Question

a. complete the truth table for the compound statement: \\(p \lor \sim q\\)

\

$$\begin{array}{|c|c|c|c|} \\hline p & q & \\sim q & p \\lor \\sim q \\\\ \\hline t & t & ? & ? \\\\ \\hline t & f & ? & ? \\\\ \\hline f & t & ? & ? \\\\ \\hline f & f & ? & ? \\\\ \\hline \\end{array}$$

b. complete the truth table for the compound statement: \\(\sim p \land \sim q\\)

\

$$\begin{array}{|c|c|c|c|c|} \\hline p & q & \\sim p & \\sim q & \\sim p \\land \\sim q \\\\ \\hline t & t & ? & ? & ? \\\\ \\hline t & f & ? & ? & ? \\\\ \\hline f & t & ? & ? & ? \\\\ \\hline f & f & ? & ? & ? \\\\ \\hline \\end{array}$$

Explanation:

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<reasoning>

Evaluate negation columns

\[

$$\begin{aligned} &\sim p: \text{negates } p \implies \text{maps } (T, T, F, F) \to (F, F, T, T)\\ &\sim q: \text{negates } q \implies \text{maps } (T, F, T, F) \to (F, T, F, T) \end{aligned}$$

\]

Evaluate disjunction column for part a

\[

$$\begin{aligned} &p \lor \sim q \text{ is False only when both } p \text{ and } \sim q \text{ are False:}\\ &\text{Row 1: } T \lor F = T\\ &\text{Row 2: } T \lor T = T\\ &\text{Row 3: } F \lor F = F\\ &\text{Row 4: } F \lor T = T \end{aligned}$$

\]

Evaluate conjunction column for part b

\[

$$\begin{aligned} &\sim p \land \sim q \text{ is True only when both } \sim p \text{ and } \sim q \text{ are True:}\\ &\text{Row 1: } F \land F = F\\ &\text{Row 2: } F \land T = F\\ &\text{Row 3: } T \land F = F\\ &\text{Row 4: } T \land T = T \end{aligned}$$

\]
</reasoning>

<answer>

Question a

\(p\)	\(q\)	\(\sim q\)	\(p \lor \sim q\)
T	F	T	T
F	T	F	F
F	F	T	T

Question b

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim p \land \sim q\)
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

</answer>

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

Answer:

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<reasoning>

Evaluate negation columns

\[

$$\begin{aligned} &\sim p: \text{negates } p \implies \text{maps } (T, T, F, F) \to (F, F, T, T)\\ &\sim q: \text{negates } q \implies \text{maps } (T, F, T, F) \to (F, T, F, T) \end{aligned}$$

\]

Evaluate disjunction column for part a

\[

$$\begin{aligned} &p \lor \sim q \text{ is False only when both } p \text{ and } \sim q \text{ are False:}\\ &\text{Row 1: } T \lor F = T\\ &\text{Row 2: } T \lor T = T\\ &\text{Row 3: } F \lor F = F\\ &\text{Row 4: } F \lor T = T \end{aligned}$$

\]

Evaluate conjunction column for part b

\[

$$\begin{aligned} &\sim p \land \sim q \text{ is True only when both } \sim p \text{ and } \sim q \text{ are True:}\\ &\text{Row 1: } F \land F = F\\ &\text{Row 2: } F \land T = F\\ &\text{Row 3: } T \land F = F\\ &\text{Row 4: } T \land T = T \end{aligned}$$

\]
</reasoning>

<answer>

Question a

\(p\)	\(q\)	\(\sim q\)	\(p \lor \sim q\)
T	F	T	T
F	T	F	F
F	F	T	T

Question b

\(p\)	\(q\)	\(\sim p\)	\(\sim q\)	\(\sim p \land \sim q\)
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

</answer>

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