Operations and families · Definition

Intersection of families

set theoryoperations and families

Before you start you need to know

Intersection of families extends the intersection of two sets to any number of sets at once.

Definition

The intersection of a nonempty family \(\mathcal{F}\) of sets is the set \(\bigcap \mathcal{F}\) of objects that belong to every member of the family.

\[\bigcap \mathcal{F} = \{x \mid \forall A \in \mathcal{F},\ x \in A\}\]

If the family is \(\mathcal{F} = \{A_i : i \in I\}\), the intersection \(\bigcap_{i \in I} A_i\) collects the \(x\) that lie in every \(A_i\). For a family of two sets it coincides with \(A \cap B\). Since an element is excluded as soon as it fails in a single member, \(\bigcap \mathcal{F} \subseteq A\) for every \(A \in \mathcal{F}\).

Worked example: let \(\mathcal{F} = \{\{1,2,3\}, \{2,3,4\}, \{3,4,5\}\}\). The 3 lies in all three sets, so \(\bigcap \mathcal{F} = \{3\}\). The 2 and the 4 are excluded, because each of them is missing from at least one of the sets.

Typical mistake: confusing the intersection of a family with its union, which collects the elements that belong to some member. That is also why the family must be nonempty: without a fixed universe set, the intersection of the empty family is not determined.

Think about it

Why is the intersection of the empty family undetermined without a fixed universe?

Because the condition \(\forall A \in \varnothing\) holds vacuously for every \(x\), so the intersection would be the whole universe, which is not fixed.

Exercises 10

  1. What does \(x \in \bigcap \mathcal{F}\) mean?
  2. If \(\mathcal{F} = \{A, B\}\), what is \(\bigcap \mathcal{F}\)?
  3. Which statement is true for every nonempty family \(\mathcal{F}\)?
  4. Given \(\mathcal{F} = \{\{a,b\},\{b,c\}\}\), what is \(\bigcap \mathcal{F}\)?
  5. Let \(\mathcal{F} = \{\{1,2,3,4\}, \{2,4,6\}, \{4,5,6\}\}\). What is \(\bigcap \mathcal{F}\)?
  6. What distinguishes \(x \in \bigcap \mathcal{F}\) from \(x \in \bigcup \mathcal{F}\)?
  7. Let \(\mathcal{F} = \{\{1,\ldots,k\} : k \in \{3,5,7\}\}\). What is \(\bigcap \mathcal{F}\)?
  8. Let \(\mathcal{F}\) be a nonempty family and \(\mathcal{G} \subseteq \mathcal{F}\) a nonempty subfamily. What relation holds between \(\bigcap \mathcal{F}\) and \(\bigcap \mathcal{G}\)?
  9. Let \(A_n = \{x \in \mathbb{N} : x \geq n\}\) for each \(n \in \mathbb{N}\), with \(\mathbb{N} = \{1,2,3,\ldots\}\), and \(\mathcal{F} = \{A_n : n \in \mathbb{N}\}\). What is \(\bigcap \mathcal{F}\)?
  10. Which of these families has empty intersection, but every pair of its members has nonempty intersection?

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.