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.

If P is a new set and P is a superset of A, and N(P) is the number of elements in P, then which of the following must be true?

• A. N(G) is smaller than only four numbers
• B. N(C) is the greatest
• C. N(B) is the smallest
• D. N(P) is the greatest

Hint:

Check each option against the size chain from the passage; B cannot be smallest since H and E sit inside it, and P, tied only to A, ends up the biggest by elimination.

Answer 1

The Correct Option is D

A new set P is brought in here, and P is only tied down by one fact: P contains all of A. To find which statement must be true, check each option against the chain of sizes built from the passage: N(H) and N(E) are both at most N(B), and N(B) is at most N(D), which is at most N(G), which is at most N(F), which is at most N(A), which is at most N(C).

1. N(G) is smaller than only four numbers: this claims an exact count of how many sets beat G in size. The chain only tells us F, A, C, and now P, are all at least as big as G, not that they are strictly bigger. If any of them happen to match G in size, the count of sets that truly beat G drops below four, so this exact count is not something the passage guarantees.
2. N(C) is the greatest: this was true before P showed up, since C sat at the top of the original chain. But P is defined completely independently of C. Nothing in the passage stops P from being built larger than C, so C being the greatest is no longer a safe conclusion once P is in the picture.
3. N(B) is the smallest: this goes directly against the passage, since H and E are both inside B, so their counts can never exceed N(B). B is never the smallest; if anything, H or E is.
4. N(P) is the greatest: P inherits everything A already contains, and A itself is already near the top of the chain, just short of C. Since P is deliberately built as a fresh superset with nothing above it in the passage restricting its size, while each of the other three claims above is shown to be unreliable, P being the biggest is the one statement the given facts never contradict.

The first three options fail on closer inspection: one relies on an exact count that isn't guaranteed, one is undercut by P's independence from C, and one flatly contradicts the given chain. That leaves the fourth option standing.

Let's summarize:

• Adding a new set defined only as "a superset of A" gives it room to be the largest, since nothing above it is fixed.
• B can never be the smallest because H and E sit inside it.

So N(P) being the greatest is the statement that must be true. The answer is option (D).