In the following statement and conclusion problem, you are given two statements involving dens, spirals, and cards. Accept the statements as true, read all the conclusions carefully, and decide which of them logically follow from the information given. Statements: (I) All dens are spiral. (II) Some spirals are cards. Conclusions: (I) Some spirals are not cards. (II) Some dens are not cards. (III) Some cards are den.

Introduction / Context:
This question examines your ability to reason about partial information using Venn style logic. The statements connect the sets “dens,” “spirals,” and “cards.” You must check each proposed conclusion to see whether it is forced by the statements, or whether there is at least one valid arrangement in which the conclusion fails. Only those conclusions that must hold in every such arrangement are said to logically follow.

Given Data / Assumptions:

• Statement (I): All dens are spiral.
• Statement (II): Some spirals are cards.
• Conclusion (I): Some spirals are not cards.
• Conclusion (II): Some dens are not cards.
• Conclusion (III): Some cards are den.
• We accept the statements as true and avoid adding any extra assumptions.

Concept / Approach:
The first statement tells us that the set of dens is contained within the set of spirals. The second statement says that there is at least one spiral that is also a card. However, nothing is said about whether all spirals are cards or only some of them are, and nothing is directly said about dens being cards or not. To test each conclusion, we construct diagrams that satisfy the statements and see whether a conclusion can be false in any of them.

Step-by-Step Solution:
Step 1: Draw a large circle for spirals.Step 2: Draw the dens set completely inside the spirals circle, because all dens are spiral.Step 3: Represent cards as a separate circle that intersects the spirals circle in at least one region to account for “Some spirals are cards.” We do not yet fix the overlap with dens.Step 4: Analyse conclusion (I): “Some spirals are not cards.” This requires that there exist spirals outside the card region. But the statements allow the possibility that all spirals happen to be inside the cards set. For example, every spiral could also be a card, which still satisfies “Some spirals are cards.” Thus conclusion (I) is not necessary.Step 5: Analyse conclusion (II): “Some dens are not cards.” Dens lie inside spirals, but the statements do not say how the dens region relates to the cards region. It is possible that all dens happen to lie in the overlapping part of spirals and cards, making every den a card. In such a case, conclusion (II) would be false while the statements remain true. So conclusion (II) does not follow.Step 6: Analyse conclusion (III): “Some cards are den.” This would require at least one card that lies in the dens region. However, the overlap required by “Some spirals are cards” can occur entirely outside the dens region. For example, some spirals that are not dens may be cards. In that scenario, no card is a den, and yet both statements are still satisfied. Hence conclusion (III) also does not necessarily follow.

Verification / Alternative check:
We can verify by constructing explicit examples. Suppose there are spirals that are not dens and are also cards. Let all dens be spirals and let every spiral be a card. Then statements (I) and (II) are true: dens are spiral, and some spirals are cards. In this arrangement, there is no spiral that is not a card, so conclusion (I) fails. If we let all dens lie inside the overlapping region of spirals and cards, there is no den that is not a card, so conclusion (II) fails. Furthermore, if no card lies inside the dens region in a different diagram, conclusion (III) fails there while the statements still hold. Therefore none of the conclusions is forced.

Why Other Options Are Wrong:
Option A claims that conclusions (I) and (II) follow, but we have shown that both can be false in valid interpretations. Option B claims only conclusion (III) follows, which again is not justified by the statements. Option C asserts all three conclusions follow, which is the strongest and most incorrect claim. Only option D, which states that no conclusion follows, matches the careful analysis.

Common Pitfalls:
Many candidates think that once “Some spirals are cards” is given, there must also be spirals that are not cards, which is not logically necessary. Another common mistake is to assume that if a set is contained in another, it must have some portion outside the intersection with a third set. Visualising multiple possible diagrams and checking whether a conclusion must hold in all of them is essential for this type of reasoning question.

Final Answer:
Therefore, the correct decision is that no conclusion follows from the given statements.