Question

complete a truth table for each compound statement below.

you may want to complete the table on your own paper and insert some intermediate columns to help you arrive at the answers.

a. \sim (\sim p \lor \sim q)

\

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

b. \sim (\sim p \land q)

\

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

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 truth values for part a

\[

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

\]

Evaluate truth values for part b

\[

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

\]
</reasoning>

<answer>

Question a

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim(\sim p \lor \sim q)\)
T	F	F	T	F
F	T	T	F	F
F	F	T	T	F

Question b

\(p\)	\(q\)	\(\sim p\)	\(\sim(\sim p \land q)\)
T	F	F	T
F	T	T	F
F	F	T	T

</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 truth values for part a

\[

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

\]

Evaluate truth values for part b

\[

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

\]
</reasoning>

<answer>

Question a

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim(\sim p \lor \sim q)\)
T	F	F	T	F
F	T	T	F	F
F	F	T	T	F

Question b

\(p\)	\(q\)	\(\sim p\)	\(\sim(\sim p \land q)\)
T	F	F	T
F	T	T	F
F	F	T	T

</answer>

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