TITLE: Complement Set of Cartesian product
QUESTION [6 upvotes]: I have a hard time knowing what it means for a thing to be a complement of a cartesian coordinate.
So let's say I have the arbitrary non empty sets $A$ and $B$:
$$ (A \times B)^c$$ 
does this equal the individual parts:
$$(A \times B)^c = A^c \times B^c?$$
If so then why?

REPLY [15 votes]: Assume $A{\times}B\subseteq X{\times}Y$ and so using set builder notation: $$A{\times}B = \{(x,y)\in X{\times}Y\mid x\in A~\wedge~ y\in B\}$$
Due to de Morgan's Laws of Dual Negation, the complement (relative to $X{\times}Y$) would therefore be : $$(A{\times}B)^{\complement_{X{\times}Y}} = \{(x,y)\in X{\times}Y\mid x\notin A~\vee~ y\notin B\}$$
Thus we have $$(A{\times}B)^{\complement_{X{\times}Y}} = (A^{\complement_X}{\times}Y) \cup (X{\times}B^{\complement_Y}) \cup (A^{\complement_X}{\times}Y^{\complement_Y})$$