**Title: Exploring the Algebra of Sets: A Proof or Counterexample Challenge** **Objective:** Explore the given equation using set theory and determine its validity by either providing a counterexample or proving it mathematically. **Problem Statement:** Either provide a counterexample or prove using the algebra of sets: \[ \forall A, B, C \subseteq U \left( A \setminus (B \setminus C) = ((A \setminus B) \cup (A \cap C)) \right). \] **Instructions:** 1. **Understand the Set Operations**: - \( A \setminus B \) represents the set of elements in \( A \) that are not in \( B \). - \( A \cap C \) is the intersection of sets \( A \) and \( C \), consisting of all elements that are common to both sets. - \( (B \setminus C) \) indicates elements in \( B \) that are not in \( C \). 2. **Analyze the Equation**: - The goal is to evaluate whether the left side of the equation \( A \setminus (B \setminus C) \) is equal to the right side \(((A \setminus B) \cup (A \cap C))\) for all subsets \( A, B, C \) of a universal set \( U \). 3. **Methods**: - **Proof**: Utilize properties of set algebra such as distributive, associative, and De Morgan’s laws to simplify and compare both sides of the equation. - **Counterexample**: Find specific sets \( A, B, C \) such that the equation does not hold. 4. **Evaluate your Findings**: - Determine if the expression is universally true (proven) or if a specific counterexample can be found (disproven). 5. **Select an Option**: - Proven - Disproven Engage with this problem to enhance understanding of set theory concepts and strengthen logical reasoning skills. ### Logical Equivalences, Inferences, and Fallacies #### Logical Equivalences | Statement | Property | |-------------------------------------|-----------------| | \( p \lor p \equiv p \) | Idempotence | | \( p \land p \equiv p \) | Idempotence | | \( p \lor q \equiv q \lor p \) | Commutativity | | \( p \land q \equiv q \land p \) | Commutativity | | \( p \lor (q \lor r) \equiv (p \lor q) \lor r \) | Associativity | | \( p \land (q \land r) \equiv (p \land q) \land r \) | Associativity | | \( p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) \) | Distributivity | | \( p \land (q \lor r) \equiv (p \land q) \lor (p \land r) \) | Distributivity | | \( p \lor (p \land q) \equiv p \) | Absorptivity | | \( p \land (p \lor q) \equiv p \) | Absorptivity | | \( p \lor \bot \equiv p \) | Identity | | \( p \land \top \equiv p \) | Identity | | \( p \lor (\neg p) \equiv \top \) | Complementarity | | \( p \land (\neg p) \equiv \bot \) | Complementarity | | \( p \lor \top \equiv \top \) | Dominance | | \( p \land \bot \equiv \bot \) | Dominance | | \( \neg (\neg p) \equiv p \) | Involution | | \( \neg (\top) \equiv \bot \) | Exclusivity | | \( \neg (\bot) \equiv \top \) | Exclusivity | | \( \neg (p \lor q) \equiv (\neg p) \land (\neg q) \) | De