The Power Set of a Set
Definition
Definition: If \(A\) is a set, the power set of \(A\), written \(\mathop{\mathcal{P}}(A)\), is the set whose elements are all subsets of \(A\). In set builder notation, \[ \mathop{\mathcal{P}}(A)=\{X: X\subseteq A\} \]
Example
\[A=\{0,1,3\}\].
\[\mathop{\mathcal{P}}(A)=\{\emptyset,\{0\},\{1\},\{3\},\{0,1\},\{0,3\},\{1,3\},\{0,1,3\}\}\]
Example
\[\mathop{\mathcal{P}}(\emptyset)=\{\emptyset\}\]
Notice that \(|\emptyset|=0\) and \(|\mathop{\mathcal{P}}(\emptyset)|=2^{0}=1\).
Example
\(\mathop{\mathcal{P}}(\{a\})=\{\emptyset,\{a\}\}\)
Example – some common mistakes
\(\mathop{\mathcal{P}}(1)\) makes no sense because \(1\) is not a set.
Example – some common mistakes 2
\(\mathop{\mathcal{P}}(\{1,\{1,2\}\}=\{\emptyset,\{1\},\{\{1,2\}\},\{1,\{1,2\}\}\}\). Notice that \(\{1,2\}\) is not an element of \(\mathop{\mathcal{P}}(\{1,\{1,2\}\})\) but \(\{\{1,2\}\}\) is.
Infinite case
THe power set \(\mathop{\mathcal{P}}(\mathbb{N})\) is very large and can be identified with infinite sequences of \(I\)’s and \(O\)’s.
The set \(\mathop{\mathcal{P}}(\mathbb{R}^2)\)
\(\mathop{\mathcal{P}}(\mathbb{R}^2)\) is huge and includes every graph of every function plus lots of other things, more than we can really comprehend.
Problem 1.4.15
What is \(\mathop{\mathcal{P}}(A\times B)\) if \(|A|=m\) and \(|A|=n\)?