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. It is followed by a more detailed discussion.
Figure6.1.Subsets by Matt Farmer and Stephen Steward
Definition6.2.
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 \(B\text{.}\)”
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 \(B\text{.}\)
We give some examples for the use of the relations \(\subseteq\) and \(\not\subseteq\text{.}\)
Example6.4.Subsets.
\(\{1,2\}\subseteq \{1,2,4,9\}\text{,}\) because \(1\in\{1,2,4,9\}\) and \(2\in\{1,2,4,9\}\text{.}\)
\(\{1,2\}\subseteq \mathbb{N}\text{,}\) because \(1\in\mathbb{N}\) and \(2\in\mathbb{N}\text{.}\)
\(\{1,2\}\not\subseteq \{1,3,4,9\}\text{,}\) because \(2\not\in \{1,3,4,9\}\text{.}\)
\(\{2\}\subseteq \{2,3\}\text{,}\) because \(2\in \{2,3\}\text{.}\)
\(\{2,3\}\subseteq\{2,3\}\text{,}\) because \(2\in \{2,3\}\) and \(3\in \{2,3\}\text{.}\)
\(\{\{2\},7\}\subseteq\{\{2\},\{2,3\},5,6,7\}\text{,}\) 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 \(\not\subseteq\text{.}\)
Example6.5.Usage of \(\in\) and \(\subseteq\).
\(3\in\{1,2,3\}\text{,}\) as the number 3 is in the set containing the numbers 1, 2, and 3.
\(\{3\}\subseteq\{1,2,3\}\text{,}\) as each element, namely the number 3, of the set \(\{3\}\) is in the set containing the numbers 1, 2, and 3.
\(\{3\}\not\in\{1,2,3\}\text{,}\) as the set \(\{3\}\) is not in the set containing the numbers 1, 2, and 3.
\(\{3\}\not\subseteq\{\{1\},\{2\},\{3\}\}\text{,}\) as the number 3 is not element of the set containing the sets \(\{1\}\text{,}\)\(\{2\}\text{,}\) and \(\{3\}\text{.}\)
\(\{1,2\}\subseteq\{1,2,3,4\}\text{,}\) 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.
Theorem6.6.
For all sets \(A\) we have: \(\{\} \subseteq A\text{.}\)
We give examples of subset relations involving the empty set.
Example6.7.The empty set as a subset.
\(\displaystyle \{\} \subseteq \{2,3\}\)
\(\displaystyle \{\} \subseteq \{\}\)
For any set \(A\) each element in \(A\) is also an element of \(A\text{,}\) thus:
Theorem6.8.
For all sets \(A\) we have: \(A\subseteq A\text{.}\)
Furthermore, if two sets are both subsets of each other, they contain the same elements and hence are equal.
Theorem6.9.
For all sets \(A\) and \(B\) we have: If \(A\subseteq B\) and \(B\subseteq A\) then \(A=B\text{.}\)
In Checkpoint 6.10 decide whether sets are subsets of other sets, and if not, select a reason why.
Checkpoint6.10.Is this a subset ? If no, 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.
when \(C:=\lbrace 11\rbrace\) and \(D: = \lbrace \rbrace\)
when \(C:=\lbrace 11\rbrace\) and \(D: = \lbrace 10,11\rbrace\)
when \(C:=\lbrace 6, 7,8\rbrace\) and \(D: = \lbrace 8, 7, 6 \rbrace\)
when \(C:=\lbrace 10, 7,6,8\rbrace\) and \(D: = \lbrace 8, 7, 6 \rbrace\)
