Ethen Blackwell

2022-07-25

Is the Cartesian product of $\left(A×B{\right)}^{c}$ a subset of ${A}^{c}×{B}^{c}$?

nezivande0u

Expert

Suppose $A\subseteq P$ and $B\subseteq Q$. Then the Cartesian product $A×B$ is a subset of the Cartesian product $P×Q$.
We know that $\left(x,y\right)\in A×B$ 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, $\left(A×B{\right)}^{c}=\left\{\left(x,y\right)\in P×Q:\left(x,y\right)\ne \in A×B\right\}=\left({A}^{c}×Q\right)\cup \left(P×{B}^{c}\right)$
However, the sets $\left({A}^{c}×Q\right)$ and $\left(P×{B}^{c}\right)$ 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:
$\left(A×B{\right)}^{c}=\left({A}^{c}×B\right)\cup \left(A×{B}^{c}\right)\cup \left({A}^{c}×{B}^{c}\right)$. Thus $\left(A×B{\right)}^{c}$ is not a subset of ${A}^{c}×{B}^{c}$ On the contrary, ${A}^{c}×{B}^{c}$ is a subset of $\left(A×B{\right)}^{c}$

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