Translate each sentence into propositional logic using the propositional variables given after the sentence.
Show the work, not just the answer
The restaurant served chicken, and either peas or carrots. (C, P, T)
Making butter is a necessary condition for the farmer to go to the market and make a profit. (B, M, P)
Patrons may have corn and potatoes if, and only if, they do not order carrots. (C, P, T)
If the restaurant runs out of cheesecake, then you can have a mea n of chicken and pie and ice cream. (G, S, D, C)
A farmer keeps goats in a pen and sheep in a pen only if the dogs and cat are kept inside. (G, S, D, C)
Either the farmer shears the sheep and milks the cows, or he slops the pigs and walks the dog. (S, C, P, D)
If the farmer shears the sheep, then he makes the wool, and if he milks the cows, then he makes the butter. (S, W, C, B)
If the farmer goes to the marker, then she makes a profit, and her wife is happy. (M, P, W)
Plato believed in the theory of forms, and Aristotle held that there are four kinds of causes, but Parmenides thought that only the one exists. (P, A, R)
If Thales reduced everything to water, then Democritus was an atomist if and only if Heraclitus claimed that the world is constantly in flux. (T, D, H)
If Plato believed in the theory of forms or Democritus was an atomist, then Aristotle help that there are four kinds of causes or Parmenides thought that only the one exists. (P, D, A, R)

Translate the following statements into symbolic form using uppercase letters to represent affirmative English statements. Example: Suppose you are given the statement “If Facebook makes us narcissistic, then either Twitter or LinkedIn relieves our loneliness.” This would be translated F ⊃
11.(Bill Gates does not support malaria research unless Warren Buffet does.
12. Mercedes will introduce a hybrid model only if Lexus and BMW do.
13. Mariah Carey sings pop and either Elton John sings rock or Diana Krall sings jazz.
14. Either Mariah Carey sings pop and Elton John sings rock or Diana Krall sings jazz.
\15. Not both Jaguar and Porsche make motorcycles.
16. Both Jaguar and Porsche do not make motorcycles.
17. Either Apple or Samsung makes smartphones.
18. Not either Ferrari or Maserati makes economy cars.
19. Neither Ferrari nor Maserati makes economy cars.
20. Either Ferrari or Maserati does not make economy cars(no subject)
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Determine the truth values of the following symbolized statements. Let A, B, and C be true and X, Y, and Z be false. Circle your answer.
1. A● X ●
2. B ∼Y
3. X¡∼Y
4. ∼C ¡ Z
5. B⊃∼Z
6. Y⊃∼A
7. ∼X⊃Z
8. BKY
9. ∼CKZ
10. ∼(A ∼Z)
11. ∼B¡●(Y⊃A)
12. A⊃∼(Z¡∼Y)
13.(A Y)¡(∼Z C) ●
14. ∼(X¡∼B) (∼Y¡A) ●
●●
15. (Y⊃C) ∼(B⊃∼X)

IV. When possible, determine the truth values of the following symbolized statements. Let A and B be true, Y and Z false. P and Q have unknown truth values. If the truth value of the statement cannot be determined, write “undetermined.”
1. A¡P
2. Q¡Z
3. Q● Y
4. Q● A
5. P⊃B
6. Z⊃Q
7. A⊃P
8. PK∼P
9. (P⊃A)⊃Z
10. (P⊃A)K(Q⊃B)

Determine the truth values of the following symbolized statements. Let A, B, and C be true and X, Y, and Z be false. Circle your answer.
1. A● X ●
2. B ∼Y
3. X¡∼Y
4. ∼C ¡ Z
5. B⊃∼Z
6. Y⊃∼A
7. ∼X⊃Z
8. BKY
9. ∼CKZ
10. ∼(A ∼Z)
11. ∼B¡●(Y⊃A)
12. A⊃∼(Z¡∼Y)
13.(A Y)¡(∼Z C) ●
14. ∼(X¡∼B) (∼Y¡A) ●
●●
15. (Y⊃C) ∼(B⊃∼X)