# Intersection with Complement is Empty iff Subset

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## Theorem

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

where:

- $S \subseteq T$ denotes that $S$ is a subset of $T$
- $S \cap T$ denotes the intersection of $S$ and $T$
- $\O$ denotes the empty set
- $\complement$ denotes set complement.

### Corollary

- $S \cap T = \O \iff S \subseteq \relcomp {} T$

## Proof

\(\ds S\) | \(\subseteq\) | \(\ds T\) | ||||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds S \setminus T\) | \(=\) | \(\ds \O\) | Set Difference with Superset is Empty Set | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds S \cap \map \complement T\) | \(=\) | \(\ds \O\) | Set Difference as Intersection with Complement |

$\blacksquare$

## Also presented as

Some sources present this as an alternative definition of a subset:

*Set $A$ is a subset of set $B$ if set $A$ contains no member that is not also in set $B$*

but this is not mainstream.

## Also see

## 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 $3.3 \ \text{(b)}$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B vi}$ - 1971: Patrick J. Murphy and Albert F. Kempf:
*The New Mathematics Made Simple*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Subsets: Definition: $1.7$