Consider the following three statements: i. Some roses are red. ii. All red flowers fade quickly. iii. Some roses fade quickly. Which of the following statements can be logically inferred from the above 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) is true and (ii) is true, then (iii) is true. D. If (i) and (ii) are false, then (iii) is false.

GATE 2018 · General Aptitude · Statements Follow · medium

Answer: Only option C is a valid logical inference. Answer: C.

1. Verify option C: (i) true AND (ii) true => (iii) true: Let R = roses, Red = red flowers, F = fade quickly. Premise i: some roses are red, so (R intersect Red) is non-empty. Let x be a rose that is red. Premise ii: all red flowers fade quickly, so x (being red) also fades quickly. Therefore x is both a rose and fades quickly. Hence (R intersect F) is non-empty, which is exactly statement iii. So if both i and ii are true, iii must be true.
2. Disprove options A and B: (i) true, (ii) false does not force (iii) either way: If ii is false, then some red flowers do NOT fade quickly, but that says nothing definitive about roses fading. The red roses might or might not fade for other reasons. We can construct a model where (i) is true and (iii) is true (roses fade for non-color reasons), and another where (i) is true and (iii) is false. So neither 'iii is false' (option A) nor 'iii is true' (option B) is guaranteed.
3. Disprove option D: (i) false AND (ii) false does not force (iii) false: Even if no roses are red (i false) and not all red flowers fade (ii false), roses might still fade quickly due to non-color reasons. For example, all roses could fade quickly due to being delicate flowers regardless of color. So (iii) can be true even when both (i) and (ii) are false. Option D is INVALID.