Subset Equivalences

Definitions

In the following:

$S \subseteq T$ denotes that $S$ is a subset of $T$

$S \cup T$ denotes the union of $S$ and $T$

$S \cap T$ denotes the intersection of $S$ and $T$

$S \setminus T$ denotes the set difference between $S$ and $T$

$\O$ denotes the empty set

$\mathbb U$ denotes the universal set

$\complement$ denotes set complement.

Union with Superset is Superset‎

$S \subseteq T \iff S \cup T = T$

Intersection with Subset is Subset‎

$S \subseteq T \iff S \cap T = S$

Set Difference with Superset is Empty Set‎

$S \subseteq T \iff S \setminus T = \O$

Intersection with Complement is Empty iff Subset

$S \subseteq T \iff S \cap \map \complement T = \O$

Complement Union with Superset is Universe

$S \subseteq T \iff \map \complement S \cup T = \mathbb U$

Set Complement inverts Subsets

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

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection: Theorem $1$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Exercise $3.3 \ \text{(a)}$