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Section 6.1 Subsets
Large sets are often easier to work with when we break them into smaller sets. We introduce relations for stating when the elements of one set are contained in another set. The subset relation lets us compare sets in ways other than equality.
The video in
Figure 6.1.1 introduces subsets. A more detailed discussion follows.
Figure 6.1.1. Subsets by Matt Farmer and Stephen Steward
Definition 6.1.2 .
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
\(B\text{.}\) ”
We give examples that use the relations
\(\subseteq\) and
\(\not\subseteq\text{.}\)
Example 6.1.3 . 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 examples that use the relations
\(\in\text{,}\) \(\notin\text{,}\) \(\subseteq\text{,}\) and
\(\not\subseteq\text{.}\)
Example 6.1.4 . 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.
Theorem 6.1.5 .
For all sets
\(A\) we have:
\(\{\} \subseteq A\text{.}\)
We give examples of subset relations involving the empty set.
Example 6.1.6 . 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:
Theorem 6.1.7 .
For all sets
\(A\) we have:
\(A\subseteq A\text{.}\)
Also, if two sets are both subsets of each other, they contain the same elements and hence are equal.
Theorem 6.1.8 .
For all sets
\(A\) and
\(B\) we have: If
\(A\subseteq B\) and
\(B\subseteq A\) then
\(A=B\text{.}\)
