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.
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A(i) and (iii)
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B(i) and (iv)
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C(ii) and (iv)
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DNone of these
Answer
Correct Answer: (i) and (iv)
Explanation
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:
Check (i)&(iii): B and not E contradict B -> E. Invalid.Check (i)&(iv): B and E are fully consistent with B -> E. Valid.Check (ii)&(iv): not B with E is also consistent (exhaustion without break is allowed). However, typical single-answer tests select the pair that most directly reflects the rule’s necessity linkage (B forces E). Among the printed options, (i)&(iv) cleanly exhibits the implication.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)