Question

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

\

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

Explanation:

<pre_analysis>
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"step_cot",
"table_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Conditional Statements"
]
}
</pre_analysis>

<reasoning>

Evaluate the disjunction column \(p \lor q\)

\[

$$\begin{aligned} &\text{Row 1: } \text{T} \lor \text{T} \equiv \text{T} \\ &\text{Row 2: } \text{T} \lor \text{F} \equiv \text{T} \\ &\text{Row 3: } \text{F} \lor \text{T} \equiv \text{T} \\ &\text{Row 4: } \text{F} \lor \text{F} \equiv \text{F} \end{aligned}$$

\]

Evaluate the conditional column \(\sim q

ightarrow (p \lor q)\)
\[

$$\begin{aligned} &\text{Row 1: } \text{F} ightarrow \text{T} \equiv \text{T} \\ &\text{Row 2: } \text{T} ightarrow \text{T} \equiv \text{T} \\ &\text{Row 3: } \text{F} ightarrow \text{T} \equiv \text{T} \\ &\text{Row 4: } \text{T} ightarrow \text{F} \equiv \text{F} \end{aligned}$$

\]
</reasoning>

<answer>

\(p\)

\(q\)

\(\sim q\)

\(p \lor q\)

\(\sim q

ightarrow (p \lor q)\) |

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

</answer>

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

Answer:

<pre_analysis>
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"step_cot",
"table_formatter",
"explain_then_answer",
"quick"
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"concepts_used": [],
"new_concepts": [],
"current_concepts": [
"Truth Tables",
"Logical Operators",
"Conditional Statements"
]
}
</pre_analysis>

<reasoning>

Evaluate the disjunction column \(p \lor q\)

\[

$$\begin{aligned} &\text{Row 1: } \text{T} \lor \text{T} \equiv \text{T} \\ &\text{Row 2: } \text{T} \lor \text{F} \equiv \text{T} \\ &\text{Row 3: } \text{F} \lor \text{T} \equiv \text{T} \\ &\text{Row 4: } \text{F} \lor \text{F} \equiv \text{F} \end{aligned}$$

\]

Evaluate the conditional column \(\sim q

ightarrow (p \lor q)\)
\[

$$\begin{aligned} &\text{Row 1: } \text{F} ightarrow \text{T} \equiv \text{T} \\ &\text{Row 2: } \text{T} ightarrow \text{T} \equiv \text{T} \\ &\text{Row 3: } \text{F} ightarrow \text{T} \equiv \text{T} \\ &\text{Row 4: } \text{T} ightarrow \text{F} \equiv \text{F} \end{aligned}$$

\]
</reasoning>

<answer>

\(p\)

\(q\)

\(\sim q\)

\(p \lor q\)

\(\sim q

ightarrow (p \lor q)\) |

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

</answer>

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