Set Complement inverts Subsets/Proof 2

Theorem

$S \subseteq T \iff \map \complement T \subseteq \map \complement S$

Proof

\(\ds S\)

\(\subseteq\)

\(\ds T\)

\(\ds \leadstoandfrom \ \ \)

\(\ds (x \in S\)

\(\implies\)

\(\ds x \in T)\)

Definition of Subset

\(\ds \leadstoandfrom \ \ \)

\(\ds (x \notin T\)

\(\implies\)

\(\ds x \notin S)\)

Rule of Transposition

\(\ds \leadstoandfrom \ \ \)

\(\ds (x \in \map \complement T\)

\(\implies\)

\(\ds x \in \map \complement S)\)

Definition of Set Complement

\(\ds \map \complement T\)

\(\subseteq\)

\(\ds \map \complement S\)

Definition of Subset

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6.7$: Subsets