Subset Equivalences
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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 $3.3 \ \text{(a)}$