Question

determine whether the compound statement

\\(p \wedge q) \
ightarrow p\\

is a tautology, self-contradiction, or neither of these by analyzing the statement in a truth table.

first, complete the truth table:

\

$$\begin{tabular}{|c|c|c|c|} \\hline \\(p\\) & \\(q\\) & \\(p \\wedge q\\) & \\((p \\wedge q) \ ightarrow p\\) \\\\ \\hline t & t & & \\\\ \\hline t & f & & \\\\ \\hline f & t & & \\\\ \\hline f & f & & \\\\ \\hline \\end{tabular}$$

what type of statement is \\((p \wedge q) \
ightarrow p\\)?

• tautology
• self-contradiction
• neither of these

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"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",
"Tautology"
]
}
</pre_analysis>

<reasoning>

Evaluate the conjunction column \(p \land q\)

\[

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

\]

Evaluate the conditional column \((p \land q)

ightarrow p\)
\[

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

\]

Classify the compound statement

\[

$$\begin{aligned} &\text{All truth values in the final column are True (T).} \\ &\text{Therefore, the statement is a tautology.} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(p\)

\(q\)

\(p \land q\)

\((p \land q)

ightarrow p\) |

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

Question 2

<mcq-correct>tautology</mcq-correct>
<mcq-option>self-contradiction</mcq-option>
<mcq-option>neither of these</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 2,
"skills_matched": [
"step_cot",
"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",
"Tautology"
]
}
</pre_analysis>

<reasoning>

Evaluate the conjunction column \(p \land q\)

\[

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

\]

Evaluate the conditional column \((p \land q)

ightarrow p\)
\[

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

\]

Classify the compound statement

\[

$$\begin{aligned} &\text{All truth values in the final column are True (T).} \\ &\text{Therefore, the statement is a tautology.} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(p\)

\(q\)

\(p \land q\)

\((p \land q)

ightarrow p\) |

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

Question 2

<mcq-correct>tautology</mcq-correct>
<mcq-option>self-contradiction</mcq-option>
<mcq-option>neither of these</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Truth Tables"
]
}
</post_analysis>