= O 0RO Gl M Provide detailed explanations and justifications for each answer. e Include all truth tables and steps in your work. If an invalid argument has all true premises, then the conclusion must be false. True or false? Explain your answer. Provide argument(s) and logical expression to justify your answer. Prove whether the following pairs of logical expressions are equivalent or not. For each pair, construct a truth table and analyze the results: AY(PAQ)V(-PAQ)and Q B) (P Q) Rand P VR 3. You are designing a security system for a building with the following requirements: The alarm should sound if the door is opened without the correct code being entered. The alarm should not sound if the correct code is entered, and the door is opened. e If the door remains closed, the alarm should not sound, regardless of the code. A) Translate these requirements into a logical expression using P (door is opened) and Q (correct code is entered). B) Construct a truth table for your expression and identify any potential issues with the system. Define mutually exclusive events in the context of logic. Provide two real-world examples of mutually exclusive events and translate them into logical expressions. OO0V Qe[ 5. Using the XOR operator, construct a logical expression that captures the following scenario: e "A light bulb will turn on if exactly one of the two switches is flipped, but not both." A) Construct a truth table for the expression. B) Analyze the truth table and explain how the XOR operator captures the exclusivity of the events. 6. Read the arguments below and write down which ones are valid, and which are not. Then translate each of them into formal logic by replacing the statements with abstract symbols A and B and make a truth table. A) If I press the gas pedal, the car will accelerate. But I didn't press the gas pedal, so the car will not accelerate. B) IfIread the news, I will not be misinformed. But I was misinformed, so I did not read the news. C) Ifthe moon is made of cheese, it will taste like cheese. But the moon is not made of cheese, so it will not taste like cheese. 7. Construct a logical expression that represents the following scenario: o "Ifit rains or it snows, then the game will be canceled. If the game is not canceled, then the team will practice. The team will only practice if it is not snowing." A) Express the scenario using logical connectives. B) Construct a truth table for your logical expression. 8. Explain the role of truth tables in verifying the validity of logical arguments. Include an example to support your discussion. 9. Identify the denial of the antecedent from the following choices. A) (A-B)A-B) A B) (A-B)AA)B C) (A-B)AA) B D) (AB)AB) A 9. o F Identify the denial of the antecedent from the following choices. A) (AB)AB) A B) (AB)AA)B C) (A-B)AA) B D) (AB)AB) A .In \"((B) A (A B)) (A)\