COS2661 - Assignment 3 2023

Question 1
Give the tautological equivalence of the following quantified sentences.

1.1
∀x (dog(x) ∧ Brown(x) → hairy(x))

Explanation:
• ∀x: For every object x.
• dog(x): x is a dog.
• Brown(x): x is brown.
• hairy(x): x is hairy.
• →: Implication, meaning "implies" or "if...then."
• ∧: Conjunction, meaning "and."

1.2
∃x (Corgi(x) ∧ ∀y (Person(y) → Loves(x, y)))

Explanation:
• ∃x: There exists an object x.
• (Corgi(x) ∧ ∀y (Person(y) → Loves(x, y))): The object x is a Corgi and it loves every
individual y who is a person.

1.3
∀x ∃y (Corgi(y) ∧ Loves(x, y))

Explanation:
• ∀x: For all x (for every person).
• ∃y: There exists a y (at least one Corgi).
• Corgi(y): y is a Corgi.
• Loves(x, y): x loves y.

1.4
∀x ∀y (Pancakes(x) ∧ Pancakes(y) ∧ x ≠ y) → TasteSimilar(x, y)

Explanation:
• "For all x and y, if x and y are both pancakes and they are not the same (x ≠ y), then x
and y taste similar."

• In simpler terms, the statement asserts that every pair of distinct pancakes taste similar to
each other.
1.5
∀x ∃y₁ ∃y₂ (Person(x) → (y₁ ≠ y₂ ∧ Knows(x, y₁) ∧ Knows(x, y₂)))

Explanation:
• ∀x - For all x (for every person x)
• ∃y₁ - There exists a person y₁
• ∃y₂ - There exists another person y₂