Question

complete the truth table for the compound statement: \\((p \lor q) \land (\sim p \lor \sim q)\\)

\

$$\begin{array}{|c|c|c|c|c|c|c|} \\hline p & q & \\sim p & \\sim q & (p \\lor q) & (\\sim p \\lor \\sim q) & (p \\lor q) \\land (\\sim p \\lor \\sim q) \\\\ \\hline t & t & ? & ? & ? & ? & ? \\\\ \\hline t & f & ? & ? & ? & ? & ? \\\\ \\hline f & t & ? & ? & ? & ? & ? \\\\ \\hline f & f & ? & ? & ? & ? & ? \\\\ \\hline \\end{array}$$

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Truth Tables",
"Logical Operators",
"Truth Table Construction"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Truth Table Construction"
]
}
</pre_analysis>

<reasoning>

Evaluate the first truth table columns

\[

$$\begin{array}{c|c|c|c|c|c|c} p & q & \sim p & \sim q & (\sim p \wedge q) & (p \vee q) & (\sim p \wedge q) \vee (p \vee q) \\ \hline \text{T} & \text{T} & \text{F} & \text{F} & \text{F} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{F} & \text{F} \end{array}$$

\]

Evaluate the second truth table columns

\[

$$\begin{array}{c|c|c|c|c|c|c} p & q & \sim p & \sim q & (p \vee q) & (\sim p \vee \sim q) & (p \vee q) \wedge (\sim p \vee \sim q) \\ \hline \text{T} & \text{T} & \text{F} & \text{F} & \text{T} & \text{F} & \text{F} \\ \text{T} & \text{F} & \text{F} & \text{T} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{F} \end{array}$$

\]
</reasoning>

<answer>

Question 1

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\((\sim p \wedge q)\)	\((p \vee q)\)	\((\sim p \wedge q) \vee (p \vee q)\)
T	F	F	T	F	T	T
F	T	T	F	T	T	T
F	F	T	T	F	F	F

Question 2

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\((p \vee q)\)	\((\sim p \vee \sim q)\)	\((p \vee q) \wedge (\sim p \vee \sim q)\)
T	F	F	T	T	T	T
F	T	T	F	T	T	T
F	F	T	T	F	T	F

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Truth Tables",
"Logical Operators",
"Truth Table Construction"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Truth Table Construction"
]
}
</pre_analysis>

<reasoning>

Evaluate the first truth table columns

\[

$$\begin{array}{c|c|c|c|c|c|c} p & q & \sim p & \sim q & (\sim p \wedge q) & (p \vee q) & (\sim p \wedge q) \vee (p \vee q) \\ \hline \text{T} & \text{T} & \text{F} & \text{F} & \text{F} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} & \text{T} & \text{F} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{F} & \text{F} \end{array}$$

\]

Evaluate the second truth table columns

\[

$$\begin{array}{c|c|c|c|c|c|c} p & q & \sim p & \sim q & (p \vee q) & (\sim p \vee \sim q) & (p \vee q) \wedge (\sim p \vee \sim q) \\ \hline \text{T} & \text{T} & \text{F} & \text{F} & \text{T} & \text{F} & \text{F} \\ \text{T} & \text{F} & \text{F} & \text{T} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{F} & \text{T} & \text{F} \end{array}$$

\]
</reasoning>

<answer>

Question 1

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\((\sim p \wedge q)\)	\((p \vee q)\)	\((\sim p \wedge q) \vee (p \vee q)\)
T	F	F	T	F	T	T
F	T	T	F	T	T	T
F	F	T	T	F	F	F

Question 2

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\((p \vee q)\)	\((\sim p \vee \sim q)\)	\((p \vee q) \wedge (\sim p \vee \sim q)\)
T	F	F	T	T	T	T
F	T	T	F	T	T	T
F	F	T	T	F	T	F

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>