4. Using only the and the ^ operators, find a logical expression that is equivalent 5. Prove that the expression you found for question 4 above is equivalent to Gpv-q) r by using logical equivalences. Show all your work. 6. Prove that the expression you found for question 4 above is equivalent to GpV-a) r by using truth tables. Show all your work. 7. Determine if the following expressions are tautologies, contradictions, or contingencies by using truth tables. Show all your work. a. (png) (pV4) 8. Let P(x) be the predicate "x is a penguin", B(x) be the predicate "x is a bird", and S(x) be the predicate "x eats squid. Translate the following expressions into English. The universe of discourse is all animals. 3x(P(x)^-B(x)) a. 9. Negate the following predicate logic statements using the quantifier negation rules discussed in class. Show all your work and ensure that no negation operations appear before any of the quantifiers in the expression you create. a, Every penguin looks good in black