Problem 14 - Book of Proof

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 \(\mathscr{P}(A) \cap \mathscr{P}(B)=\mathscr{P}(A \cap B)\).

Step-by-Step Solution

Answer

The statement is false. There are cases where \(\mathscr{P}(A) \cap \mathscr{P}(B)\) does not equal to \(\mathscr{P}(A \cap B)\).

Step 1: Understand the Terms

Power set \(\mathscr{P}(A)\) of a set \(A\) is a set of all of its subsets. Thus, it contains every subset that can be created from set \(A\). The intersection of two sets \(A\) and \(B\) (denoted \(A \cap B\)) is the set that contains all elements that \(A\) and \(B\) have in common.

Step 2: Evaluate Left Side of the Equation

\(\mathscr{P}(A) \cap \mathscr{P}(B)\) represents the intersection of the power sets of \(A\) and \(B\), which can be interpreted as the set of all common subsets between \(A\) and \(B\).

Step 3: Evaluate Right Side of the Equation

\(\mathscr{P}(A \cap B)\) represents the power set of the intersection of \(A\) and \(B\), which can be interpreted as the set of every subset that can be formed from the common elements between \(A\) and \(B\).

Step 4: Compare Both Sides of the Equation

Comparing the interpretations from Step-2 and Step-3, it is clear that \(\mathscr{P}(A) \cap \mathscr{P}(B)\) is not always equal to \(\mathscr{P}(A \cap B)\). For a simple disproof, consider \(A = \{1\}\) and \(B = \{2\}\). Then \(\mathscr{P}(A) \cap \mathscr{P}(B) = \{\emptyset\}\) but \(\mathscr{P}(A \cap B) = \{\emptyset, \{1, 2\}\}\). Since \(\{\emptyset\} ≠ \{\emptyset, \{1, 2\}\}\), the statement is false.

Key Concepts

Power Set

In set theory, the power set of a given set is a fundamental concept. The power set, denoted \( \mathscr{P}(A) \), contains all possible subsets of a set \( A \). This includes the empty set and \( A \) itself.
For example, if \( A = \{1, 2\} \), the power set \( \mathscr{P}(A) \) will be:

• \( \emptyset \)
• \( \{1\} \)
• \( \{2\} \)
• \( \{1, 2\} \)

The number of subsets in a power set is always \( 2^n \), where \( n \) is the number of elements in \( A \).
Understanding power sets is crucial because they help extend the idea of sets into more complex mathematical frameworks.

Intersection of Sets

When talking about the intersection of sets, we refer to finding the common elements between two sets. Given two sets, say \( A \) and \( B \), the intersection is represented by \( A \cap B \).
This effectively forms a new set that consists only of elements found in both \( A \) and \( B \).
For instance, if \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), the intersection \( A \cap B \) will be the set \( \{2, 3\} \) as these are the elements common to both sets.
Intersections play a key role in various real-world applications, such as database search queries where you need to filter results containing several criteria at once.

Subsets

A subset is a set where all its elements are contained within another set. Essentially, if every element of a set \( A \) is also a part of set \( B \), then \( A \) is a subset of \( B \), denoted as \( A \subseteq B \).
For example, consider the set \( B = \{1, 2, 3, 4\} \). Subsets of \( B \) include:

• \( \emptyset \)
• \( \{1\} \)
• \( \{1, 2\} \)
• \( B \) itself.

If a subset lacks some of the elements from set \( B \), it is called a "proper subset" and is denoted by \( A \subset B \) (without the equal sign).
Understanding subsets is crucial in proving set-related theorems as they allow you to explore relationships between sets and their possible combinations.