Question

Each of the following statements is either true or false. If a statement is true, prove it. If a statement is false, disprove it. These exercises are cumulative, covering all topics addressed in Chapters \(1-9 .\) If \(A\) and \(B\) are sets, then \(\mathscr{P}(A)-\mathscr{P}(B) \subseteq \mathscr{P}(A-B)\).

Short answer

The statement is false. A counterexample is provided with sets A={1,2} and B={3,4}, demonstrating that \(\mathscr{P}(A)-\mathscr{P}(B) \not\subseteq \mathscr{P}(A-B)\).

Step-by-step solution

Understanding statement

Let two sets A and B are given. This step is understanding the statement \(\mathscr{P}(A)-\mathscr{P}(B) \subseteq \mathscr{P}(A-B) \). In words, it is saying that, power set of A minus the power set of B is a subset of the power set of A minus B.

Interpretation of \(\mathscr{P}(A)-\mathscr{P}(B)\)

The \(\mathscr{P}(A)-\mathscr{P}(B)\) means the set of all subsets of A that are not subsets of B. It includes all the sets which are in power set of A but not in power set of B.

Interpretation of \(\mathscr{P}(A-B)\)

\(\mathscr{P}(A-B)\) represents all subsets of the set A-B (set A minus set B). This means all sets obtained from the elements in A which are not in B.

Disproving the statement

Assume that A={1,2} and B={3,4}. Then the power set of A is \(\mathscr{P}(A) =\{ \emptyset, \{1\}, \{2\}, \{1,2\}\}\) and the power set of B is \(\mathscr{P}(B)=\{\emptyset, \{3\}, \{4\}, \{3,4\}\}\). Thus, \(\mathscr{P}(A)-\mathscr{P}(B) =\{ \{1\}, \{2\}, \{1,2\}\}\) because these elements are not present in \(\mathscr{P}(B)\). Also, A-B={1,2} and \(\mathscr{P}(A-B) =\{ \emptyset, \{1\}, \{2\}, \{1,2\}\}\). Therefore, clearly \(\mathscr{P}(A)-\mathscr{P}(B) \not\subseteq \mathscr{P}(A-B)\), which disproves the statement.