Question

1. construct a truth table for the given statement and select the final answer column.

\\ (p \
ightarrow \sim q) \vee (\sim q \wedge p) \\

\

$$\begin{array}{|c|c|c|c|c|c|} \\hline p & q & \\sim q & p \ ightarrow \\sim q & \\sim q \\wedge p & (p \ ightarrow \\sim q) \\vee (\\sim q \\wedge p) \\\\ \\hline t & t & & & & \\\\ \\hline t & f & & & & \\\\ \\hline f & t & & & & \\\\ \\hline f & f & & & & \\\\ \\hline \\end{array}$$

a.
f
t
t
t

b.
t
f
f
f

c.
f
f
t
t

d.
t
t
f
t

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Truth Tables",
"Mathematical Logic"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Mathematical Logic",
"Truth Value Analysis"
]
}
</pre_analysis>

<reasoning>

Evaluate negation of q

Using the Truth Tables knowledge point
\[

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

\]

Evaluate conditional statement

Using the Mathematical Logic knowledge point
\[

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

\]

Evaluate conjunction statement

Using the Mathematical Logic knowledge point
\[

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

\]

Evaluate final disjunction

Using the Truth Tables knowledge point
\[

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

\]
</reasoning>

<answer>
<mcq-correct>(A) F
T
T
T</mcq-correct>
<mcq-option>(B) T
F
F
F</mcq-option>
<mcq-option>(C) F
F
T
T</mcq-option>
<mcq-option>(D) T
T
F
T</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Truth Tables",
"Mathematical Logic"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Mathematical Logic",
"Truth Value Analysis"
]
}
</pre_analysis>

<reasoning>

Evaluate negation of q

Using the Truth Tables knowledge point
\[

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

\]

Evaluate conditional statement

Using the Mathematical Logic knowledge point
\[

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

\]

Evaluate conjunction statement

Using the Mathematical Logic knowledge point
\[

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

\]

Evaluate final disjunction

Using the Truth Tables knowledge point
\[

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

\]
</reasoning>

<answer>
<mcq-correct>(A) F
T
T
T</mcq-correct>
<mcq-option>(B) T
F
F
F</mcq-option>
<mcq-option>(C) F
F
T
T</mcq-option>
<mcq-option>(D) T
T
F
T</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>