# 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 $3.3 \ \text{(a)}$