Set Difference with Superset is Empty Set
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Theorem
- $S \subseteq T \iff S \setminus T = \O$
where:
- $S \subseteq T$ denotes that $S$ is a subset of $T$
- $S \setminus T$ denotes the set difference between $S$ and $T$
- $\O$ denotes the empty set.
Proof
\(\ds \) | \(\) | \(\ds S \setminus T = \O\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \neg \paren {\exists x: x \in S \land x \notin T}\) | Definition of Empty Set | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \forall x: \neg \paren {x \in S \land x \notin T}\) | De Morgan's Laws (Predicate Logic) | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \forall x: x \notin S \lor x \in T\) | De Morgan's Laws: Disjunction of Negations | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \forall x: x \in S \implies x \in T\) | Rule of Material Implication | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds S \subseteq T\) | Definition of Subset |
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 5$: Complements and Powers
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.6$. Difference and complement: Example $19$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 8 \ \text{(a)}$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.5$: Complementation
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.2$: Boolean Operations: Problem $\text{A}.2.1$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.11$