Answer
$for\,\,all\,\,sets\,\,A,B\,,C\,and\,\,D,if\,A\cap C=\varnothing \\
then\,(A\times B)\cap (C\times D)=\varnothing \\
To\,\,prove\,\,that\,\,a\,\,set\,\,(A\times B)\cap (C\times D)\,\,is\,\,equal\,\,to\,\,the\,\,empty\,\,set\,\, \varnothing ,\\ prove\,\,that\,\,(A\times B)\cap (C\times D)\,\,has\,\,no\,\,elements. \\ To\,\,
do\,\,this, suppose\,\,\\(A\times B)\cap (C\times D) \,\,has\,\,an\,element\,\,and\,\,derive\,\,a\,\,contradiction \\
(x,y)\in (A\times B)\cap (C\times D)\\
(x,y)\in (A\times B)\,and\,(x,y)\in(C\times D)\\
by\,\,def.\,of\,cartesian\,product\,\\
x\in A\,\,and\,x\in C \\
by\,\,def.\,\,of\,\,inter\! section\\
x\in A\cap C \Rightarrow A\cap C\neq \varnothing \\
but A\cap C=\varnothing \\
(this\,\,is\,\,a\,\,contradiction)\\
\therefore (A\times B)\cap (C\times D)=\varnothing
$
Work Step by Step
$for\,\,all\,\,sets\,\,A,B\,,C\,and\,\,D,if\,A\cap C=\varnothing \\
then\,(A\times B)\cap (C\times D)=\varnothing \\
To\,\,prove\,\,that\,\,a\,\,set\,\,(A\times B)\cap (C\times D)\,\,is\,\,equal\,\,to\,\,the\,\,empty\,\,set\,\, \varnothing ,\\ prove\,\,that\,\,(A\times B)\cap (C\times D)\,\,has\,\,no\,\,elements. \\ To\,\,
do\,\,this, suppose\,\,\\(A\times B)\cap (C\times D) \,\,has\,\,an\,element\,\,and\,\,derive\,\,a\,\,contradiction \\
(x,y)\in (A\times B)\cap (C\times D)\\
(x,y)\in (A\times B)\,and\,(x,y)\in(C\times D)\\
by\,\,def.\,of\,cartesian\,product\,\\
x\in A\,\,and\,x\in C \\
by\,\,def.\,\,of\,\,inter\! section\\
x\in A\cap C \Rightarrow A\cap C\neq \varnothing \\
but A\cap C=\varnothing \\
(this\,\,is\,\,a\,\,contradiction)\\
\therefore (A\times B)\cap (C\times D)=\varnothing
$
