COS2661 Assignment 3 guidelines and answers 2023

Question 1

Question 1.1:

∀x (Brown(x) ∧ dog(x) → hairy(x))



Question 1.2:

∃x (Corgi(x) ∧ ∀y (Loves(x, y)))



Question 1.3:

∀x ∃y (Corgi(y) ∧ Loves(x, y))



Question 1.4:

∀x ∀y (Pancakes(x) ∧ Pancakes(y) → TasteSimilar(x, y))

, Question 1.5:

∀x ∃y,z (Person(x) ∧ Knows(x, y) ∧ Knows(x, z) ∧ y ≠ z)



Question 2

Question 2.1:

There exists an actor who does not play in "Days and Years" and does not watch "Days and Years".



Question 2.2:

Faro plays in "Fourth Street" or Faro plays in "Enemies or not?", but not both.



Question 2.3:

If "Fourth Street" is not a soap opera and "Days and Years" is a soap opera, then Faro or Rudo
watches "Enemies or not?".



Question 2.4:

For every actor, if they play in "Enemies or not?" and do not watch "Enemies or not?", then they play
in "Fourth Street".



Question 2.5:

Faro watches "Fourth Street" and Rudo watches "Fourth Street" if and only if Faro plays in "Fourth
Street" and Rudo plays in "Fourth Street".



Question 3

To show that the given expressions are logically equivalent, we can apply De Morgan's laws:



De Morgan's laws state:

1) ¬(P ∧ Q) ≡ ¬P ∨ ¬Q (De Morgan's law for conjunction)

2) ¬(P ∨ Q) ≡ ¬P ∧ ¬Q (De Morgan's law for disjunction)



Using De Morgan's laws, we can rewrite the original expression as follows:



¬(∃x Red(x) ∧ ∀y Cubes(y))