DHARMPROPOSITIONAL LOGIC 147
5.2 Let R be “He is richer” and let C be “He has a car”. Write each of the following statements in the
symbolic form using R and C.
(i) He is richer and has a car.
(ii) He is richer but not has a car.
(iii) It is not true that he is poorer and has a car.
(iv) He is neither richer nor has a car.
(v) It is true that he is poorer and not has a car.
(vi) He is richer so he has a car therefore he is not poor.
5.3 Construct the truth tables of the following prepositions,
(i) ~ (P → ~ Q) (ii) ~ (P ∧ Q) ∨ ~ (Q ↔ P)
(iii) (P → Q) ↔ (~ P ∨ Q) (iv) (P ∧ (Q → P)) → Q.
5.4 Find the truth values of the following propositions under the given truth values of P and Q as
True and R and S as False.
(i) (P → Q) ∨ ~ (P ↔ ~ Q) (ii)(P ∨ (Q → (R ∨ ~ P))) (Q ∨ ~ S)
(iii) (P → ((~ Q ∧ R) ∧ ~ (Q ∨ (~ P ↔ Q))).
5.5 Show the following equivalence,
(i) (P → Q) → R ⇔ (~ P ∨ Q) → R ⇔ (P ∧ Q) → R
(ii)P → (Q ∨ R) ⇔ (P → Q) ∨ (P → R)
(iii) ~ (P ∧ Q) ⇔ P ∧ ~ Q
(iv) (P ∨ Q) ∧ (~ P ∧ (~ P ∧ Q)) ⇔ (~ P ∧ Q)
(v)P ∧ (Q ↔ R) ⇔ P ∧ ((Q → R) ∧ (R → Q))
(vi)P → (Q ∨ R) ⇔ (P ∧ ~ Q) → R
(vii) (P → R) ∧ (Q → R) ⇔ (P ∨ Q) → R.
5.6 Show that following formulas are tautology :
(i) (((R → C) ∧ R) → C) (ii)(P ∧ Q) → P
(iii)B ∨ (B → C) (iv) ((A → B) ∧ (B → C)) → (A → C)
(v) (P → (P ∨ Q)) (vi) (A → B) ∨ (A → ~ B)
(vii) (A → (B ∧ C)) → (((B → (D ∧ E)) → (A → D)).
5.7 Determine the truth values of the following composite statements :
(i) If 2 < 5, then 2 + 3 ≠ 5.
(ii) It is true that 6 + 6 = 6 and 3 + 3 = 6.
(iii) If Delhi is the capital of India then Washington is the capital of US.
(iv) If 1 + 1 ≠ 2, then it is not true that 3 + 3 = 8 if and only if 2 + 2 = 4.
5.8 From the given premises show the validity of the following arguments, which drives the conclu-
sion shown on right.
(i)P → Q, Q → R/∴ P → R
(ii) (A → B) C, A ∧ D, B ∧ T/∴ C
(iii) (P → Q) ∧ (P → R), ~ (Q ∧ R), S ∨ P/∴ S
(iv)A → B, (A → (A ∧ B)) → C/∴ C
(v)Q ∨ (R → S), (R → (R ∧ S)) → (T ∨ U), (T → Q) ∧ (U → V) /∴ (Q ∨ V)
(vi) (E ∨ F) → G, H → (I ∧ G), /∴ (E → G) ∧ (H → I)
(vii)M → J, J → ~ H, ~ H → ~ T, ~ T → M, M → ~ H /∴ ~ T
(ix) (H → J) ∧ (J → K), (I ∨ K) → L, ~ L /∴ ~ (H ∨ J)
(x) (H → J) ∧ (J → K), (H ∨ J), (H → ~ K) ∧ (J → ~ I), (I ∧ ~ K) → L, K → (I ∨ M)
/∴ L ∨ M