1.1.2: Subsets and Power Sets
We will often want to compare sets. And one obvious kind of comparison one might make is as follows: everything in one set is in the other too . This situation is sufficiently important for us to introduce some new notation.
If every element of a set \(A\) is also an element of \(B\) , then we say that \(A\) is a subset of \(B\) , and write \(A \subseteq B\) . If \(A\) is not a subset of \(B\) we write \(A \not\subseteq B\) . If \(A \subseteq B\) but \(A \neq B\) , we write \(A \subsetneq B\) and say that \(A\) is a proper subset of \(B\) .
Every set is a subset of itself, and \(\emptyset\) is a subset of every set. The set of even numbers is a subset of the set of natural numbers. Also, \(\{ a, b \} \subseteq \{ a, b, c \}\) . But \(\{ a, b, e \}\) is not a subset of \(\{ a, b, c \}\) .
The number \(2\) is an element of the set of integers, whereas the set of even numbers is a subset of the set of integers. However, a set may happen to both be an element and a subset of some other set, e.g., \(\{0\} \in \{0, \{0\}\}\) and also \(\{0\} \subseteq \{0, \{0\}\}\) .
Extensionality gives a criterion of identity for sets: \(A = B\) iff every element of \(A\) is also an element of \(B\) and vice versa. The definition of “subset” defines \(A \subseteq B\) precisely as the first half of this criterion: every element of \(A\) is also an element of \(B\) . Of course the definition also applies if we switch \(A\) and \(B\) : that is, \(B \subseteq A\) iff every element of \(B\) is also an element of \(A\) . And that, in turn, is exactly the “vice versa” part of extensionality. In other words, extensionality entails that sets are equal iff they are subsets of one another.
\(A = B\) iff both \(A \subseteq B\) and \(B \subseteq A\) .
Now is also a good opportunity to introduce some further bits of helpful notation. In defining when \(A\) is a subset of \(B\) we said that “every element of \(A\) is …,” and filled the “ \(\dots\) ” with “an element of \(B\) ”. But this is such a common shape of expression that it will be helpful to introduce some formal notation for it.
\((\forall x \in A)\phi\) abbreviates \(\forall x(x \in A \rightarrow \phi)\) . Similarly, \((\exists x \in A)\phi\) abbreviates \(\exists x(x \in A \land \phi)\) .
Using this notation, we can say that \(A \subseteq B\) iff \((\forall x \in A)x \in B\) .
Now we move on to considering a certain kind of set: the set of all subsets of a given set.
The set consisting of all subsets of a set \(A\) is called the power set of \(A\) , written \(\Pow{A}\) . \[\Pow{A} = \Setabs{B}{B \subseteq A}\nonumber\]
What are all the possible subsets of \(\{ a, b, c \}\) ? They are: \(\emptyset\) , \(\{a \}\) , \(\{b\}\) , \(\{c\}\) , \(\{a, b\}\) , \(\{a, c\}\) , \(\{b, c\}\) , \(\{a, b, c\}\) . The set of all these subsets is \(\Pow{\{a,b,c\}}\) : \[\Pow{\{ a, b, c \}} = \{\emptyset, \{a \}, \{b\}, \{c\}, \{a, b\}, \{b, c\}, \{a, c\}, \{a, b, c\}\}\nonumber\]
List all subsets of \(\{a, b, c, d\}\) .
Show that if \(A\) has \(n\) elements, then \(\Pow{A}\) has \(2^n\) elements.