2.3 Collections of Sets
Now that we know how to combine pairs of sets, we can inductively define unions and intersections for a finite or infinite number of sets.
When we are dealing with more than one or two related, but distinct sets, we often use another set as an index set in order to more easily describe and distinguish the sets in the collection. The most common indexing set used is the set of natural numbers, \(\mathbb{N}=\{1, 2, 3, \ldots \}\). Then we use this indexing set as a label for the sets that we are considering. So for each natural number, there is a corresponding set, \(A_n\), usually related to the value of \(n\) in some way.
Example 2.11 Let the natural numbers, \(\mathbb{N}=\{1, 2, 3, \ldots \}\), be an indexing set. For each natural number, \(n\), we define \[A_n = [n,n+1),\] which is an interval in the real numbers. In this situation, we have \[A_1 = [1,2), \: A_2 = [2,3), \: A_3 = [3,4),\: \ldots \] In this case, the use of \(\mathbb{N}\) tells us that we have an infinite set of sets. We use the indexing set to express the form of a general interval. When we assign an index value (such as \(n=3\)), we are effectively identifying a particular interval in the sequence of intervals.
Using indexing sets, we can then define the union and intersection of a collection of sets.
Definition 2.8 Let \(S\) be an indexing set and let \(\left\{ A_i\right\}_{i\in S}\) be an indexed non-empty family of sets. Then we define the union and intersection of the family of sets as \[\bigcup_{i\in S} A_i = \left\{ x \middle \vert x\in A_i \mbox{ for some } i\in S\right\} \quad \mbox{ and } \quad \bigcap_{i\in S} A_i = \left\{ x\middle \vert x\in A_i \mbox{ for all } i \in S\right\}.\]
Example 2.12 Continuing with our previous example where \(A_n= [n,n+1)\) for each \(n\in \mathbb{N}\), we can write \[\bigcup_{i\in \{1,2,3\}} A_i = [1,4) \quad \mbox{and} \quad \bigcap_{i\in \{1,2,3\}} A_i = \emptyset\] since the sets \(A_1\), \(A_2\), \(A_3\), and \(A_4\) do not overlap. If we extend this to the entire indexing set, then \[\bigcup_{i\in \mathbb{N}} A_i = [1,\infty) \quad \mbox{and} \quad \bigcap_{i\in \mathbb{N}} A_i = \emptyset.\]
An indexed collection of sets \(\{A_i\}_{i\in S}\) is called mutually disjoint if, for any \(i,j\in S\) with \(i\neq j\), \(A_i \cap A_j = \emptyset\). The sets in the previous example are mutually disjoint since \([n,n+1) \cap [m,m+1) = \emptyset\) if \(m\neq n\).
Example 2.13 For each positive integer \(n\), (\(n\in \mathbb{N}\)), let \[S_n= \left\{x\in \mathbb{R}\middle \vert \frac{-1}{n} < x < \frac{1}{n} \right\}.\]
\[S_1=(-1,1)\]
\[S_2=(\frac{-1}{2}, \frac{1}{2})\]
\[S_3=(\frac{-1}{3}, \frac{1}{3})\]
Then we can see that for any \(i<j\), we have that \(S_i\cap S_j = S_j\) and \(S_i \cup S_j = S_i\).
We can also take the union and intersections over the entire collection of sets. \[\bigcup_{n\in \mathbb{N}} S_n = (-1,1) \quad \mbox{ and } \quad \bigcap_{n\in \mathbb{N}} S_n = \{0\}.\]
The above example is also an example of what is called a nested collection of sets.
Definition 2.9 An indexed collection of sets, \(\left\{ A_i\right\}_{i\in I}\), with an order on \(I\) is called nested if either \[A_i \subseteq A_j \mbox{ whenever } i<j \quad \mbox{or} \quad A_i \supseteq A_j \mbox{ whenever } i<j.\] When the indexing set is the natural numbers this is often denoted by \[A_0 \subseteq A_1 \subseteq A_2 \subseteq \cdots \quad \mbox{ or } \quad A_0 \supseteq A_1 \supseteq A_2 \supseteq \cdots.\]
Note: the student is strongly encourage to complete the related exercises to see addition examples of nested collections of sets. We can also see from the definitions that a collection of sets cannot be both nested and mutually disjoint.
Using the ideas of indexing sets and families of sets, we can generalize De Morgan’s Laws to a general collection of sets, the proofs of which are very similar to the proof in the case of two sets.
Theorem 2.7 Let \(I\) be an indexing set and let \(\{A_i\}_{i\in I}\) be a collection of sets that are all subsets of the same universal set. Then \[\left( \bigcup_{i\in I} A_i \right)^c = \bigcap_{i \in I} A_i^c \quad \mbox{and} \quad \left( \bigcap_{i\in I} A_i \right)^c = \bigcup_{i \in I} A_i^c\]
2.3.1 Exercises
For each of the following collections of sets:
\[\displaystyle{\mathcal{A} = \left\{ \left[ \frac{1}{n},n\right) \right\}_{n=2,3,4,\ldots }}\]
\[\displaystyle{\mathcal{B} = \left\{ \left( n,\infty \right) \right\}_{n=0,1,2,3,4,5,\ldots} }\]
\[\displaystyle{\mathcal{C} = \left\{ \left[ -n, n \right] \right\}_{n=0,1,2,3,4,5,\ldots }}\]
\[\displaystyle{\mathcal{D} = \left\{ [x,x+1)\right\}_{x\in \mathbb{R}}}\]
\[\displaystyle{\mathcal{E} = \left\{ \{z\in \mathbb{C}\middle \vert|z|=r\}\right\}_{r\in \mathbb{R}^+} }\]
\[\displaystyle{\mathcal{F} = \left\{ \{n\in \mathbb{Z}\middle \vert n=3k+j \mbox{ for some } k\in \mathbb{Z}\}\right\}_{j=0,1,2}}\]
- Determine if the sets are mutually disjoint
- Determine if the collection is nested
- Find the union of the collection