https://file.aiquickdraw.com/mathbot/file/1731837068504okn30r4b.jpg

Let's work on the questions from the image one by one:

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### **1. Construct truth table for \([(P \lor Q) \land \neg R] \leftrightarrow Q\)**

We need to construct the truth table step by step:

1. **Columns for \(P, Q, R\) (all possible combinations).**
2. **Compute \(P \lor Q\).**
3. **Compute \(\neg R\).**
4. **Compute \((P \lor Q) \land \neg R\).**
5. **Compare \((P \lor Q) \land \neg R\) with \(Q\) using the biconditional (\(\leftrightarrow\)).**

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### **2. Show that \( Q \lor (P \land \neg Q) \lor (\neg P \land \neg Q)\) is a tautology**

- A tautology is a statement that is always true, regardless of the truth values of its components.
- Here we need to:
  1. Simplify the formula step by step to see if it simplifies to a tautology.
  2. Alternatively, construct a truth table for all possible truth values of \(P, Q\).

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### **3. Show that \(P \land (\neg Q \lor R)\) and \(P \lor (Q \land \neg R)\) are logically not equivalent**

- Logical equivalence means that both expressions have the same truth values in all cases.
- To verify:
  1. Construct truth tables for both expressions step by step.
  2. Compare the truth values row by row to identify if there are any differences.

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### **4. Obtain the disjunctive normal form of \(\neg (P \lor Q) \leftrightarrow (P \land Q)\)**

- The disjunctive normal form (DNF) is a standard form where the expression is represented as a disjunction (\(\lor\)) of conjunctions (\(\land\)).
- Steps:
  1. Simplify \(\neg (P \lor Q)\).
  2. Simplify \((P \land Q)\).
  3. Compute the biconditional (\(\leftrightarrow\)).
  4. Rewrite in DNF format.

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Would you like me to solve these step by step, starting with question 1?

1. Construct truth table for [(P ∨ Q) ∧ ¬R] ↔ Q
2. Show that the formula Q ∨ (P ∧ ¬Q) ∨ (¬P ∧ ¬Q) is a tautology
3. Show that P ∧ (¬Q ∨ R) and P ∨ (Q ∧ ¬R) are logically not equivalent
4. Obtain the disjunctive normal form of ¬(P ∨ Q) ↔ (P ∧ Q)

Explore solutions to propositional logic problems including constructing truth tables, identifying tautologies, checking logical equivalences, and obtaining disjunctive normal forms. Detailed steps and solutions are provided for undergraduate and graduate-level students.

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