# Relative Complement inverts Subsets

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## Contents

## Theorem

Let $S$ be a set.

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

Then:

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

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

## Proof

\(\displaystyle A\) | \(\subseteq\) | \(\displaystyle B\) | |||||||||||

\(\displaystyle \leadstoandfrom \ \ \) | \(\displaystyle A \cap B\) | \(=\) | \(\displaystyle A\) | 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 |

$\blacksquare$

## Also known as

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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $3.3 \ \text{(e)}$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: $\text{(k)}$ - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.2$: Operations on Sets: Exercise $1.2.2 \ \text{(ii)}$