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?

The correct option is: If (i) and (ii) are true, then (iii) is true.

To understand why this logical inference is correct, let us analyze the premises step-by-step when both statement (i) and statement (ii) are assumed to be true:

1. Analyze Statement (i): "Some roses are red."
This statement tells us that there exists a non-empty set of flowers that are both "roses" and "red". In other words, there is at least one entity, let us call it x, such that x is a rose and x is red.

2. Analyze Statement (ii): "All red flowers fade quickly."
This is a universal statement. It means that if any object is a red flower, it must belong to the set of things that fade quickly. Since a red rose is a type of red flower, any red rose must also fade quickly.

3. Combine the Premises:
From Statement (i), we established that there is at least one rose, x, that is red.
From Statement (ii), since x is red (and is a flower/rose), x must fade quickly.
Therefore, this specific rose, x, fades quickly.

4. Evaluate Statement (iii): "Some roses fade quickly."
Since we have shown that there is at least one rose (our entity x) that fades quickly, we can logically conclude that "Some roses fade quickly" is a true statement.

Thus, if (i) and (ii) are true, then (iii) must be true.