Why is the Cartesian product of a set $A$ and empty set an empty set? [duplicate]
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 $?
Solution 1:
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 $A\times B\neq\emptyset$.