Difficulty: Medium
Correct Answer: Only conclusion II follows
Explanation:
Introduction / Context:
This question is a logic puzzle that uses the familiar shapes squares, rectangles, and triangles in an abstract way. Even though the statements conflict with actual geometry, you must treat them as logically true for the purpose of the problem. Your task is to judge whether either of the two conclusions logically follows from these artificial relations.
Given Data / Assumptions:
Concept / Approach:
The first statement says that the sets “squares” and “rectangles” are completely disjoint. The second statement says that the set of rectangles is fully contained within the set of triangles. Thus any rectangle is automatically a triangle. For the conclusions, we must see whether any intersection between triangles and squares is forced and whether we can guarantee the existence of at least one triangle that is a rectangle.
Step-by-Step Solution:
Step 1: Draw a large set representing triangles.Step 2: Place the rectangles set completely inside the triangles set, because all rectangles are triangles.Step 3: Place the squares set completely outside the rectangles set, because no square is a rectangle as per statement 1.Step 4: Evaluate conclusion I: “Some triangles are squares.” This requires at least one object that is both triangle and square. However, the statements do not link squares directly to triangles. Squares may or may not belong to the triangle set. It is logically possible to draw the squares set entirely outside the triangles set, making their intersection empty, while both statements remain true. Therefore conclusion I does not necessarily follow.Step 5: Evaluate conclusion II: “Some triangles are rectangles.” Statement 2, “All rectangles are triangles,” means every rectangle is also a triangle. Under typical exam conventions, we consider that rectangles exist. If at least one rectangle exists, then that rectangle is both a rectangle and a triangle, so some triangles are rectangles. Hence conclusion II follows in that standard interpretation.
Verification / Alternative check:
Symbolically, statement 2 can be written as rectangle implies triangle. If there exists at least one rectangle, call it r, then r is a rectangle and therefore r is a triangle. This directly satisfies conclusion II that some triangles are rectangles. Statement 1, that no square is a rectangle, does not prevent squares from being triangles or non triangles; it only says they cannot be rectangles. Since nothing forces squares to be triangles, we cannot accept conclusion I as a necessary outcome.
Why Other Options Are Wrong:
Option A claims that only conclusion I follows, but we have seen that squares might be placed completely outside the triangle set without violating the statements. Option C claims that both conclusions follow, which is too strong because conclusion I is not guaranteed. Option D says that neither conclusion follows, which ignores the direct implication contained in statement 2 when we assume the existence of rectangles. Only option B, which accepts conclusion II alone, is consistent with the usual reasoning conventions used in such exam questions.
Common Pitfalls:
One common mistake is to be influenced by real geometry, where every square is also a rectangle. Here, the question explicitly defines a different abstract relationship. Another mistake is to overlook the existence implied by the presence of a set like rectangles in such questions. While strict symbolic logic can treat a universal statement as vacuously true even for an empty set, exam questions of this type normally assume that the mentioned categories are non empty unless stated otherwise.
Final Answer:
Therefore, under the stated assumptions, only conclusion II follows from the given statements.
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