Chapter 6 - Set Theory - Exercise Set 6.3 - Page 372: 18

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

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