Mathematical Foundation of Computer Science

DHARM

148 MATHEMATICAL FOUNDATION OF COMPUTER SCIENCE

5.9 For the given premises determine a suitable conclusion so that the argument is valid,

(i)P → ~ Q, ~ P → R(ii)P → ~ Q, R → P, Q

(iii) P, P ∧ R, P → Q, Q → S(iv)P → (R ∧ S), ~ (R ∧ S), ~ P → S

5.10 Prove argument is valid

1. (H → J) ∨ (J → K) 2. (I ∧ K) → L


2. ~ L /∴ ~ (H ∨ J)
   5.11 Prove formula (P → (P ∨ Q) is a tautology.
   5.12 Determine the interpretation for which argument is invalid


3. J → (K → L) 2. K → (~ L → M)


4. (L ∨ M) → N ∴ ~ (J → N)
   5.13 Prove argument is valid


5. Q ∨ (R → S) 2. (R → (R ∧ S)) → (T ∨ U)


6. (T → Q) ∧ (U → V) ∴Q ∨ V
   5.14 (i) No academician is wealthy. Some scientists are wealthy.
   /∴ (a) some scientists are academicians.
   (b) Some academician is not scientists.
   (ii) All artists are interesting people. Some philosophers are mathematicians. Some agents are
   salesmen. Only uninteresting people are salesmen.
   /∴ (a) some philosophers are not salesmen.
   (b) Salesmen are not mathematicians.
   (c) Some agents are not philosophers.
   (d) Artists are salesmen but they are not interesting people.
   5.15 Let {1, 2, 3, 4, 5} be the universal set, determine the truth value of each of the statements,
   (i)(∃x) (∀y), x^2 < y + 1 (ii)(∀x) (∃y), x^2 + y^2 = 25
   (iii)(∃x) (∀y) (∃z), x^2 > y + z (iv)(∃x) (∃y) (∃z), x^2 + y^2 = 2z^2
   5.16 Negate the following statements :
   (i)(∃x) P(x) → (∀y) Q(y)(ii)(∃x) P(x) ∧ (∀y) Q(y)
   (iii)~ (∃x) P(x) ∨ (∀y) ~ Q(y)(iv)(∀x) (∃y) [P(x) ∨ Q(y)]
   (v)~ (∃x) ~ P(x) → (∀x) P(x)(vi)(∀x) [P(x) ∨ Q(y)]
   5.17 Show that following are valid formulas :
   (i)(∃y)[(∃x)F(x) → F(y)] (ii)(∃y)[P(y) → (∀x)P(x)]
   (iii)(∃x)H(x) ∨ ~ (∃x) H(x)(iv)(∀x)F(x) ∨ (∀x)G(x)(x)] → (∀x) [F(x) ∨ G(x)]
   5.18 Show that from given premises drives the conclusion shown on right.
   (i)(∀x)[H(x) → M(x)] /∴ (∃x)[H(x) ∧ M(x)]
   (ii)(∀x)[H(x) → M(x)], (∃y)H(y)/∴ (∃x)[H(x) ∧ M(x)]
   (iii)(∃x)[H(x) ∧ M(x)] → (∃y)[P(y) ∧ Q(y)], (∃y)[P(y) ∧ ~ Q(y)] /∴ (∀x)[H(x) → ~ M(x)]
   5.19 Symbolizes and prove the validity of the following arguments :
   (i) No mortal are perfect. All human are mortal. Therefore no human are perfect.
   (ii) Himalaya is large. Therefore every thing is large.
   (iii) All human are mortal. Jorge is human. Therefore Jorge is mortal.
   (iv) Not every thing is edible. Therefore nothing is edible.
   (v) All cats are animals. Some dogs are animals. Therefore some cats are dogs.