Chapter 6 - Set Theory - Exercise Set 6.3 - Page 372: 21

$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\ P(A \times B) = P(A) \times P(B)\,\,\\ this\,\,is\,\,\,false\\ counter\,\,example\,\,:\\ A=\left \{ 7 \right \},B=\left \{ 8 \right \}\\ A\times B=\left \{ \left ( 7,8 \right ) \right \}\\ p(A\times B)=\left \{ \varnothing ,\left \{ \left ( 7,8 \right ) \right \} \right \}\\ P(A)=\left \{ \varnothing ,\left \{ 7 \right \} \right \}\\ P(B)=\left \{ \varnothing ,\left \{ 8 \right \} \right \}\\ P(A)\times p(B)=\left \{ (\varnothing ,\varnothing ),(\varnothing ,\left \{8\right \}),(\left \{ 7 \right \},\varnothing ),(\left \{ 7 \right \},\left \{ 8 \right \}) \right \}\\ so\,\,P(A \times B) \neq P(A) \times P(B)\,\,\\ $

$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\ P(A \times B) = P(A) \times P(B)\,\,\\ this\,\,is\,\,\,false\\ counter\,\,example\,\,:\\ A=\left \{ 7 \right \},B=\left \{ 8 \right \}\\ A\times B=\left \{ \left ( 7,8 \right ) \right \}\\ p(A\times B)=\left \{ \varnothing ,\left \{ \left ( 7,8 \right ) \right \} \right \}\\ P(A)=\left \{ \varnothing ,\left \{ 7 \right \} \right \}\\ P(B)=\left \{ \varnothing ,\left \{ 8 \right \} \right \}\\ P(A)\times p(B)=\left \{ (\varnothing ,\varnothing ),(\varnothing ,\left \{8\right \}),(\left \{ 7 \right \},\varnothing ),(\left \{ 7 \right \},\left \{ 8 \right \}) \right \}\\ so\,\,P(A \times B) \neq P(A) \times P(B)\,\,\\ $