COM2216 TEST 2
20 October 2022
Answer all questions.
1. Give formal proofs of the following:
(i) (A ˄ B) ˅ (A ˄ C) from the premises A and B ˅ C
(5)
(ii) A from the premise (A ˄ B) ˅ A
(5)
(iii) {(A ˄ C) ˅ (D ˄ C), B } |= C ˄ B
(5)
2. Write down De Morgan's laws for quantifiers.
(4)
3. Prove the following :
(i) ∀x (S(x) → ¬P(x)) ↔ ¬∃x (S(x) ˄ P(x))
(4)
(ii) ∃x (S(x) ˄ P(x)) ↔ ¬∀x (S(x) → ¬P(x))
(4)
4. Let N(x) be the statement "x has visited North Dakota," where the domain consists of the
students in your school. Express each of these quantifications in English.
(i) ∃xN(x)
(1)
(ii) ∀xN(x)
(1)
(iii) ¬∃xN(x)
(1)
(iv) ∃x ¬N(x)
(1)
(v) ¬∀x N(x)
(2)
(vi) ∀x¬N(x)
(2)