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Let $A \times \emptyset = \{(x,y)| x\in A, y \in \emptyset \}$. We know there is no element in $\emptyset$. But how does it follow that $A \times \emptyset = \emptyset $?


marked as duplicate by Rahul, Micah, Asaf Karagila, Henry T. Horton, Alexander Gruber Mar 5 '13 at 1:39

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Claim: $A\times B=\emptyset$ iff $A=\emptyset$ or $B=\emptyset$

Proof: If $A=\emptyset$ or $B=\emptyset$, then there is no $(a,b)$ such that $a\in A$ and $b\in B$. Therefore $A\times B$, which is the set of these pairs, is empty.

If $A\neq\emptyset$ and $B\neq\emptyset$, there exists $a\in A$ and $b\in B$, thus $(a,b)\in A\times B$. Therefore $A\times B\neq\emptyset$.


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