Examine the statements: I. I watch TV only if I am bored. (Watching TV ⇒ Bored) II. I am never bored when I have my brother’s company. (Brother’s company ⇒ Not bored) III. Whenever I go to the theatre, I take my brother along. (Theatre ⇒ Brother’s company) Which conclusion is valid?
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AIf I am bored, I watch TV
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BIf I am bored, I seek my brother’s company
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CIf I am not with my brother, then I watch TV
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DIf I am not bored, I do not watch TV
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EIf I go to the theatre, I must be bored
Answer
Correct Answer: If I am not bored, I do not watch TV
Explanation
Introduction / Context:We must convert verbal conditionals into logical implications and identify a conclusion that necessarily follows. The key phrase “only if” establishes a necessary condition.
Given Data / Assumptions:
- I: WatchTV ⇒ Bored (necessary condition = “bored”).
- II: Brother ⇒ Not bored.
- III: Theatre ⇒ Brother.
Concept / Approach:From “P only if Q” we get P ⇒ Q. The valid contrapositive is ¬Q ⇒ ¬P. Hence from I we also obtain: Not bored ⇒ Do not watch TV.
Step-by-Step Solution:
• Option D is the contrapositive of I and is therefore valid: Not bored ⇒ Not watch TV.• Option A (“If bored, watch TV”) is the converse; it does not follow from “only if.”• Option B introduces seeking brother’s company; not implied.• Option C claims absence of brother ⇒ watch TV; not stated.• Option E claims theatre implies boredom; but Theatre ⇒ Brother ⇒ Not bored, which actually blocks TV watching during theatre times.Verification / Alternative check:Construct a timeline: During theatre (with brother), not bored; thus by I’s contrapositive, no TV—consistent with D.
Why Other Options Are Wrong:They confuse converse with contrapositive or add unstated behaviors.
Common Pitfalls:Misinterpreting “only if,” assuming “if and only if,” or overlooking the contrapositive.
Final Answer:If I am not bored, I do not watch TV.