Questions: Is the argument valid?

Is the argument valid?
Transcript text: Is the argument valid?
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Solution

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Solution Steps

Step 1: Identify the logical statements
  • The given truth table includes two propositions \( p \) and \( q \).
  • The table shows the truth values for \( p \), \( q \), \( \neg p \), \( \neg q \), \( p \land q \), \( p \lor q \), \( \neg p \lor \neg q \), and \( (p \land q) \rightarrow \neg (p \lor q) \).
Step 2: Analyze the truth table
  • The truth table provides the truth values for each combination of \( p \) and \( q \):
    • \( p \) and \( q \) both true (T, T)
    • \( p \) true and \( q \) false (T, F)
    • \( p \) false and \( q \) true (F, T)
    • \( p \) and \( q \) both false (F, F)
Step 3: Determine the validity of the argument
  • To determine if the argument is valid, we need to check if the conditional statement \( (p \land q) \rightarrow \neg (p \lor q) \) is true in all cases.
  • From the truth table:
    • When \( p \) and \( q \) are both true, \( (p \land q) \) is true and \( \neg (p \lor q) \) is false, making the conditional false.
    • In all other cases, the conditional is true.

Final Answer

  • The argument is not valid because the conditional statement \( (p \land q) \rightarrow \neg (p \lor q) \) is not true in all cases.
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