Question

complete the truth table for \\(\sim p \
ightarrow \sim q\\).

\\(\

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

\\)

Explanation:

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

<reasoning>

Determine the truth values for \(\sim p\) and \(\sim q\)

\[

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

\]

Evaluate the conditional statement \(\sim p

ightarrow \sim q\)
\[

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

\]

Construct the complete truth table

\[

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

\]
</reasoning>

<answer>

\(p\)

\(q\)

\(\sim p\)

\(\sim q\)

\(\sim p

ightarrow \sim q\) |

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

</answer>

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

Answer:

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

<reasoning>

Determine the truth values for \(\sim p\) and \(\sim q\)

\[

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

\]

Evaluate the conditional statement \(\sim p

ightarrow \sim q\)
\[

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

\]

Construct the complete truth table

\[

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

\]
</reasoning>

<answer>

\(p\)

\(q\)

\(\sim p\)

\(\sim q\)

\(\sim p

ightarrow \sim q\) |

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

</answer>

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