High School

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Consider the following statements:

- P: "If the alarm rings, then the security guard will arrive."
- Q: "The security guard has arrived."
- R: "The alarm did not ring."

Determine the truth values of P, Q, and R under different scenarios. Additionally, evaluate the truth value of the compound statement \((P \land \neg R) \lor (Q \land \neg P)\).

Answer :

The evaluations show that the compound statement (P∧¬R)∨(Q∧¬P) is true only in Scenario 1, and false in the other scenarios.

Let's analyze the truth values of the statements P, Q, and R under different scenarios and then evaluate the compound statement [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex].

Statements:

  1. P: "If the alarm rings, then the security guard will arrive."
  2. Q: "The security guard has arrived."
  3. R: "The alarm did not ring."

Scenarios and Truth Values:

1. Scenario 1:

The alarm rings, and the security guard arrives.

  • P: True (because the security guard arrived when the alarm rang)
  • Q: True
  • R: False (because the alarm rang)

2. Scenario 2:

The alarm rings, but the security guard does not arrive.

  • P: False (because the security guard did not arrive when the alarm rang)
  • Q: False
  • R: False (because the alarm rang)

3. Scenario 3:

The alarm does not ring, and the security guard arrives.

  • P: True (because P is vacuously true when the alarm does not ring)
  • Q: True
  • R: True (because the alarm did not ring)

4. Scenario 4:

The alarm does not ring, and the security guard does not arrive.

  • P: True (because P is vacuously true when the alarm does not ring)
  • Q: False
  • R: True (because the alarm did not ring)

Evaluating the Compound Statement [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex]:

1. Scenario 1:

  • P: True
  • Q: True
  • R: False
  • [tex]\( \neg R \)[/tex]: True
  • [tex]\( P \land \neg R \)[/tex]: True
  • [tex]\( \neg P \)[/tex]: False
  • [tex]\( Q \land \neg P \)[/tex]: False
  • [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex]: True

2. Scenario 2:

  • P: False
  • Q: False
  • R: False
  • [tex]\( \neg R \)[/tex]: True
  • [tex]\( P \land \neg R \)[/tex]: False
  • [tex]\( \neg P \)[/tex]: True
  • [tex]\( Q \land \neg P \)[/tex]: False
  • [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex]: False

3. Scenario 3:

  • P: True
  • Q: True
  • R: True
  • [tex]\( \neg R \)[/tex]: False
  • [tex]\( P \land \neg R \)[/tex]: False
  • [tex]\( \neg P \)[/tex]: False
  • [tex]\( Q \land \neg P \)[/tex]: False
  • [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex]: False

4. Scenario 4:

  • P: True
  • Q: False
  • R: True
  • [tex]\( \neg R \)[/tex]: False
  • [tex]\( P \land \neg R \)[/tex]: False
  • [tex]\( \neg P \)[/tex]: False
  • [tex]\( Q \land \neg P \)[/tex]: False
  • [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex]: False

Evaluation:

Scenario 1: [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex] is True.

Scenario 2: [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex] is False.

Scenario 3: [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex] is False.

Scenario 4: [tex]\( (P \land \neg R) \lor (Q \land \neg P) \)[/tex] is False.

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