Question

for the conditional statement \\(t \
ightarrow s\\), identify the inverse, converse, contrapositive, and original statement in symbols.

• \\(t \

ightarrow s\\)

• \\(s \

ightarrow t\\)

• \\(\sim t \

ightarrow \sim s\\)

• \\(\sim s \

ightarrow \sim t\\)

Explanation:

<pre_analysis>
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"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Conditional Statements"
],
"new_concepts": [
"Converse Inverse Contrapositive"
],
"current_concepts": [
"Conditional Statements",
"Converse Inverse Contrapositive"
]
}
</pre_analysis>

<reasoning>

Identify the original statement

\[
t
ightarrow s
\]

Identify the converse and inverse

\[

$$\begin{aligned} &\text{Converse: } s ightarrow t\\ &\text{Inverse: } \sim t ightarrow\ \sim s \end{aligned}$$

\]

Identify the contrapositive

\[
\sim s
ightarrow\ \sim t
\]
</reasoning>

<answer>
For the conditional statement \(t
ightarrow s\), identify the inverse, converse, contrapositive, and original statement in symbols.

• <blank>Original statement</blank> \(t

ightarrow s\)

• <blank>Converse</blank> \(s

ightarrow t\)

• <blank>Inverse</blank> \(\sim t

ightarrow\ \sim s\)

• <blank>Contrapositive</blank> \(\sim s

ightarrow\ \sim t\)
</answer>

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

Answer:

<pre_analysis>
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"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Conditional Statements"
],
"new_concepts": [
"Converse Inverse Contrapositive"
],
"current_concepts": [
"Conditional Statements",
"Converse Inverse Contrapositive"
]
}
</pre_analysis>

<reasoning>

Identify the original statement

\[
t
ightarrow s
\]

Identify the converse and inverse

\[

$$\begin{aligned} &\text{Converse: } s ightarrow t\\ &\text{Inverse: } \sim t ightarrow\ \sim s \end{aligned}$$

\]

Identify the contrapositive

\[
\sim s
ightarrow\ \sim t
\]
</reasoning>

<answer>
For the conditional statement \(t
ightarrow s\), identify the inverse, converse, contrapositive, and original statement in symbols.

• <blank>Original statement</blank> \(t

ightarrow s\)

• <blank>Converse</blank> \(s

ightarrow t\)

• <blank>Inverse</blank> \(\sim t

ightarrow\ \sim s\)

• <blank>Contrapositive</blank> \(\sim s

ightarrow\ \sim t\)
</answer>

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