Question

complete the truth table for the statement \\((a \land \sim b) \
ightarrow c\\).

\

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

Explanation:

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

<reasoning>

Evaluate the sub-expression \(A \land \sim B\)

\[

$$\begin{array}{ccc|c} A & B & \sim B & A \land \sim B \\ \hline \text{T} & \text{T} & \text{F} & \text{F} \\ \text{T} & \text{T} & \text{F} & \text{F} \\ \text{T} & \text{F} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{F} & \text{F} \\ \text{F} & \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{T} & \text{F} \end{array}$$

\]

Evaluate the conditional statement \((A \land \sim B)

ightarrow C\)
\[

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

\]
</reasoning>

<answer>

\(A\)

\(B\)

\(C\)

\((A \land \sim B)

ightarrow C\) |

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

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

<reasoning>

Evaluate the sub-expression \(A \land \sim B\)

\[

$$\begin{array}{ccc|c} A & B & \sim B & A \land \sim B \\ \hline \text{T} & \text{T} & \text{F} & \text{F} \\ \text{T} & \text{T} & \text{F} & \text{F} \\ \text{T} & \text{F} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{F} & \text{F} \\ \text{F} & \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{T} & \text{F} \end{array}$$

\]

Evaluate the conditional statement \((A \land \sim B)

ightarrow C\)
\[

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

\]
</reasoning>

<answer>

\(A\)

\(B\)

\(C\)

\((A \land \sim B)

ightarrow C\) |

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

</answer>

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