Question

use the equivalency

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

to select the statement that is logically equivalent to ...

\it is not true that this course covers logic and it covers finance.\

this course covers finance, or this course does not cover logic.
this course covers logic, or this course does not cover finance.
this course does not cover logic, and this course does not cover finance.
this course does not cover logic, or this course does not cover finance.
this course does not cover logic, and this course covers finance.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"De Morgans Laws",
"Logical Statement"
],
"new_concepts": [],
"current_concepts": [
"De Morgans Laws",
"Logical Statement"
]
}
</pre_analysis>

<reasoning>

Define the propositional variables

\[

$$\begin{aligned} p&: \text{"this course covers logic"} \\ q&: \text{"it covers finance"} \end{aligned}$$

\]

Translate the given statement into symbolic form

\[
\sim(p \wedge q)
\]

Apply De Morgan's Law to find the equivalent statement

\[

$$\begin{aligned} \sim(p \wedge q) &\equiv \sim p \vee \sim q \\ \sim p \vee \sim q&: \text{"This course does not cover logic, or this course does not cover finance."} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>This course covers finance, or this course does not cover logic.</mcq-option>
<mcq-option>This course covers logic, or this course does not cover finance.</mcq-option>
<mcq-option>This course does not cover logic, and this course does not cover finance.</mcq-option>
<mcq-correct>This course does not cover logic, or this course does not cover finance.</mcq-correct>
<mcq-option>This course does not cover logic, and this course covers finance.</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"De Morgans Laws"
]
}
</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": [
"De Morgans Laws",
"Logical Statement"
],
"new_concepts": [],
"current_concepts": [
"De Morgans Laws",
"Logical Statement"
]
}
</pre_analysis>

<reasoning>

Define the propositional variables

\[

$$\begin{aligned} p&: \text{"this course covers logic"} \\ q&: \text{"it covers finance"} \end{aligned}$$

\]

Translate the given statement into symbolic form

\[
\sim(p \wedge q)
\]

Apply De Morgan's Law to find the equivalent statement

\[

$$\begin{aligned} \sim(p \wedge q) &\equiv \sim p \vee \sim q \\ \sim p \vee \sim q&: \text{"This course does not cover logic, or this course does not cover finance."} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>This course covers finance, or this course does not cover logic.</mcq-option>
<mcq-option>This course covers logic, or this course does not cover finance.</mcq-option>
<mcq-option>This course does not cover logic, and this course does not cover finance.</mcq-option>
<mcq-correct>This course does not cover logic, or this course does not cover finance.</mcq-correct>
<mcq-option>This course does not cover logic, and this course covers finance.</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"De Morgans Laws"
]
}
</post_analysis>