Is the Cartesian product of ( A × B ) c a subset of A...

Suppose $A\subseteq P$ and $B\subseteq Q$. Then the Cartesian product $A\times B$ is a subset of the Cartesian product $P\times Q$.
We know that $(x,y)\in A\times B$ if both $x\in A$ and $y\in B$ are true, which fails if either $x\ne \in A$ or $y\ne \in B$ or both. Thus, $(A\times B{)}^c=\{(x,y)\in P\times Q:(x,y)\ne \in A\times B\}=(A^c\times Q)\cup (P\times B^c)$
However, the sets $(A^c\times Q)$ and $(P\times B^c)$ are not disjoint sets so that the pairs where both elements are not in the corresponding subsets are being double counted. This can be corrected as under:
$(A\times B{)}^c=(A^c\times B)\cup (A\times B^c)\cup (A^c\times B^c)$. Thus $(A\times B{)}^c$ is not a subset of $A^c\times B^c$ On the contrary, $A^c\times B^c$ is a subset of $(A\times B{)}^c$