Q.4 Fact: If it rains, then the field is wet.
Read the following statements:
(i) It rains
(ii) The field is not wet
(iii) The field is wet
(iv) It did not rain
Which one of the options given below is NOT logically possible, based on the given fact?
(A) If (iii), then (iv). (B) If (i), then (iii).
(C) If (i), then (ii). (D) If (ii), then (iv).

The given fact “If it rains, then the field is wet” forms a conditional statement in propositional logic. This translates to P → Q, where P is “it rains” (statement i) and Q is “the field is wet” (statement iii). Option (C) is not logically possible because it directly contradicts this fact.

Logical Notation

Assign propositions clearly:

• P: It rains (i)

• ¬Q: The field is not wet (ii)

• Q: The field is wet (iii)

• ¬P: It did not rain (iv)

The fact P → Q means rain guarantees a wet field. Truth table for P → Q: true except when P is true and Q is false.

Option Analysis

Evaluate each option against P → Q:

Option (A): If (iii), then (iv) → If Q, then ¬P
Field wet implies no rain. Possible—field could be wet from irrigation, not rain. Does not violate P → Q.

Option (B): If (i), then (iii) → If P, then Q
Rain implies wet field. This restates the given fact exactly, so fully possible.

Option (C): If (i), then (ii) → If P, then ¬Q
Rain implies field not wet. Impossible—directly denies P → Q (the one false case in truth table).

Option (D): If (ii), then (iv) → If ¬Q, then ¬P
Field not wet implies no rain. This is the contrapositive of P → Q (logically equivalent), so possible.

Truth Table Verification

Only P true and Q false (option C scenario) makes the fact false.

Why C Fails

Option C assumes rain (P true) but dry field (¬Q true), creating P ∧ ¬Q. This is the negation of P → Q, hence impossible under the given fact. All other combinations preserve the implication.