B$ .

Terminology

If $\blue A$ is a subset of the set $\green B$ , then we also say that $\blue A$ is included in $\green B$ , that $\green B$ includes $\blue A$ , that $\blue A$ is contained in $\green B$ , or that $\green B$ contains $\blue A$ . The symbol $\subseteq$ is also called inclusion.

Venn diagram

Subsets can also be displayed in a Venn diagram as an oval within an oval.

Let $\blue A$ and $\green B$ be sets.

If $\blue A \subseteq \green B$ , then this inclusion is visualized by drawing the oval corresponding to $\blue A$ inside the oval corresponding to $\green B$ .

If $\orange C \not\subseteq \green B$ , then this non-inclusion is visualized by making sure the oval corresponding to $\orange C$ does not lie in the oval corresponding to $\green B$ . The elements in $\orange C$ but not in $\green B$ are drawn as points inside the oval of $\orange C$ and outside the oval of $\green B$ .

Example

As before, we take $\blue A = \{1, 4, 6\}$ , $\green B = \{1, 2, 3, 4, 5, 6\}$ , and $\orange C = \{2.5, \pi, 5\}$ . In the Venn diagram, we see that $\blue A \subseteq \green B$ because the oval of $\blue A$ lies in the oval of $\green B$ . Also, the points of $2.5$ and $\pi$ of $\orange C$ do not lie in $\blue A$ and are drawn outside the oval of $\blue A$ .

Proper subsets

Every set is a subset of itself because every element of a set $\blue A$ is of course an element of $\blue A$ itself.

If a set $\blue A$ is a subset of a set $\green B$ with $\blue A \neq \green B$ , then we call $\blue A$ a proper subset of $\green B$ . It is written as $\blue A \subset \green B$ . The symbol $\subset$ is referred to as a proper inclusion or a strict inclusion.

Example

Consider sets $\blue A = \{1, 4, 6\}$ and $\green B = \{1, 2, 3, 4, 5, 6\}$ . We see that $\blue A \subset \green B$ .

Equality

Recall that two sets $\blue A$ and $\green B$ are equal if and only if every element of $\blue A$ is an element of $\green B$ and conversely every element of $\green B$ is an element of $\blue A$ . Therefore, $\blue A=\green B$ if and only if $\blue A \subseteq \green B$ and $\green B\subseteq \blue A$ .

Example

Let $\blue A$ be the set $\{1, 2, 3\}$ and $\green B$ be the set $\{3, 2, 1\}$ . Here $\blue A$ is equal to $\green B$ , because $\blue A \subseteq \green B$ and $\green B \subseteq \blue A$ .

Examples

Among the special sets of numbers, we have the inclusions ${\mathbb N}\subseteq {\mathbb Z}$ , ${\mathbb Z}\subseteq {\mathbb Q}$ , and ${\mathbb Q}\subseteq {\mathbb R}$ .

Empty set

The empty set is a subset of every set.

In order to prove this statement, we let $\blue A$ be an arbitrary set. We will show that $\emptyset\subseteq A$ . We do this by establishing that every element of $\emptyset$ is also an element of $\blue A$ . In the language of logic this means that the statement

$a \in \emptyset \Rightarrow a \in \blue A$ is always true. But this is trivially satisfied since $a \in \emptyset$ is always false.

Transitivity

Containment of set is transitive in the sense that, for all sets $\blue A$ , $\green B$ and $\orange C$ , we have

$(\blue A \subseteq \green B\text{ and }\green B \subseteq \orange C)\ \Rightarrow\ \blue A \subseteq \orange C$

For example, if $\blue A = \{1, 2, 3\}$ , $\green B = \{1,2,3, 4\}$ , and $\orange C = \{1, 2, 3, 4, 6, 8\}$ ,
then $\blue A \subseteq \green B$ and $\green B \subseteq \orange C$ , so $\blue A \subseteq \orange C$ .

Consider the sets $A = \{ 3, 6, 20\}$ and $B = \{3, 4, 6, 10, 13, 20\}$ , is it true that $A \subseteq B$ ?

Yes

It is true that $A \subseteq B$ . Because $A = \{ 3, 6, 20\}$ is contained in $B = \{3, 4, 6, 10, 13, 20\}$ . So every element of $A$ is an element of $B$ .

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