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 $T$
• $\varnothing$ 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

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

Intersection of Complement with Subset is Empty

$S \subseteq T \iff S \cap \complement \left({T}\right) = \varnothing$

Complement Union with Superset is Universe

$S \subseteq T \iff \complement \left({S}\right) \cup T = \mathbb U$

Complements Invert Subsets

$S \subseteq T \iff \complement \left({T}\right) \subseteq \complement \left({S}\right)$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $3.3 \ \text{(a)}$