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

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 ($\leftrightarrow$).

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 $P, Q$.

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.

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.

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