"Relation" versus "relationship"

To me, the main difference is that relationship is broader than relation:

• both can mean “the way in which two or more concepts, objects, or people are connected”;
• in addition, relationship can mean “the state of being connected”.

So, in “She has a close relationship with her daughter”, because you're talking about the fact a particular existing connection, I wouldn't use “relation”.

In your second example, if “the relationship between poor housing and health problems” is merely an academic study of the possible correlation, then “relation” would fit well.

In the third, again, the “special relationship” is a particular, existing connection, so “relation” wouldn't fit the bill. In addition, “special relationship” is a well established phrase since post-World War II times.

While relation and relationship refer to the connection between things, relation shades more toward the way things are connected, while relationship refers to the connection itself. The difference is not spacious.

Example:

"The size of the targets bore no relation to their importance." [NOAD]

This is different from

"The two friends enjoyed a very close relationship."

There's other meanings as well...

"Fred has had [sexual] relations with Sally."

"I hate having to invite my poor relations to parties, because they always steal the silverware."

In the database sense, without being pedantic about the academically correct definitions of "Relation" and "Relationship", especially at the physical level when we are discussing tables (not tuples) but in keeping with the English meaning ...

• A real (table) or derived (columns from more than one table) is a Relation

  • but most developers will cringe at, it is not commonly understood
  • therefore we commonly call them tables or derived tables

• and therefore the Relationships can be simply Relations between tables

  • both defined (as Foreign Keys) and not (derived, as per other data values, such as related by date or time or some other value)
  • personally I would capitalise the formally defined Relations and leave the derived or projected relations uncapitalised.

Relationship means "bond" while relation means "connection", used in formal and informal lexicon.