◀ ▲ ▶Branches / Set-theory / Definition: Set Intersection

Definition: Set Intersection

Let $A$ and $B$ be sets. Based on the conjunction operation "\(\wedge\)", the set intersection of \(A\) and \(B\) is defined as \[A\cap B:=\{x | x\in A\wedge x\in B\}.\] The intersection contains all elements \(x\), which are contained in both, \(A\) and \(B\). It can be visualized by a Venn diagram like this:

Examples

1. Let $A=\{1,2,3,4\}$ and let $B=\{3,4,5,6\}.$ Then the intersection set is $A\cap B=\{3,4\}$. Please note that we do not have to list the repeating elements twice in the intersection set.
2. The intersection of the set of countries of Europe and the set of the countries of Asia is the set of all countries belonging to both, Asia and Europe, i.e. the set $\{\text{Azerbaijan}, \text{Georgia}, \text{Kazakhstan}, \text{Russia},\text{Turkey}\}$¹.

Table of Contents

1. Definition: Disjoint Sets
2. Proposition: Intersection of a Set With Another Set is Subset of This Set
3. Proposition: Set Intersection is Commutative
4. Proposition: Set Intersection is Associative

Mentioned in:

Axioms: 1
Corollaries: 2
Definitions: 3 4 5 6 7
Explanations: 8
Lemmas: 9
Parts: 10
Proofs: 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Propositions: 30 31 32 33 34 35 36 37 38 39

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

1. Reinhardt F., Soeder H.: "dtv-Atlas zur Mathematik", Deutsche Taschenbuch Verlag, 1994, 10th Edition
2. Kohar, Richard: "Basic Discrete Mathematics, Logic, Set Theory & Probability", World Scientific, 2016

Footnotes