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.