Answer
$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\
if\,\,A \cap C = B \cap C\,\,and\,A \cup C = B \cup C\,\,\\
then\,\,A = B.\\
this\,\,is\,\,true:\\
proof:\\
to\,\,prove\,\,this\,we\,must\,show\,that\,:\\
A\subseteq B\,\,\,and\,\,B\subseteq A \\
proof\,\,(A\subseteq B)\\
x\in A \Rightarrow x\in C \,or\, x\notin C \\
first\,\,case\,\,(x\in A\,\,and\,\,x\in C)\\
x\in A\,\,and\,\,x\in C\Rightarrow x\in A\cap C \\
\because A \cap C = B \cap C \\
\therefore x\in A\cap C\Rightarrow x\in B \cap C\Rightarrow x\in B \\
second\,\,case\,\,:(x\in A \,\,and\,\,x\notin C)\\
x\in A \,\,and\,\,x\notin C \Rightarrow x\in A\cup C \\
\because A \cup C = B \cup C \\
\therefore x\in B\cup C\Rightarrow x\in B \,\,(as\,\,x\notin C) \\
so\,\,in\,\,both\,\,cases\,\,:\\
x\in A \Rightarrow x\in B \\
\therefore A\subseteq B
$
$proof\,\,(B\subseteq A)\\
x\in B \Rightarrow x\in C \,or\, x\notin C \\
first\,\,case\,\,(x\in B\,\,and\,\,x\in C)\\
x\in B\,\,and\,\,x\in C\Rightarrow x\in B\cap C \\
\because A \cap C = B \cap C \\
\therefore x\in B\cap C\Rightarrow x\in A \cap C\Rightarrow x\in A \\
second\,\,case\,\,:(x\in B \,\,and\,\,x\notin C)\\
x\in B \,\,and\,\,x\notin C \Rightarrow x\in B\cup C \\
\because A \cup C = B \cup C \\
\therefore x\in A\cup C\Rightarrow x\in A \,\,(as\,\,x\notin C) \\
so\,\,in\,\,both\,\,cases\,\,:\\
x\in B \Rightarrow x\in A \\
\therefore B\subseteq A \\
\because B\subseteq A\,\,and\,\,A\subseteq B \\
\therefore A=B
$
Work Step by Step
$For\,\,all\,\,sets\,\,A,\,B,\,\,and\,C\,\,\\
if\,\,A \cap C = B \cap C\,\,and\,A \cup C = B \cup C\,\,\\
then\,\,A = B.\\
this\,\,is\,\,true:\\
proof:\\
to\,\,prove\,\,this\,we\,must\,show\,that\,:\\
A\subseteq B\,\,\,and\,\,B\subseteq A \\
proof\,\,(A\subseteq B)\\
x\in A \Rightarrow x\in C \,or\, x\notin C \\
first\,\,case\,\,(x\in A\,\,and\,\,x\in C)\\
x\in A\,\,and\,\,x\in C\Rightarrow x\in A\cap C \\
\because A \cap C = B \cap C \\
\therefore x\in A\cap C\Rightarrow x\in B \cap C\Rightarrow x\in B \\
second\,\,case\,\,:(x\in A \,\,and\,\,x\notin C)\\
x\in A \,\,and\,\,x\notin C \Rightarrow x\in A\cup C \\
\because A \cup C = B \cup C \\
\therefore x\in B\cup C\Rightarrow x\in B \,\,(as\,\,x\notin C) \\
so\,\,in\,\,both\,\,cases\,\,:\\
x\in A \Rightarrow x\in B \\
\therefore A\subseteq B
$
$proof\,\,(B\subseteq A)\\
x\in B \Rightarrow x\in C \,or\, x\notin C \\
first\,\,case\,\,(x\in B\,\,and\,\,x\in C)\\
x\in B\,\,and\,\,x\in C\Rightarrow x\in B\cap C \\
\because A \cap C = B \cap C \\
\therefore x\in B\cap C\Rightarrow x\in A \cap C\Rightarrow x\in A \\
second\,\,case\,\,:(x\in B \,\,and\,\,x\notin C)\\
x\in B \,\,and\,\,x\notin C \Rightarrow x\in B\cup C \\
\because A \cup C = B \cup C \\
\therefore x\in A\cup C\Rightarrow x\in A \,\,(as\,\,x\notin C) \\
so\,\,in\,\,both\,\,cases\,\,:\\
x\in B \Rightarrow x\in A \\
\therefore B\subseteq A \\
\because B\subseteq A\,\,and\,\,A\subseteq B \\
\therefore A=B
$