Given that statement p is false, determine whether the following statements are true or false.
Understand the Problem
The question involves logical statements and asks to determine the truth of several statements given that statement p is false. It appears to relate to propositional logic.
Answer
1. Indeterminate 2. True 3. Indeterminate 4. True 5. True
Answer for screen readers
- Indeterminate (depends on $q$)
- True when $q = F$
- Indeterminate (depends on $q$)
- True
- True
Steps to Solve
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Identify the truth values Given that statement $p$ is false, denote it as $p = F$. We need to evaluate each logical statement under this condition.
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Analyze statement 1: $\sim p \land q$ Since $p = F$, then $\sim p = T$. However, the truth of this statement depends on $q$. Thus, we cannot determine its overall truth without knowing the value of $q$.
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Analyze statement 2: $\sim (p \lor q) \iff (\sim p \land \sim q)$ We have $\sim p = T$. So, it reduces to: $$ \sim (F \lor q) \iff (T \land \sim q) $$ This simplifies to: $$ \sim q \iff (T \land \sim q) $$ This statement can be true if $q = F$ (making both sides true).
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Analyze statement 3: $\sim (p \lor q) \land (\sim p \lor r)$ With $\sim p = T$, we rewrite the expression: $$ \sim (F \lor q) \land (T \lor r) $$ This reduces to: $$ \sim q \land (T \lor r) $$ The truth of this statement depends on $q$.
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Analyze statement 4: $\sim p \lor q$ Here we have: $$ T \lor q $$ This is always true regardless of $q$.
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Analyze statement 5: $\sim p \lor \sim (q \land r)$ This simplifies to: $$ T \lor \sim (q \land r) $$ This is also always true regardless of $q$ or $r$.
- Indeterminate (depends on $q$)
- True when $q = F$
- Indeterminate (depends on $q$)
- True
- True
More Information
In propositional logic, the truth values of statements depend greatly on the variables involved and their interactions through logical operations. Establishing the truth of each compound statement often requires further information about individual statements, in this case $q$ and $r$.
Tips
- Misinterpreting the truth of compound statements without examining all variables involved.
- Overlooking that a single false variable can affect an entire statement composed of logical operations.
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