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Databases relationships are quantified with the following categories:

• One-to-one relationships
• One-to-many relationships
• Many-to-many relationships

We’ll discuss each of these relationships and provide an example. If you think of a family structure when thinking about relationships, you’re ahead of the game. When you spend time alone with one parent, that’s a specific type of relationship; when you spend time with both your parents, that’s another one. If you bring in a significant partner and all of you—your parents, you, and your partner—do something together, that’s another relationship. This is identical to the bucket analogy. All those different types of relationships are like specific buckets that hold the dynamics of your relationships. In the database world, this is the data you’ve created.

What is that bucket analogy the write is talking about?

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    source – Nico Aug 14 '14 at 16:22
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    It's an analogy that the author made a few sentences previously: "Think of the relationship as a repository or bucket, and each bucket of data has a specific relationship." (emphasis added) – Hellion Aug 14 '14 at 16:24
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    The bucket analogy is weak, if you are really learning relational database design I would suggest you stick to spreadsheets and the like. Buckets become useful only in the context of non-relational data. – Phil Aug 14 '14 at 18:35
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    It doesn't make any sense to me. They don't explain how a relationship is like a bucket. They might as well have said that a relationship is like an orange. This book received very poor reviews. It seems that it is far below O'Reilley's usual high standard. – Dangph Aug 14 '14 at 19:45
  • I do not have a citation for this: A "bucket" is an abstract logical space that constrains information, so as to delimit it from other possible permutations. Roughly akin to "context," "iteration," "model," "scenario." – Seamus Nanatchk Oct 29 '14 at 3:33
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@Nico, thank you for the source!

To answer your question, the bucket analogy the writer is talking about is, as several commenters to your original post have stated, previously mentioned in the text:

Now that you have separate tables that store related data, you need to think about the number of items in each table that relate to the number of items in another table. This is all about relationships and the type of relationships data falls into. Think of the relationship as a repository or bucket, and each bucket of data has a specific relationship.

That being said this passage seems to be rather poorly written. Several commenters to your original post have stated as much. I would find a different source to learn from if I were you.

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