Question

Choose the correct choice(s) regarding the following propositional logic assertion S:

S : ((P ∧ Q)→ R)→ ((P ∧ Q)→ (Q → R))

1. The antecedent of S is logically equivalent to the consequent of S.
2. S is a tautology
3. S is a contradiction
4. S is neither a tautology nor a contradiction.

Answer 1

Correct Answer - Option :

Concept

• An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause.
• A consequent is the second half of a hypothetical proposition.

Symbol:

∧ = AND = .

V = OR = +

¬ = NOT ≡ ̅ 

1 = TRUE

Formula:

A → B = ¬ A ∨ B = ¬ A + B

¬ (A.B) = A̅ + B̅

Calculation:

Antecedent of S:

(P ∧ Q)→ R ≡   ¬ (P.Q) + R ≡ P̅ + Q̅  + R 

Consequent of S:

(P ∧ Q)→ (Q → R) ≡ ¬ (P.Q) + (¬ Q + R

≡ P̅ + Q̅ + Q̅ + R ≡ P̅ + Q̅  + R 

Hence the antecedent of S is logically equivalent to the consequent of S.

S = ((P ∧ Q)→ R)→ ((P ∧ Q)→ (Q → R))

S = ¬ (¬ (P.Q) + R) + (¬ (P.Q) + (¬ Q + R))

S = P.Q.¬R + ¬ P + ¬ Q. + (¬ Q + R)

S = P.Q.R̅ + P̅ + Q̅  + Q̅ + R

S = P.Q.R̅ +  P̅ + Q̅ + R   

S = (P̅ + P)(P̅ + Q.R̅ ) + Q̅ + R     // A + A̅B = A+ B 

S = P̅ + Q.R̅  + Q̅ + R

S = P̅ + Q̅ + R̅  + R

S = 1  = TRUE                                     // A + A̅ = 1

S is a tautology