COS2661 Formal logic 2 assignment 2 memo

COS2661 Assignment 2 Memo

Question 1
1.1 (2 Marks)
∃x ¬P(x) (2 Marks) (There exists an x such that P(x) is false)
Tautological equivalence:
∃x ¬P(x) ≡ ¬∀x P(x)
The statement "There exists an x such that P(x) is false" is logically equivalent to saying "It is
not true that P(x) holds for all x."
This uses the quantifier negation rule:
¬∀x P(x) ≡ ∃x ¬P(x)
¬∃x P(x) ≡ ∀x ¬P(x)
Final Answer:
∃x ¬P(x) ≡ ¬∀x P(x)


1.2 (2 Marks)
∀x ¬P(x) (2 Marks) (For all x, P(x) is false)
Tautological equivalence:
∀x ¬P(x) ≡ ¬∃x P(x)
Explanation:
The statement "For all x, P(x) is false" means that there does not exist any x for which P(x) is
true.
This is based on the quantifier negation rule:
¬∃x P(x) ≡ ∀x ¬P(x)
¬∀x P(x) ≡ ∃x ¬P(x)
Final Answer:
∀x ¬P(x) ≡ ¬∃x P(x)
Because denying the existence of an element for which P(x) is true is the same as asserting
that P(x) is false for all elements.

, 1.3 (4 Marks)




Question 2
2.1 Student(david, cc) → (Teacher(patty) ∧ Teach(patty, david)) (3 Marks)
2.2 ¬Student(patty, cc) → Principal(themba) ∧ Head(themba, cc) (3 Marks)
2.3 If Cathy is a student at Cyprian College, David and Patty are, respectively, the principal
and teacher teaching her. (3 Marks)
2.4 All students at Cyprian College are being taught by Patty or Cathy. (3 Marks)