7. There is a spaceship where every passenger has exactly one role. Each passenger can either be a Crewmate or an Imposter. A Crewmate always tells the truth, and an Imposter always lies. For each question, determine the role of Person A and the role of Person B, or write “Cannot be determined” for that person if there is not enough information. Explain your reasoning for full credit! (You can use a truth table or just plain English to explain.) 

a) Person A says “I am a Crewmate, or B is a Crewmate,” and Person B says “A is a Crewmate if I am an Imposter.”
b) Person A says “I am an Imposter, and Person B is a Crewmate,” and Person B says nothing.
c) Person A says “Both Person B and I are Imposters,” and Person B says “At least one of us is a Crewmate.”

8. Show that (p →q) ∨¬p ≡p →q using logical equivalences.
Cite the laws of equivalences used to reach each step. 

9. Show that ¬((p ∧q) ∨p) →¬p is a tautology using:
For part a, include all intermediate columns.
For part b, cite the laws of equivalence used to reach each step.
a) a truth table 
b) logical equivalences

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