Five Venn diagrams showing various intersections and separations between sets P and Q.

Let’s solve this step by step.

We are given five Venn diagrams labeled (i) through (v), each showing two sets: P and Q, with numbers inside them. We need to figure out which diagram correctly represents the relationship between set P and set Q based on the numbers they contain.

But wait — actually, looking at the problem again, it seems we’re being asked to identify which diagram matches a specific condition — but the user didn’t state what that condition is! Hmm.

Wait — perhaps this is a standard question where you have to match the correct Venn diagram to a description like “P and Q overlap” or “P is inside Q”, etc., but since no description was given, maybe the task is to interpret what each diagram shows?

Actually, let me re-read the original request. The user said: “Solve the problem accurately.” But there’s no explicit question in the image or text. However, in many such problems, the goal is to identify which diagram correctly represents a given set relationship — for example, if someone says “Set P contains {0,1,3,5,8} and Set Q contains {2,5,8,9}, which diagram shows that?”

Looking at diagram (i):

- Left circle (P): has 0, 1, 3, and also shares 5 and 8 with Q → so P = {0,1,3,5,8}
- Right circle (Q): has 2, 9, and shares 5 and 8 → so Q = {2,5,8,9}
- Overlap: {5,8}

That makes sense for intersecting sets.

Diagram (ii):

- Outer circle is Q: contains 6, 3, and inner circle P which has 8,2 → so P ⊂ Q? Wait, label says outer is Q, inner is P? Actually, labels: outer circle is labeled Q, inner small circle is labeled P? No — look: in (ii), the big circle is labeled Q, and inside it there's a smaller circle labeled P? Wait no — actually, in (ii), the big circle is labeled Q, and inside it there’s a small circle with 8 and 2, and outside that small circle but still inside Q are 6 and 3. And the small circle is labeled P? Let me check:

In (ii):
- Big circle: labeled Q → contains 6, 3, and the small circle
- Small circle inside: labeled P → contains 8, 2
→ So P is subset of Q? But then why is 8 and 2 only in P? That would mean P = {8,2}, Q = {6,3,8,2} → yes, P ⊂ Q.

But wait — in standard notation, if P is inside Q, then all elements of P are in Q. Here, 8 and 2 are in both? No — in diagram (ii), the small circle (P) has 8 and 2, and those are NOT in the rest of Q? Actually, in Venn diagrams, if a number is inside the small circle, it’s in P; if it’s in the big circle but not small, it’s in Q only. So:

In (ii):
- P = {8,2}
- Q = {6,3,8,2} → because 8 and 2 are inside P which is inside Q, so they are also in Q.
→ So P ⊆ Q.

Similarly, diagram (iii):
- Big circle P: contains 3, 8, and small circle Q which has 4,5
→ So Q ⊂ P? Because Q is inside P.
P = {3,8,4,5}, Q = {4,5}

Diagram (iv):
- Two separate circles: P = {5,6,7,8}, Q = {2,3,4} → disjoint sets.

Diagram (v):
- Two separate circles: Q = {3,4,2,1,6}, P = {5,7} → disjoint.

Now — perhaps the actual question (which might be implied or missing) is: “Which diagram shows that P and Q have common elements?” Then answer is (i).

Or “Which shows P is a subset of Q?” → (ii)

“Which shows Q is a subset of P?” → (iii)

“Which shows P and Q are disjoint?” → (iv) and (v)

But since the user didn’t specify, I think there might be a mistake — unless this is part of a larger context.

Wait — perhaps the original problem was to match sets to diagrams? For example, if P = {0,1,3,5,8} and Q = {2,5,8,9}, then diagram (i) is correct.

Maybe that’s it. Let me assume that the intended question is: Given P = {0,1,3,5,8} and Q = {2,5,8,9}, which diagram represents this?

Then:

- Elements only in P: 0,1,3
- Only in Q: 2,9
- In both: 5,8

That’s exactly diagram (i).

So I’ll go with that.

Final Answer: (i)