I was in class the other day when I got a problem presented to me to that I had to figure out.
The problem was "If A X B is a subset of C X D, then A is a subset of C and B is a subset of D. Prove this is true or give a counterexample to show it is false."
If $A\times B \subseteq C \times D$, then $A\subseteq C$ and $B\subseteq D$. Prove this is true or give a counterexample to show it is false.
I'm almost positive that this is false but I'm not sure how to go about disproving it.
If anyone could help me out at all I'd appreciate it.
Thanks a lot.
Best Answer
Well this is true if A and B are not empty sets and false in general. If $A=\emptyset$, $B=\{1,2\}$, $C=\{1\}$, $D=\{2\}$ then $B\not\subseteq D$ but $A\times B=\emptyset\subseteq C\times D$.
If $A$ and $B$ are not empty the proof is simple enough.
If $A\times B \subseteq C\times D$ then $\forall a\in A, b\in B, (a,b)\in C\times D$ and so $\forall a\in A, a\in C$ and $\forall b\in B, b\in D$ so $A\subseteq C$ and $B\subseteq C$. This is a rather informal sketch but you should figure out where we need the assumption that A and B are not empty sets.