Construct the truth tables for the following formulas:
(a) (PQ) (b) P(QP) (c) (P(PQ)Q) (d) (PQ)(PQ) 

Let's break down each formula and construct the corresponding truth tables step by step.

### (a) \(\neg(\neg P \lor \neg Q)\)
We will construct the truth table for the negation of the disjunction of \(\neg P\) and \(\neg Q\).

| \(P\) | \(Q\) | \(\neg P\) | \(\neg Q\) | \(\neg P \lor \neg Q\) | \(\neg(\neg P \lor \neg Q)\) |
|:---:|:---:|:---:|:---:|:---:|:---:|
| T   | T   | F   | F   | F   | T   |
| T   | F   | F   | T   | T   | F   |
| F   | T   | T   | F   | T   | F   |
| F   | F   | T   | T   | T   | F   |

### (b) \(P \land (Q \land P)\)
This is the conjunction of \(P\) and the conjunction of \(Q\) and \(P\).

| \(P\) | \(Q\) | \(Q \land P\) | \(P \land (Q \land P)\) |
|:---:|:---:|:---:|:---:|
| T   | T   | T   | T   |
| T   | F   | F   | F   |
| F   | T   | F   | F   |
| F   | F   | F   | F   |

### (c) \((P \land (P \to Q)) \to Q\)
We need to evaluate the truth values of \(P \land (P \to Q)\), then see the truth value for \((P \land (P \to Q)) \to Q\).

| \(P\) | \(Q\) | \(P \to Q\) | \(P \land (P \to Q)\) | \((P \land (P \to Q)) \to Q\) |
|:---:|:---:|:---:|:---:|:---:|
| T   | T   | T   | T   | T   |
| T   | F   | F   | F   | T   |
| F   | T   | T   | F   | T   |
| F   | F   | T   | F   | T   |

### (d) \((P \to Q) \leftrightarrow (\neg P \lor Q)\)
This formula checks the biconditional (↔) of \(P \to Q\) and \(\neg P \lor Q\).

| \(P\) | \(Q\) | \(\neg P\) | \(P \to Q\) | \(\neg P \lor Q\) | \((P \to Q) \leftrightarrow (\neg P \lor Q)\) |
|:---:|:---:|:---:|:---:|:---:|:---:|
| T   | T   | F   | T   | T   | T   |
| T   | F   | F   | F   | F   | T   |
| F   | T   | T   | T   | T   | T   |
| F   | F   | T   | T   | T   | T   |

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Do you need further details or explanations on any of the truth tables?

### Related questions:
1. How is a biconditional operator \(\leftrightarrow\) evaluated in a truth table?
2. What is the significance of negation in logical expressions?
3. Why do logical conjunctions (\(\land\)) return false when any operand is false?
4. How does the conditional (\(\to\)) behave differently from biconditional (\(\leftrightarrow\))?
5. Can disjunction (\(\lor\)) ever return false when at least one operand is true?

### Tip: 
When constructing truth tables, always start by evaluating the simplest components (negation, conjunction, etc.) and build the more complex expressions step by step.

Construct the truth tables for the following formulas:
(a) (PQ)
(b) P(QP)
(c) (P(PQ)Q)
(d) (PQ)(PQ)

Explore the step-by-step construction of truth tables for logical formulas including (PQ), P(QP), (P(PQ)Q), and (PQ)(PQ). This page covers key concepts in propositional logic, negation, conjunction, disjunction, and biconditional operations. Suitable for introductory logic students.

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