Question

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

\

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

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Truth Tables",
"Logical Operators",
"Conditional Statements",
"Truth Table Construction"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Conditional Statements",
"Truth Table Construction"
]
}
</pre_analysis>

<reasoning>

Evaluate negations of q and r

\[

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

\]

Evaluate conditional statement

\[

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

\]

Evaluate final disjunction

\[

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

\]
</reasoning>

<answer>

\(p\)

\(q\)

\(r\)

\(\sim q\)

\(\sim q

ightarrow p\) | \(\sim r\) | \((\sim q
ightarrow p) \lor \sim r\) |

T	T	T	F	T	F	T
T	T	F	F	T	T	T
T	F	T	T	T	F	T
T	F	F	T	T	T	T
F	T	T	F	T	F	T
F	T	F	F	T	T	T
F	F	T	T	F	F	F
F	F	F	T	F	T	T

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Truth Tables",
"Logical Operators",
"Conditional Statements",
"Truth Table Construction"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Conditional Statements",
"Truth Table Construction"
]
}
</pre_analysis>

<reasoning>

Evaluate negations of q and r

\[

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

\]

Evaluate conditional statement

\[

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

\]

Evaluate final disjunction

\[

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

\]
</reasoning>

<answer>

\(p\)

\(q\)

\(r\)

\(\sim q\)

\(\sim q

ightarrow p\) | \(\sim r\) | \((\sim q
ightarrow p) \lor \sim r\) |

T	T	T	F	T	F	T
T	T	F	F	T	T	T
T	F	T	T	T	F	T
T	F	F	T	T	T	T
F	T	T	F	T	F	T
F	T	F	F	T	T	T
F	F	T	T	F	F	F
F	F	F	T	F	T	T

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>