Intersection with Empty Set
Theorem
The intersection of any set with the empty set is itself the empty set:
- $S \cap \O = \O$
Proof
| \(\ds S \cap \O\) | \(\subseteq\) | \(\ds \O\) | Intersection is Subset | |||||||||||
| \(\ds \O\) | \(\subseteq\) | \(\ds S \cap \O\) | Empty Set is Subset of All Sets | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds S \cap \O\) | \(=\) | \(\ds \O\) | Definition 2 of Set Equality |
$\blacksquare$
Also see
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 8 \ \text{(b)}$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $1$: Exercise $1 \ \text {(f)}$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B ii}$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 5 \ \text{(c)}$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.6$: Set Identities and Other Set Relations: Exercise $2 \ \text{(b)}$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 2$
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.11$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets: $\text {(iii)}$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): intersection: $\text {(i)}$
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(iii)}$