Q:

What is Cartesian product for two empty sets.

If there are two sets $A$ and $B$ and both are null sets(or empty sets). What is $A\times B$ ? Is it also a null set? 

A:

$$\begin{array}{rcll}
\varnothing \times \varnothing
&=& \{(x,y) \ | \ x \in \varnothing \land y \in \varnothing\} & \text{defn. of Cartesian product} \\
&=& \{(x,y) \ | \ \bot \land \bot\} & \text{defn. of empty set} \\
&=& \{(x,y) \ | \ \bot\} & \text{defn. of conjunction} \\
&=& \varnothing & \text{defn. of empty set} \\
\end{array}$$

