Operations and families · Definition

Union of families

set theoryoperations and families

Before you start you need to know

It extends the union of two sets to any collection of sets, indexed or not.

Definition

The union of a family \(\{A_i\}_{i \in I}\) is the set of objects that belong to at least one of the \(A_i\).

\[\bigcup_{i \in I} A_i = \{x \mid \exists\, i \in I,\ x \in A_i\}\]

If \(I = \{1,2\}\), the union is \(A_1 \cup A_2\): the general definition extends the union of two sets to any index set, even an infinite one. If \(I = \varnothing\), the union is empty, because there is no \(A_i\) to belong to.

Worked example: if \(A_1 = \{1,2\}\), \(A_2 = \{2,3\}\) and \(A_3 = \{3,4\}\), then \(\bigcup_{i=1}^{3} A_i = \{1,2,3,4\}\). Each element of the union belongs to at least one of the \(A_i\), and it need not belong to all of them.

Typical mistakes: requiring membership in every set, which describes the intersection \(\bigcap_i A_i\) instead, or forgetting the empty-index case. Another frequent confusion is taking the union of a family to be the family itself: \(\bigcup \mathcal{F}\) is a set of elements, not a set of sets.

Think about it

To think about: why is the union over the empty index set the empty set?

Because no object can belong to some \(A_i\) when there are no indices, so the union has no elements.

Exercises 10

  1. What does \(x \in \bigcup_{i \in I} A_i\) mean?
  2. Let \(A_1 = \{1,2\}\), \(A_2 = \{2,3\}\) and \(A_3 = \{3,4\}\). What is \(\bigcup_{i=1}^{3} A_i\)?
  3. With the same \(A_1, A_2, A_3\) as before, what is \(\bigcap_{i=1}^{3} A_i\)?
  4. What is \(\bigcup_{i \in \varnothing} A_i\) for a family indexed by the empty set?
  5. For any family \(\{A_i\}_{i \in I}\), which statement is true?
  6. If \(I = \{1, 2\}\), which set is \(\bigcup_{i \in I} A_i\)?
  7. Let \(A_n = \{n, n+1\}\) for \(n \in \{1,2,3\}\). How many elements does \(\bigcup_{n} A_n\) have?
  8. Let \(A_n = \{n\}\) for \(n \in \{1, 2, \dots, 10\}\). What is \(\bigcup_{n} A_n\)?
  9. Let \(\mathcal{F} = \{\{1,2\}, \{2,3\}\}\), a family of two sets. What is \(\bigcup \mathcal{F}\)?
  10. Let \(A_k = \{k, 2k\}\) for \(k \in \{1,2,3\}\). What is \(\bigcup_{k} A_k\)?

References

  • Herbert B. Enderton, Elements of Set Theory, 1st edition, Academic Press, 1977.
  • Thomas Jech, Set Theory (The Third Millennium Edition), 3rd millennium edition, Springer, 2003.