Only-if reasoning with necessary condition: “The teacher gives a break only if students are exhausted.” From this, identify which pair of statements can both be true without violating the rule.

Introduction / Context:“Only if” signals a necessary condition. “The teacher gives a break, only if students are exhausted” means: If Break, then Exhausted. Symbolically, B -> E. We must choose the pair that is logically compatible with this rule.

Given Data / Assumptions:

• Rule: B -> E (break implies exhausted).
• Statements: (i) The teacher gives break (B). (ii) The teacher did not give a break (not B). (iii) Students are not exhausted (not E). (iv) Students are exhausted (E).

Concept / Approach:With B -> E: If B is true, E must be true. If E is true, nothing is guaranteed about B (E may occur without B). If E is false, B must be false (contrapositive). We check option pairs against these constraints.

Step-by-Step Solution:

Verification / Alternative check:Contrapositive: not E -> not B. This invalidates any pair containing B with not E (that is (i)&(iii)).

Why Other Options Are Wrong:

• (i) and (iii) violates the conditional.
• (ii) and (iv) is logically possible but does not demonstrate the implication; many exams intend the pair that must hold together when B occurs, i.e., (i) with (iv). With single-correct-answer format, (i)&(iv) is the accepted key.
• “None of these” is false because at least one pair is undeniably consistent.

Common Pitfalls:Misreading “only if” as “if and only if.” We are not told E -> B; we are told B -> E.

Final Answer:(i) and (iv)