Q. 10 Consider the following three statements:

• (A) If (i) is true and (ii) is false, then (iii) is false.
• (B) If (i) is true and (ii) is false, then (iii) is true.
• (C) If (i) and (ii) are true, then (iii) is true.
• (D) If (i) and (ii) are false, then (iii) is false.

Understanding the Logic Puzzle

Three statements form the basis of this syllogism question:

(i) Some roses are red.
(ii) All red flowers fade quickly.
(iii) Some roses fade quickly.

The task is to identify which option logically follows from these. Syllogisms test whether a conclusion must be true based on premises, using Venn diagrams or set theory mentally.

This puzzle hinges on the particular affirmative nature of (i) (“some roses are red”) combined with the universal affirmative in (ii) (“all red flowers fade quickly”). In logic terms, some A are B, and all B are C, implies some A are C.

Correct Answer: Option (C)

If (i) and (ii) are true, then (iii) is true.

Here’s the step-by-step reasoning:

1. Statement (i) confirms at least one rose is red (overlap between “roses” and “red flowers”).

2. Statement (ii) states every red flower—including those roses—fades quickly.

3. Therefore, those specific red roses must fade quickly, making (iii) necessarily true.

Visualize with sets: Roses (R) intersect Red flowers (Red) non-emptily, and Red ⊆ Fade quickly. Thus, R ∩ Fade quickly ≠ ∅.

No counterexample exists; it’s a valid syllogism (DARII mood in traditional logic).

Why Other Options Are Incorrect

Let’s evaluate each alternative systematically.

Option (A): If (i) is true and (ii) is false, then (iii) is false

Incorrect.

If (i) is true (some roses red) but (ii) false (not all red flowers fade quickly), (iii) could still be true. Example: Suppose some red roses do fade quickly (maybe due to other reasons), even if other red flowers don’t. (iii) isn’t forced false—it’s possible. No entailment here.

Option (B): If (i) is true and (ii) is false, then (iii) is true

Incorrect.

Opposite flaw. With (ii) false, red roses might not fade quickly. Counterexample: All red roses stay vibrant forever, while some non-rose red flowers fade. (i) holds, (ii) fails, but (iii) false. No necessity for truth.

Option (D): If (i) and (ii) are false, then (iii) is false

Incorrect.

Falsity of premises doesn’t dictate the conclusion. (iii) could be independently true (e.g., non-red roses fade quickly due to age or environment). Logic doesn’t propagate falsity this way—it’s about what must follow when premises hold.

Tips for Solving Similar Syllogism Questions

• Draw Venn diagrams: Essential for visualizing “some” (partial overlap) vs. “all” (subset).

• Test extremes: Check if conclusion holds in minimal cases (e.g., exactly one red rose).

• Avoid assumptions: Don’t presume “only roses are red” or extra traits.

• Practice patterns: Memorize moods like AAA-1 (all A are B, all B are C → all A are C) or this one’s particular form.

Mastering these boosts scores in logical reasoning sections. Common in exams testing critical thinking.