We define the following predicates: L(x, y)=x likes y F(x, y)=x is friends with y where...

Transcribed Image Text:

We define the following predicates: L(x, y)=x likes y F(x, y)=x is friends with y where the domain for x and y is all students in CS 2800. For each of the English statements in (a) and (b), translate them into quantified state- ments using the proper quantifiers. For (c) and (d), select the answer that best matches the description. Make sure to justify your answer. 2 (a) There is a student that everyone likes (including themselves). How does your answer change if you want to express there is a student that everyone (excluding themselves) likes? (b) There is a student that is friends with everyone that they like. (c) For any two distinct students, there is some student that they both like. i. Vavy(xy L(x, z) AL(y, z)) ii. Vavy(xyz(L(x, z) ^ L(y, z))) iii. VxVy(xyz (L(x, z) ^ L(y, z))) iv. 3zVaVy(x + y^ (L(x, z) ^ L(y, z))) v. 2VxVy((L(x, z) ^ L(y, z)) ⇒ xy) (d) xVy(F(x, y) ⇒ (Vz(F(y, z) ⇒ L(x, z)))) i. Some student is friends with everyone their friends like. ii. Some student is liked by all of their friends' friends. iii. Some student likes everyone that their friends like. iv. Some student has a friend that is friends with everyone the student likes. v. Some student likes all of their friends' friends. We define the following predicates: L(x, y)=x likes y F(x, y)=x is friends with y where the domain for x and y is all students in CS 2800. For each of the English statements in (a) and (b), translate them into quantified state- ments using the proper quantifiers. For (c) and (d), select the answer that best matches the description. Make sure to justify your answer. 2 (a) There is a student that everyone likes (including themselves). How does your answer change if you want to express there is a student that everyone (excluding themselves) likes? (b) There is a student that is friends with everyone that they like. (c) For any two distinct students, there is some student that they both like. i. Vavy(xy L(x, z) AL(y, z)) ii. Vavy(xyz(L(x, z) ^ L(y, z))) iii. VxVy(xyz (L(x, z) ^ L(y, z))) iv. 3zVaVy(x + y^ (L(x, z) ^ L(y, z))) v. 2VxVy((L(x, z) ^ L(y, z)) ⇒ xy) (d) xVy(F(x, y) ⇒ (Vz(F(y, z) ⇒ L(x, z)))) i. Some student is friends with everyone their friends like. ii. Some student is liked by all of their friends' friends. iii. Some student likes everyone that their friends like. iv. Some student has a friend that is friends with everyone the student likes. v. Some student likes all of their friends' friends.