Empty set and power sets

Question

The empty set is a member of $P({a,b}) \times P({p,q})$. True or false?

My first instinct was false, since the empty set is a member of each power set individually, but when multiplied together, you get {0,0}, which I'm not sure represents the empty set. But my counter argument is that the empty set is a member of the power set of anything, right?

Answer 1

Your first instinct was right: $P(a,b) \times P(p,q)$ contains $(\emptyset, \emptyset)$, but not $\emptyset$. $(\emptyset, \emptyset)$ is not the empty set, so $P(a,b) \times P(p,q)$ does not contain the empty set.

And yes, every powerset contains the empty set, but $P(a,b) \times P(p,q)$ is not a powerset, it's the cartesian product of two powersets.