M = {a,b,c}, N = {a}

\[M \subseteq M\] \[a \not\subseteq N\] \[a \in N\] \[P = \{\{a, b\}, c, d\}\] \[\{a, b\} \not\subseteq P\] \[\{\{a, b\}\} \subseteq P\] \[\{a, b\} \in P\] \[|\emptyset| = 0\] \[|\{\emptyset\}| = 1\] \[\emptyset \not= \{\emptyset\}\] \[\emptyset \not\subseteq \{\emptyset\}\] \[\emptyset \in \{\emptyset\}\] \[|\mathcal{P}(A)| = 2^{|A|}\]

S = {1}

\[\mathcal{P}(S) = \{\emptyset, \{1\}\}\] \[\mathcal{P}(S)\ \cap S = \emptyset\] \[\mathcal{P}(\mathcal{P}(S)) = \{\emptyset, \{1\}, \{\emptyset\}, \{\emptyset, \{1\}\}\}\]

Set difference

\[B\ \backslash\ A = \{x \in B\ |\ x \not\in A\}\]

Compliment

\[\bar A = \{x \in U\ |\ x \not\in A\}\]

Symmetric Difference

\[A \Delta B = \{x\ |\ (x \in A \vee x \in B)\ \wedge x \not\in (A \cap B)\}\]