The compound proposition you provided is a bit unclear due to formatting, but I understand it as:

$(p \lor \neg q) \rightarrow (p \land q)$

We need to construct the truth table for this expression. Let's break it down step by step.

Components:

1. $p$: Proposition $p$.
2. $q$: Proposition $q$.
3. $\neg q$: The negation of $q$.
4. $p \lor \neg q$: $p$ OR $\neg q$.
5. $p \land q$: $p$ AND $q$.
6. $(p \lor \neg q) \rightarrow (p \land q)$: The implication from $p \lor \neg q$ to $p \land q$.

Truth Table Construction:

Let's construct the truth table by considering all possible truth values for $p$ and $q$.

$p$	$q$	$\neg q$	$p \lor \neg q$	$p \land q$	$(p \lor \neg q) \rightarrow (p \land q)$
T	T	F	T	T	T
T	F	T	T	F	F
F	T	F	F	F	T
F	F	T	T	F	F

Explanation:

1. First Row:

   • $p = T$, $q = T$
   • $\neg q = F$
   • $p \lor \neg q = T \lor F = T$
   • $p \land q = T \land T = T$
   • $(p \lor \neg q) \rightarrow (p \land q) = T \rightarrow T = T$

2. Second Row:

   • $p = T$, $q = F$
   • $\neg q = T$
   • $p \lor \neg q = T \lor T = T$
   • $p \land q = T \land F = F$
   • $(p \lor \neg q) \rightarrow (p \land q) = T \rightarrow F = F$

3. Third Row:

   • $p = F$, $q = T$
   • $\neg q = F$
   • $p \lor \neg q = F \lor F = F$
   • $p \land q = F \land T = F$
   • $(p \lor \neg q) \rightarrow (p \land q) = F \rightarrow F = T$

4. Fourth Row:

   • $p = F$, $q = F$
   • $\neg q = T$
   • $p \lor \neg q = F \lor T = T$
   • $p \land q = F \land F = F$
   • $(p \lor \neg q) \rightarrow (p \land q) = T \rightarrow F = F$

Final Truth Table:

\hline p & q & \neg q & p \lor \neg q & p \land q & (p \lor \neg q) \rightarrow (p \land q) \\ \hline T & T & F & T & T & T \\ T & F & T & T & F & F \\ F & T & F & F & F & T \\ F & F & T & T & F & F \\ \hline \end{array}$$ Would you like any more details or have any questions? --- Here are 5 questions related to this topic: 1. What is the difference between conjunction ($$\land$$) and disjunction ($$\lor$$) in logic? 2. How does the truth value of an implication ($$\rightarrow$$) change based on its premises? 3. Can you create a truth table for $$p \rightarrow (q \lor r)$$? 4. What is the role of negation ($$\neg$$) in logical statements? 5. How does the truth table change if we reverse the implication, i.e., $$(p \land q) \rightarrow (p \lor \neg q)$$? **Tip:** In logic, an implication ($$\rightarrow$$) is false only when the first part (antecedent) is true, and the second part (consequent) is false.