Question

determine whether the compound statement is a tautology, a self-contradiction, or neither of these.

build a truth table to answer this question.

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

this is a tautology.
this is a self-contradiction.
this is neither a tautology nor a self-contradiction.

Explanation:

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

<reasoning>

Construct the truth table for the sub-expressions

\[

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

\]

Evaluate the disjunction of the sub-expressions

\[

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

\]

Classify the compound statement

\[
\text{Since all truth values in the final column are True (T), the statement is a tautology.}
\]
</reasoning>

<answer>
<mcq-correct>This is a tautology.</mcq-correct>
<mcq-option>This is a self-contradiction.</mcq-option>
<mcq-option>This is neither a tautology nor a self-contradiction.</mcq-option>
</answer>

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

Answer:

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

<reasoning>

Construct the truth table for the sub-expressions

\[

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

\]

Evaluate the disjunction of the sub-expressions

\[

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

\]

Classify the compound statement

\[
\text{Since all truth values in the final column are True (T), the statement is a tautology.}
\]
</reasoning>

<answer>
<mcq-correct>This is a tautology.</mcq-correct>
<mcq-option>This is a self-contradiction.</mcq-option>
<mcq-option>This is neither a tautology nor a self-contradiction.</mcq-option>
</answer>

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