Proving a statement

[tuanl]

To prove the statement “if all A are B, then all C are D”, it suffices to show which of the following:

A. All A are D, and all B are C
B. All A are C, and all D are B
C. All C are A, and all B are D
D. All B are C, and all A are D

I'm confused by this problem. I tried all 4 answer choices and it seems to me that only C comes close to the correct answer. But if C is correct, then we have "all(all(all C)) are D", not "all C are D" (!?). Can someone help me solve it?

 

[LochWulf]

If both assertions of choice C hold and the hypothesis of the statement is assumed, then together you have

C subset of A, A subset of B, B subset of D.

That the subset relation is transitive then establishes the conclusion. I suppose that correlates with your nested arrangement, working from the innermost set (all C) outward.

 

[tuanl]

If both assertions of choice C hold and the hypothesis of the statement is assumed, then together you have

C subset of A, A subset of B, B subset of D.

That the subset relation is transitive then establishes the conclusion. I suppose that correlates with your nested arrangement, working from the innermost set (all C) outward.

Great explanation! I like your interpretation of "all A are B" as subset of a set. Many thanks