Answer
\[(A\cap B)\], \[({{A}^{C}}\cap B),\text{ }(A\cap {{B}^{C}}),\text{ and }(A{}^{C}\cup B{}^{C}){}^{C}\] is a subset of \[(A\cup B)\].
\[(A\cap B)\] and \[(A{}^{C}\cup B{}^{C}){}^{C}\]are subsets of A and also B.
\[(A\cup B{}^{C})\] is a subset of A.
\[(A{}^{C}\cup B)\]is a subset of B.
Work Step by Step
From a Venn Diagram:
\[\begin{align}
& A\subset (A\cup B) \\
& B\subset (A\cup B) \\
\end{align}\]
\[\begin{align}
& (A\cap B)\subset A \\
& (A\cap B)\subset B \\
& (A\cap B)\subset (A\cup B) \\
\end{align}\]
\[\begin{align}
& ({{A}^{C}}\cap B)\subset (A\cup B) \\
& ({{A}^{C}}\cap B)\subset B \\
\end{align}\]
\[\begin{align}
& (A\cap {{B}^{C}})\subset (A\cup B) \\
& (A\cap {{B}^{C}})\subset A \\
\end{align}\]
\[\begin{align}
& {{({{A}^{C}}\cap {{B}^{C}})}^{C}}\subset (A\cup B) \\
& {{({{A}^{C}}\cap {{B}^{C}})}^{C}}\subset A \\
& {{({{A}^{C}}\cap {{B}^{C}})}^{C}}\subset B \\
\end{align}\]
Therefore,
A, B, \[(A\cap B)\], \[({{A}^{C}}\cap B),\text{ }(A\cap {{B}^{C}}),\text{ and }(A{}^{C}\cup B{}^{C}){}^{C}\] is a subset of \[(A\cup B)\].
\[(A\cap B)\] and \[(A{}^{C}\cup B{}^{C}){}^{C}\]are subsets of A and also B.
\[(A\cup B{}^{C})\] is a subset of A.
\[(A{}^{C}\cup B)\]is a subset of B.
