Exercise 1.4 (3)
If p and q are true and r and s are false, find the truth value of each of the following compound statements.
i) p ∧ (q ∧ r)
Ans:
p ∧ (q ∧ r) ≡T∧(T∧F)
≡T∧F
≡F
Hence Truth Value is F.
ii) (p → q) ∨ (r ∧ s)
Ans:
(p → q) ∨ (r ∧ s) ≡ (T →T)∨(F∧F)
≡ T∨F
≡ T
Hence Truth Value is T
iii) ∼ [(∼ p ∨ s) ∧ (∼ q ∧ r)]
Ans:
∼ [(∼ p ∨ s) ∧ (∼ q ∧ r)] ≡~[(~T∨F)∧(~T∧F)]
≡ ~[(F∨F)∧(F∧F)]
≡ ~[F∧F]
≡ ~[F]
≡T
Hence Truth Value is T
iv) (p → q) ↔ ∼ (p ∨ q)
Ans:
(p → q) ↔ ∼ (p ∨ q) ≡ (T→T)↔~(T∨T)
≡ T↔~T
≡ T↔F
≡ F
Hence Truth Value is F
v) [(p ∨ s) → r] ∨ ∼ [∼ (p → q) ∨ s]
Ans:
[(p ∨ s) → r] ∨ ∼ [∼ (p → q) ∨ s] ≡ [(T∨F)→F]∨~[~(T→T)∨F]
≡ (T→F)∨~[~T∨F]
≡ F∨~[F∨F]
≡ F∨~F
≡ F∨T
≡ T
Hence Truth Value is T
vi) ∼ [p ∨ (r ∧ s)] ∧ ∼ [(r ∧ ∼ s) ∧ q]
Ans:
∼ [p ∨ (r ∧ s)] ∧ ∼ [(r ∧ ∼ s) ∧ q] ≡ ~[T∨(F∧F)]∧~[(F∧~F)∧T]
≡ ~[T∨F]∧~[(F∧T)∧T]
≡ ~T∧~(F∧T)
≡ F∧~F
≡ F∧T
≡ F
Hence Truth Value is F