Question

Eight sets A, B, C, D, E, F, G and H are such that:

A is a superset of B, but subset of C.
B is a subset of D, but superset of E.
F is a subset of A, but superset of B.
G is a superset of D, but subset of F.
H is a subset of B.

N(A), N(B), N(C), N(D), N(E), N(F), N(G) and N(H) are the number of elements in the sets A, B, C, D, E, F, G and H respectively.

Which one of the following could be FALSE, but not necessarily FALSE?

• A. E is a subset of D
• B. E is a subset of C
• C. E is a subset of A
• D. E is a subset of H

Hint:

Build the chain of set sizes from the passage; E and H are never directly compared, so any statement linking them is left undecided.

Answer 1

The Correct Option is D

Step 1: Build the size chain from the passage.
Translate every subset or superset statement into an inequality on the counts N(...). "A is a superset of B" means \(N(B) \le N(A)\). "A is a subset of C" means \(N(A) \le N(C)\). "B is a subset of D" means \(N(B) \le N(D)\). "B is a superset of E" means \(N(E) \le N(B)\). "F is a subset of A, but superset of B" means \(N(B) \le N(F) \le N(A)\). "G is a superset of D, but subset of F" means \(N(D) \le N(G) \le N(F)\). "H is a subset of B" means \(N(H) \le N(B)\).

Step 2: String these into one chain.
Putting the pieces together: \(N(H) \le N(B)\) and \(N(E) \le N(B)\), and then \(N(B) \le N(D) \le N(G) \le N(F) \le N(A) \le N(C)\). So every one of B, D, G, F, A, C sits above E on the chain, since E connects straight into B and B connects all the way up to C.

Step 3: Check each option against the chain.
Option (A), "E is a subset of D": always true, since \(E \subseteq B \subseteq D\) is a direct, unbroken link.
Option (B), "E is a subset of C": always true, since \(E \subseteq B \subseteq D \subseteq G \subseteq F \subseteq A \subseteq C\) is a full chain from E to C.
Option (C), "E is a subset of A": always true, for the same reason, since A sits further up that same chain.
Option (D), "E is a subset of H": nothing in the passage ever compares E and H directly. Both happen to sit at or below B, but that alone does not tell us whether E is inside H, H is inside E, or neither contains the other. So this statement is not guaranteed true, meaning it could be false, but nothing forces it to be false either.

Final Answer:
Only "E is a subset of H" is left undecided by the given facts, so it is the one that could be false without being necessarily false. \[ \boxed{\text{Option (D)}} \]