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: $1.7$