Question

Which of the following statements are correct? Prove each correct statement. Disprove each incorrect statement by finding a counterexample. (a) \(A\) and \(B\) are disjoint if and only if \(B\) and \(A\) are disjoint. (Read the statement carefully - the order in which the sets are listed might matter') (b) \(A \cup B\) and \(C\) are disjoint if and only if both the following are true: (i) \(A\) and \(C\) are disjoint and (ii) \(B\) and \(C\) are disjoint. (c) \(A \cap B\) and \(C\) are disjoint if and only if both the following are true: (i) \(A\) and \(C\) are disjoint and (ii) \(B\) and \(C\) are disjoint. (d) \(A \cup B\) and \(C\) are disjoint if and only if one of the following is true: (i) \(A\) and \(C\) are disjoint or (ii) \(B\) and \(C\) are disjoint. (e) \(A \cap B\) and \(C\) are disjoint if and only if one of the following is true: (i) \(A\) and \(C\) are disjoint or (ii) \(B\) and \(C\) are disjoint. (f) Let \(U\) be a universal set with \(A, B \subseteq U, A\) and \(B\) are disjoint if and only if \(\bar{A}\) and \(\bar{B}\) are disjoint.

Short answer

Statements (a) and (b) are correct; statements (c), (d), (e), and (f) are incorrect.

Step-by-step solution

Analyze Statement (a)

Considering statement (a), which claims that 'A and B are disjoint if and only if B and A are disjoint'. To disprove the 'if and only if' statements, we need to check each part independently."A and B are disjoint" means \( A \cap B = \emptyset \). Due to the commutativity of intersection, \( B \cap A = \emptyset \) must also be true. Therefore, 'A and B are disjoint if and only if B and A are disjoint' is correct.

Check Statement (b)

Statement (b) claims that 'A \cup B and C are disjoint if and only if A and C are disjoint and B and C are disjoint.' \( A \cup B \) has no intersection with \( C \) when neither \( A \) nor \( B \) has any elements in \( C \), thus \((A \cup B) \cap C = \emptyset\) if and only if \( A \cap C = \emptyset \) and \( B \cap C = \emptyset \). Therefore, statement (b) is correct.

Evaluate Statement (c)

Statement (c) proposes that '(A \cap B) and C are disjoint if and only if (i) A and C are disjoint and (ii) B and C are disjoint.' Consider the case where A and B overlap significantly but both are not entirely disjoint from C. Hence, it is possible that \( A \cap C eq \emptyset \) or \( B \cap C eq \emptyset \), but \( (A \cap B) \cap C = \emptyset \). Thus, statement (c) is incorrect, as these conditions do not guarantee disjointness of \( A \cap B \) and \( C \).

Investigate Statement (d)

Statement (d) is 'A \cup B and C are disjoint if and only if one of the following is true: (i) A and C are disjoint or (ii) B and C are disjoint.' This is incorrect because for \( A \cup B \) to be disjoint with \( C \), neither \( A \) nor \( B \) should intersect with \( C \). If only one condition holds, it is possible \( (A \cup B) \) still meets \( C \) through the other set. Hence, statement (d) is false.

Verify Statement (e)

Statement (e) says '(A \cap B) and C are disjoint if and only if one of the following is true: (i) A and C are disjoint or (ii) B and C are disjoint.' The proposition here is that at least one disjointness guarantees the disjointness of intersections. However, this is misleading as \((A \cap B) \cap C = \emptyset\) might require more conditions. Consider a case where \(A\) or \(B\) partly intersects with \(C\). Therefore, statement (e) is incorrect.

Examine Statement (f)

Statement (f) suggests that 'A and B are disjoint if and only if \( \bar{A} \) and \( \bar{B} \) are disjoint.' For \( A \) and \( B \) to be disjoint, \( A \cap B = \emptyset \). Venn Diagrams or logical deduction would show \( \bar{A} \) and \( \bar{B} \) overlap all areas not covered by \( A \cap B \), which do not coincide. Thus, \( \bar{A} \cap \bar{B} eq \emptyset \) whenever \( A \cap B = \emptyset \), making this statement false.

Key concepts

Set Theory

Set Theory is a fundamental branch of mathematical logic that deals with the collection of objects, known as sets. Sets can be simple groups of numbers, letters, or even other sets. The primary operations performed on sets are union, intersection, and complement.

A set is well-defined and can be finite or infinite. The concept of belonging to a set is captured mathematically by using symbols like \( \in \) for membership. For example, if an element \( x \) is in set \( A \), we write \( x \in A \).

In Set Theory, different types of sets have unique properties and names:

• Empty Set: A set with no elements, denoted as \( \emptyset \).
• Subset: Set \( A \) is a subset of set \( B \) if all elements of \( A \) are also in \( B \), written as \( A \subseteq B \).
• Universal Set: Contains all possible elements for a particular discussion or problem, typically denoted as \( U \).

Understanding these basic concepts is crucial for accurately performing operations on sets and analyzing set relationships.

Intersection and Union of Sets

In Set Theory, understanding how sets intersect and unite is essential in determining their relationships.

Intersection of Sets: The intersection of two sets \( A \) and \( B \) is a new set containing only the elements that are in both \( A \) and \( B \), denoted as \( A \cap B \). If \( A \cap B = \emptyset \), the sets are disjoint, meaning they have no elements in common.

Union of Sets: The union of two sets \( A \) and \( B \) is a set containing all elements from \( A \), \( B \), or both, denoted as \( A \cup B \). This operation combines all elements across the involved sets.

These operations help in analyzing scenarios, such as checking the availability of mutual interests between groups (intersection) or the need for involvement in various activities (union). Recognizing these concepts is vital to solve problems related to set relationships, such as assessing if sets are disjoint or not.

Logical Statements

Logical Statements in set theory often involve evaluating the truth values or proving certain conditions about set relationships. Understanding logic helps in reasoning and constructing valid mathematical arguments.

"If and only if" statements require thorough examination. For instance, in the statement "\( A \) and \( B \) are disjoint if and only if \( B \) and \( A \) are disjoint," the statement is inherently true due to the commutative nature of set operations. To break it down:

• If \( A \cap B = \emptyset \), it implies \( B \cap A = \emptyset \).
• Conversely, if \( B \cap A = \emptyset \), then \( A \cap B = \emptyset \).

Properly dissecting logical statements aids in establishing the equivalencies or discrepancies between mathematical propositions. Mastery of logical statements involves using principles like commutativity, associativity, and distributivity to simplify problems.

Counterexamples in Math

Counterexamples are a critical concept in mathematics, used to demonstrate that a particular statement or proposition is false. By presenting a single example where the conditions don't hold, it disproves a universal claim.

For example, let's examine the proposed statement: "\( A \cup B \) and \( C \) are disjoint if only one of these holds: either \( A \) and \( C \) or \( B \) and \( C \) are disjoint." A counterexample can show that if only one of the conditions is met, \( A \cup B \) can still intersect with \( C \), thus disproving the statement.

To use a counterexample effectively:

• Choose sets that highlight the flaw in the statement.
• Demonstrate through set operations that the expected result does not arise.

Implementing counterexamples can be powerful because it requires only a single instance to demonstrate a statement's failure, thereby helping refine or discard incorrect mathematical assumptions.