Relative Complement inverts Subsets

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Let $S$ be a set.

Let $A \subseteq S, B \subseteq S$ be subsets of $S$.


$A \subseteq B \iff \relcomp S B \subseteq \relcomp S A$

where $\complement_S$ denotes the complement relative to $S$.


\(\displaystyle A\) \(\subseteq\) \(\displaystyle B\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle A \cap B\) \(=\) \(\displaystyle A\) Intersection with Subset is Subset‎
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \relcomp S {A \cap B}\) \(=\) \(\displaystyle \relcomp S A\) Relative Complement of Relative Complement
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \relcomp S A \cup \relcomp S B\) \(=\) \(\displaystyle \relcomp S A\) De Morgan's Laws: Complement of Intersection
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \relcomp S B\) \(\subseteq\) \(\displaystyle \relcomp S A\) Union with Superset is Superset


Also known as

This result can be referred to by saying that the subset operation is inclusion-inverting.