Syllogism — Bottles, Boxes, Bags, and Trays Statements: • All the bottles are boxes. • All the boxes are bags. • Some bags are trays. Conclusions: (1) Some bottles are trays. (2) Some trays are boxes. (3) All the bottles are bags. (4) Some trays are bags.

Introduction / Context:
This is a classic categorical syllogism with two universals and one particular. The goal is to see which conclusions are compelled by subset relations and which are not forced by the existence of “some” in a different subset.

Given Data / Assumptions:

• Bottles (Bo) ⊆ Boxes (Bx).
• Boxes (Bx) ⊆ Bags (Bg).
• Some Bags (Bg) are Trays (Tr): Bg∩Tr ≠ ∅.
• Conclusions to check: (1) Some Bo are Tr. (2) Some Tr are Bx. (3) All Bo are Bg. (4) Some Tr are Bg.

Concept / Approach:
Transitivity of subsets: from Bo ⊆ Bx and Bx ⊆ Bg, we get Bo ⊆ Bg. Particular intersections can be converted: if some Bg are Tr, then some Tr are Bg (but not necessarily boxes or bottles).

Step-by-Step Solution:

Verification / Alternative check:
Construct a diagram with Bo inside Bx inside Bg. Place a few Tr elements intersecting Bg but outside Bx. Premises hold; (1) and (2) are false, confirming only (3) and (4) must follow.

Why Other Options Are Wrong:

• Options including (1) or (2) assume unnecessary overlaps with bottles/boxes.
• Only the pair (3) and (4) is guaranteed.

Common Pitfalls:
Assuming that because trays intersect bags, they must also intersect every subset within bags. That is not logically required.

Final Answer:
Only (3) and (4)