Set is a structure.. or structure is a set.., which is correct?

Question

A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of associated mathematical objects is called the structure and the set is called the underlying set. (Extracted from here)

A set (with other mathematical objects such as subsets, sets of subsets, operations and relations,) is a mathematical structure if it satisfies all the requirements (axioms).

Question:

Are the above two statements equivalent? Which is correct?

My interpretation:

The above two statements are equivalent, because, a Set can be a Structure or a Structure can be a Set. They are interchangeable. They mean the same thing.

A set can form a structure together with some operations or relations. — Wuestenfux, Jun 07 '19 at 13:40
Sometimes informally we don't distinguish the set from the structure — J. W. Tanner, Jun 07 '19 at 13:51

score 3 · Answer 1 · edited Jun 07 '19 at 13:49

Both of those statements are essentially equivalent correct informal ways to describe the idea of a mathematical structure. They convey the same idea, which you can recognize when you see examples like groups, vector spaces or graphs.

Since they are only informal statements, you don't have to argue that they are both correct with a reason like "Set can be a Structure or a Structure can be a Set".

Set is a structure.. or structure is a set.., which is correct?

1 Answers1