Answer
$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\
P(A \cup B) \subseteq P(A) \cup P(B)\,\,\\
this\,\,is\,\,\,false\\
counter\,\,example\,\,:\\
A=\left \{ 7 \right \},B=\left \{ 8 \right \}\\
A\cup B=\left \{ 7,8 \right \}\\
P(A\cup B)=\left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \},\left \{ 7,8 \right \} \right \}\\
P(A)=\left \{ \varnothing ,\left \{ 7 \right \} \right \}\\
P(B)=\left \{ \varnothing ,\left \{ 8 \right \} \right \}\\
P(A)\cup P(B)=\left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \} \right \}\,\,\,\,\\
P(A\cup B)\nsubseteq P(A)\cup P(B) \\
as\,\left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \},\left \{ 7,8 \right \} \right \}\nsubseteq \left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \} \right \}$
Work Step by Step
$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\
P(A \cup B) \subseteq P(A) \cup P(B)\,\,\\
this\,\,is\,\,\,false\\
counter\,\,example\,\,:\\
A=\left \{ 7 \right \},B=\left \{ 8 \right \}\\
A\cup B=\left \{ 7,8 \right \}\\
P(A\cup B)=\left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \},\left \{ 7,8 \right \} \right \}\\
P(A)=\left \{ \varnothing ,\left \{ 7 \right \} \right \}\\
P(B)=\left \{ \varnothing ,\left \{ 8 \right \} \right \}\\
P(A)\cup P(B)=\left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \} \right \}\,\,\,\,\\
P(A\cup B)\nsubseteq P(A)\cup P(B) \\
as\,\left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \},\left \{ 7,8 \right \} \right \}\nsubseteq \left \{ \varnothing ,\left \{ 7 \right \},\left \{ 8 \right \} \right \}$