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Beginning Activity Beginning Activity 1: The Union and Intersection of a Family of Sets
A4US

In Section 5.3, we discussed various properties of set operations. We will now focus on the associative properties for set union and set intersection. Notice that the definition of “set union” tells us how to form the union of two sets. It is the associative law that allows us to discuss the union of three sets. Using the associate law, if \(A\text{,}\) \(B\text{,}\) and \(C\) are subsets of some universal set, then we can define \(A \cup B \cup C\) to be \(\left( A \cup B \right) \cup C\) or \(A \cup \left( B \cup C \right)\text{.}\) That is,

\begin{equation*} A \cup B \cup C = \left(A \cup B \right) \cup C = A \cup \left( B \cup C \right)\!\text{.} \end{equation*}

For this activity, the universal set is \(\N\) and we will use the following four sets:

\begin{align*} A \amp = \left\{ 1, 2, 3, 4, 5 \right\} \amp B \amp = \left\{ 2, 3, 4, 5, 6 \right\}\\ C \amp = \left\{ 3, 4, 5, 6, 7 \right\} \amp D \amp = \left\{ 4, 5, 6, 7, 8 \right\} \end{align*}

1.

Use the roster method to specify the sets \(A \cup B \cup C\text{,}\) \(B \cup C \cup D\text{,}\) \(A \cap B \cap C\text{,}\) and \(B \cap C \cap D\text{.}\)

2.

Use the roster method to specify each of the following sets. In each case, be sure to follow the order specified by the parentheses.

(a)

\(\left( A \cup B \cup C \right) \cup D\)

(b)

\(A \cup \left( B \cup C \cup D \right)\)

(c)

\(A \cup \left( B \cup C \right) \cup D\)

(d)

\(\left( A \cup B \right) \cup \left( C \cup D \right)\)

(e)

\(\left( A \cap B \cap C \right) \cap D\)

(f)

\(A \cap \left( B \cap C \cap D \right)\)

(g)

\(A \cap \left( B \cap C \right) \cap D\)

(h)

\(\left( A \cap B \right) \cap \left( C \cap D \right)\)

3.

Based on the work in Exercise 2, does the placement of the parentheses matter when determining the union (or intersection) of these four sets? Does this make it possible to define \(A \cup B \cup C \cup D\) and \(A \cap B \cap C \cap D\text{?}\)

We have already seen that the elements of a set may themselves be sets. For example, the power set of a set \(T\text{,}\) \(\mathcal{P}( T )\text{,}\) is the set of all subsets of \(T\text{.}\) The phrase, “a set of sets” sounds confusing, and so we often use the terms collection and family when we wish to emphasize that the elements of a given set are themselves sets. We would then say that the power set of \(T\) is the family (or collection) of sets that are subsets of \(T\text{.}\)

One of the purposes of the work we have done so far in this activity was to show that it is possible to define the union and intersection of a family of sets.

Definition.

Let \(\mathcal{C}\) be a family of sets. The union of \(\mathbf{\mathcal{C}}\) is defined as the set of all elements that are in at least one of the sets in \(\mathcal{C}\text{.}\) We write

\begin{equation*} \bigcup_{X \in \mathscr{C}}^{}X = \left\{x \in U \mid x \in X \text{ for some } X \in \mathscr{C} \right\} \end{equation*}

The intersection of \(\mathbf{\mathcal{C}}\) is defined as the set of all elements that are in all of the sets in \(\mathcal{C}\text{.}\) That is,

\begin{equation*} \bigcap_{X \in \mathscr{C}}^{}X = \left\{x \in U \mid x \in X \text{ for all } X \in \mathscr{C} \right\} \end{equation*}

For example, consider the four sets \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) used earlier in this activity and the sets

\begin{equation*} S = \left\{5, 6, 7, 8, 9 \right\} \text{ and } T = \left\{6, 7, 8, 9, 10 \right\}\text{.} \end{equation*}

We can then consider the following families of sets: \(\mathscr{A} = \left\{A, B, C , D \right\}\) and \(\mathscr{B} = \left\{A, B, C , D, S, T \right\}\text{.}\)

4.

Explain why

\begin{equation*} \bigcup_{X \in \mathscr{A}}^{}X = A \cup B \cup C \cup D \text{ and } \bigcap_{X \in \mathscr{A}}^{}X = A \cap B \cap C \cap D\text{,} \end{equation*}

and use your work in Exercise 1, Exercise 2, and Exercise 3 to determine \(\bigcup\limits_{X \in \mathscr{A}}^{}X\) and \(\bigcap\limits_{X \in \mathscr{A}}^{}X\text{.}\)

5.

Use the roster method to specify \(\bigcup\limits_{X \in \mathscr{B}}^{}X\) and \(\bigcap\limits_{X \in \mathscr{B}}^{}X\text{.}\)

6.

Use the roster method to specify the sets \(\left( \bigcup\limits_{X \in \mathscr{A}}^{}X \right)^c\) and \(\bigcap\limits_{X \in \mathscr{A}}^{}X^c\text{.}\) Remember that the universal set is \(\N\text{.}\)