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homework section 2.2: introduction to truth tables score: 12/15 answere…

Question

homework section 2.2: introduction to truth tables
score: 12/15 answered: 12/15
question 13
complete the truth table for the statement \\(a \land \sim(b \lor c)\\).
you will probably want to complete the intermediate columns on your own paper and then enter the final answer in the truth table.

\

$$\begin{tabular}{|c|c|c|c|} \\hline a & b & c & \\(a \\land \\sim(b \\lor c)\\) \\\\ \\hline t & t & t & ? \\\\ \\hline t & t & f & ? \\\\ \\hline t & f & t & ? \\\\ \\hline t & f & f & ? \\\\ \\hline f & t & t & ? \\\\ \\hline f & t & f & ? \\\\ \\hline f & f & t & ? \\\\ \\hline f & f & f & ? \\\\ \\hline \\end{tabular}$$

Explanation:

Response

<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",
"Truth Table Construction"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Truth Table Construction"
]
}
</pre_analysis>

<reasoning>

Evaluate the sub-expression \(B \lor C\) and its negation \(

eg(B \lor C)\)
\[

$$\begin{array}{ccc|c|c} A & B & C & B \lor C & eg(B \lor C) \\ \hline T & T & T & T & F \\ T & T & F & T & F \\ T & F & T & T & F \\ T & F & F & F & T \\ F & T & T & T & F \\ F & T & F & T & F \\ F & F & T & T & F \\ F & F & F & F & T \end{array}$$

\]

Evaluate the conjunction \(A \land

eg(B \lor C)\)
\[

$$\begin{array}{ccc|c|c} A & B & C & eg(B \lor C) & A \land eg(B \lor C) \\ \hline T & T & T & F & F \\ T & T & F & F & F \\ T & F & T & F & F \\ T & F & F & T & T \\ F & T & T & F & F \\ F & T & F & F & F \\ F & F & T & F & F \\ F & F & F & T & F \end{array}$$

\]
</reasoning>

<answer>

\(A\)\(B\)\(C\)\(A \land

eg(B \lor C)\) |

TTTF
TTFF
TFTF
TFFT
FTTF
FTFF
FFTF
FFFF

</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",
"Truth Table Construction"
],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Truth Table Construction"
]
}
</pre_analysis>

<reasoning>

Evaluate the sub-expression \(B \lor C\) and its negation \(

eg(B \lor C)\)
\[

$$\begin{array}{ccc|c|c} A & B & C & B \lor C & eg(B \lor C) \\ \hline T & T & T & T & F \\ T & T & F & T & F \\ T & F & T & T & F \\ T & F & F & F & T \\ F & T & T & T & F \\ F & T & F & T & F \\ F & F & T & T & F \\ F & F & F & F & T \end{array}$$

\]

Evaluate the conjunction \(A \land

eg(B \lor C)\)
\[

$$\begin{array}{ccc|c|c} A & B & C & eg(B \lor C) & A \land eg(B \lor C) \\ \hline T & T & T & F & F \\ T & T & F & F & F \\ T & F & T & F & F \\ T & F & F & T & T \\ F & T & T & F & F \\ F & T & F & F & F \\ F & F & T & F & F \\ F & F & F & T & F \end{array}$$

\]
</reasoning>

<answer>

\(A\)\(B\)\(C\)\(A \land

eg(B \lor C)\) |

TTTF
TTFF
TFTF
TFFT
FTTF
FTFF
FFTF
FFFF

</answer>

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