Why can’t I conclude that two groups overlap just because they share a category?

Because both groups can fit inside the same larger category while occupying separate parts of it. The claims say A ⊆ C and B ⊆ C: every A member is in C, and every B member is in C. They do not say that any particular member belongs to both A and B. To establish overlap, you need at least one shared member, or evidence that the intersection A ∩ B is nonempty. [S1]

Imagine C is everyone attending a community club. A is the members who play chess, and B is the members who paint. Both groups are within the club, but the statements alone do not tell you whether any club member does both. They might; they might not. A shared larger category leaves both possibilities open.

Treat the large rectangle as the shared category C. The circles represent the smaller groups. These are two possible arrangements:

Case 1: overlap                 Case 2: no overlap
+------------------+            +------------------+
| C                |            | C                |
|   ( A )          |            |   ( A ) ( B )    |
|      ( B )       |            |                  |
+------------------+            +------------------+

In both cases, A and B are inside C. Only Case 1 has a shared region, so only it shows overlap. The diagram is a way to visualize set relationships, not proof that one arrangement rather than another is true. Two subsets of one larger set can be shown as separate circles when they have no elements in common. [S2]

For a quick test, try drawing a valid counterexample: can you place A and B separately inside C without contradicting either statement? If yes, the statements do not force overlap.

Here is a fictional example. Let C be the set of three people {Mina, Jo, Lee}. Let A = {Mina} and B = {Jo}. Every member of A is in C, and every member of B is in C, but A and B share nobody. This single possible arrangement shows why the conclusion does not follow from the premises.

Compare it with another possible arrangement: A = {Mina} and B = {Mina, Jo}. Both groups are still inside C, and now Mina is in both. The original statements allow either arrangement. They do not tell you which one applies to a real situation.

Be careful not to confuse “both groups are in C” with “some member of A is in B.” The first is about where each group sits relative to C; the second is a separate claim about a shared member.

Write down exactly what the premises say, then test whether they guarantee the proposed conclusion. For “all A are C” and “all B are C,” record A ⊆ C and B ⊆ C. The overlap conclusion would require A ∩ B ≠ ∅—in words, at least one member belongs to both groups. [S1]

Now try a counterexample: use different members for A and B, with both still inside C. If that arrangement satisfies every premise but has no overlap, the conclusion is not established. If the task asks what you can conclude, say that both groups are contained in C, while overlap remains unknown. If it asks what extra evidence would settle the question, look for a named shared member or a statement that some A is also B. A diagram can make the possibilities clear, but the premises—not the drawing—determine what is justified.