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Definition: Set Union

Let $A$ and $B$ be sets. Based on the disjunction operation “\(\vee\)”, the set union of \(A\) and \(B\) is defined as \[A\cup B:=\{x | x\in A \vee x\in B\}.\]

The union is the set containing all elements \(x\) belonging either to \(A\) or to \(B\) (including those belonging to both). It can be visualized as the following Venn diagram:

Examples

  1. Let $A=\{1,2,3,4\}$ and let $B=\{3,4,5,6\}.$ Then the set union is $A\cup B=\{1,2,3,4,5,6\}$. Please note that we do not have to list the repeating elements twice in the union set.
  2. The union of the set of countries of Europe and the set of the countries of Asia equals the set of the countries belonging to Eurasia.

| | | | | created: 2017-08-12 20:59:49 | modified: 2019-07-28 19:22:53 | by: bookofproofs | references: [979], [7838]

1.Proposition: Sets are Subsets of Their Union

2.Proposition: Set Union is Commutative

3.Proposition: Set Union is Associative

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Bibliography (further reading)

[7838] Kohar, Richard: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016

[979] Reinhardt F., Soeder H.: “dtv-Atlas zur Mathematik”, Deutsche Taschenbuch Verlag, 1994, 10