Question

Let \( \Omega = \{1,2,3,4,5,6\} \). Then which of the following classes of sets is an algebra?

• A. \( \mathcal{F}_1 = \{\emptyset, \Omega, \{1,2\}, \{3,4\}, \{3,6\}\} \)
• B. \( \mathcal{F}_2 = \{\emptyset, \Omega, \{1,2,3\}, \{4,5,6\}\} \)
• C. \( \mathcal{F}_3 = \{\emptyset, \Omega, \{1,2\}, \{4,5\}, \{1,2,4,5\}, \{3,4,5,6\}, \{1,2,3,6\}\} \)
• D. \( \mathcal{F}_4 = \{\emptyset, \{4,5\}, \{1,2,3,6\}\} \)

Answer 1

The Correct Option is B

1. Definition of an Algebra: - A class \( \mathcal{F} \) of subsets of \( \Omega \) is an algebra if: \begin{itemize} \item \( \emptyset \in \mathcal{F} \) and \( \Omega \in \mathcal{F} \), \item It is closed under union and intersection, \item It is closed under complements relative to \( \Omega \). \end{itemize} 2. Analyze Each Option: - Option \( \mathcal{F}_1 \): Not closed under union. For example, \( \{1, 2\} \cup \{3, 4\} = \{1, 2, 3, 4\} \notin \mathcal{F}_1 \). - Option \( \mathcal{F}_2 \): Closed under union, intersection, and complements. For example: \[ \{1, 2, 3\}^c = \{4, 5, 6\} \in \mathcal{F}_2. \] - Option \( \mathcal{F}_3 \): Not closed under complements. For example, \( \{1, 2\}^c = \{3, 4, 5, 6\} \notin \mathcal{F}_3 \). - Option \( \mathcal{F}_4 \): Does not contain \( \Omega \), so it is not an algebra. 3. Conclusion: - \( \mathcal{F}_2 \) satisfies all the properties of an algebra.