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 and \(A \cap B=\varnothing\), then \(\mathscr{P}(A)-\mathscr{P}(B) \subseteq \mathscr{P}(A-B)\).

Short answer

The statement is true. If \(A\) and \(B\) are sets and \(A \cap B = \varnothing\), then \(\mathscr{P}(A) - \mathscr{P}(B) \subseteq \mathscr{P}(A - B)\).

Step-by-step solution

Understanding the Problem Statement

The statement to prove or disprove is: if \(A\) and \(B\) are sets and \(A \cap B = \varnothing\), then \(\mathscr{P}(A)-\mathscr{P}(B) \subseteq \mathscr{P}(A-B)\). The \(\cap\) represents intersection of sets, where \(A \cap B = \varnothing\) means that sets \(A\) and \(B\) have no elements in common. \(\mathscr{P}(A)\) and \(\mathscr{P}(B)\) denote the power sets of \(A\) and \(B\) respectively, which are the sets of all subsets of \(A\) and \(B\). \(A - B\) denotes set difference, which is the set of all elements in \(A\) that are not in \(B\). Lastly, the \( \subseteq \) symbol denotes a subset.

Proving the Statement

Let \(x\) be an arbitrary element from the set \(\mathscr{P}(A)-\mathscr{P}(B)\). This implies that \(x \subseteq A\) but \(x \nsubseteq B\). Since \(x \nsubseteq B\), there exists at least one element in \(x\) that is not in \(B\). Therefore, all elements of \(x\) must be in the set \(A - B\), which is defined as all elements in \(A\) not in \(B\). So, \(x \subseteq A - B\), which implies that \(x\) is an element of the power set of \(A - B\), \(\mathscr{P}(A - B)\). Since \(x\) was arbitrary, this is true for all \(x\) in the set \(\mathscr{P}(A) - \mathscr{P}(B)\). Therefore, we can conclude that if \(A\) and \(B\) are sets and \(A \cap B = \varnothing\), then \(\mathscr{P}(A) - \mathscr{P}(B) \subseteq \mathscr{P}(A - B)\). Hence, the statement is proven to be true.