1. Fundamentals/ICY: Databases 2013/14 Week 10 – Monday/Friday – relations, contd.
John Barnden
Professor of Artificial Intelligence
School of Computer Science
University of Birmingham, UK

2. Reminder

3. Relation from a Table   ‘9568876A’, ‘Chopples’, 37 >,
People
PERS-ID
NAME
AGE A
Chopples 37 Z
Blurp F
Rumpel 88
The relation at the moment is
  ‘ A’, ‘Chopples’, 37 >,
 ‘ Z’, ‘Blurp’, NULL >,
 ‘ F’, ‘Rumpel’, 88 > 

4. New

5. Note: ((Items in double round brackets are optional material))

6. “Relations don’t remember where they came from”
Consider a relation R on A, B, C, D, E, …
i.e., R  A  B  C  D  E  ….
Suppose A  AA, B  BB, C  CC, etc.
Then: a tuple formed from sets A, B, … is also automatically a tuple formed from AA, BB, …
That is, R  AA  BB  CC  DD  EE  ….
So R is also a relation on AA, BB, CC, DD, EE, ….
So a relation has no very tight connection to the original sets it might have been defined from, unlike the case of tables, where the attribute domains are part of the nature of the table.

7. “Arity” of Relations
A relation on two sets is binary, on three sets is ternary, …
… even when not all the sets are different.
So a relation on A and A is still binary and NOT “unary.”
The members of the relation are two-element tuples.
A relation on, say, A, B and A is ternary and not binary.
The members of the relation are three-element tuples.

8. “Arity” of Relations, contd.
A “unary relation” on A is a set of singleton tuples formed from A elements.
(But is sometyimes loosely taken to be just a subset of A.)
Unusual (though not inconceivable) to want a single-attribute table in a finalized ER model.
But one-attribute tables often arise dynamically from table operations, as you know.

9. Relations from Somewhere to Somewhere
A relation R “from” set A “to” set B is the same thing as a relation “on” A “and” B — just different terminology.
Similarly, a relation from A, B, C to D, E is the same thing as a relation on A, B, C, D, E.

10. Changing the Sets in a Relation Around
A relation R on A, B, C, D, E, say, obviously “induces” (i.e., gives rise to, in a natural way) a relation on any reordering of the sets, such as D, A, B, E, C, just by reordering each tuple in the same way.
Thus, R induces a relation from, say, D, A to B, E, C.
When there are just two sets A and B, the (only possible) reordering of the sets gives the inverse of R.

11. Removing some of the Sets in a Relation (Projection)
And we can remove some of the sets and the corresponding items from each tuple.
Given the relation on D,A,B,E,C, we can get a relation on, say, D,B,C, just by removing the second and fourth item from each tuple.
This is the mathematical operation underlying the PROJECT relational operator on tables (what I would prefer call Select-Columns or Select-Attributes).

12. Functional Relations (Partial Functions)
A relation R from A to B is functional if, for any a in A, there is AT MOST one (but perhaps no) b in B such that a, b> is in R.
So several things in A can be related to the same thing in B.
But you can’t have several things in B related to the same thing in A.
A functional relation from A to B is also called a partial function from A to B.

13. Functional Relations, contd.
Can generalize:
a relation R from A1, A2, A3 … to B1, B2, B3, …is functional if,
for each combination of things a1, a2, a3, … in A1, A2, A3, … respectively,
there is at most one b1, b2, b3, … in B1, B2, B3, … respectively such that a1, a2, a3, …, b1, b2, b3, …> is in R.

14. Functional Relations arising from Functional Dependencies in a DB
Suppose attribute X is functionally dependent on (= determined by) attributes A, B, … in a table.
Then, at any moment, the induced relation from A, B to X is a partial function from the A, B, … value domains to the X value domain.
Special case:
Consider any superkey (e.g., the primary key) of a table.
Then the relation in the table at a given moment is a partial function from the superkey’s domains to the remaining attribute domains (if any).

15. Caution
The word “partial” in the phrase “partial function” has nothing to do with the word “partial” in “partial dependency” as discussed under Normalization.
Any dependency relationship in a table gives us a partial function, irrespective of whether the dependency is also “partial” in the special sense of involving only a part of the PK.

16. ((Restriction of a Relation))
Consider a relation R from A to B,
and a subset AA of A.
Then the restriction of R to AA is the relation derived from R by restricting attention to AA,
i.e., including only tuples whose first element is in AA.
The new relation is notated R|AA

17. ((Restriction More Generally))
Consider a relation R from A, B, …C to D, E, …, F
and subsets AA of A, BB of B, …, CC of C.
Then the restriction of R to AA, BB, …, CC is the relation derived from R by restricting attention to AA, BB, …, CC
i.e., including only tuples whose first few elements are in AA, BB, …, CC respectively.
The new relation is notated R|AA, BB, …, CC

18. Totality of Relations
A relation R from A to B is total (on A) if it relates everything in A to at least one thing in B.
I.e., for every member a of A, there is at least one b in B such that  a, b > is in R.

19. Totality, contd. Can generalize:  a, b, c, …, d, e, … > is in R.
A relation R from A, B, C, … to D, E, … is total (on A, B, C, …)
if for every member a of A, b of B, c of C, etc.
there is at least one d in D, e in E, etc. such that
 a, b, c, …, d, e, … > is in R.

20. Partiality of Table Relations
The relation in a table (at a given moment), considered as a relation from any of its attribute value domains to the remaining value domains, will almost always NOT be total.
This is simply because it’s highly unlikely that all possible combinations of values from the former collection of value domains will appear in the table!!

21. Functions
A total functional relation from A to B is called a function from A to B.
Each thing in A is related to exactly one thing in B. (But two different things in A can be related to the same thing in B, and not everything in B needs to be related to anything in A. So the inverse relation is not necessarily either functional or total.)
Caution: every function is also a partial function.

22. ((From Partiality to Totality by Restriction))
We can always turn a merely-partial R from A to B into a total one by slimming A down enough! Just remove the members of A that aren’t related to anything by R, to get a new set AA. We don’t remove any tuples from R.
R (as a relation from AA to B) is total on AA.
And note that R|AA = R.
AA is called the domain of R, notated dom(R).
Not to be confused with “value domains” of DB entity attributes.
Can generalize the above to non-binary relations.

23. ((Totality contd. and “Onto”))
A relation R from A to B is onto if for everything in B there is at least one thing in A that is related by R to it. I.e.:
For every member b of B,
there is at least one a in A such that a, b> is in R.
Onto-ness is just totality in the other direction.
You can also say that R is total on B, or that the inverse of R is total.

24. Other Categories of Relation
A relation R from A to B is one-to-one (1-1) if, for any a in A, there is at most one b in B such that a, b> is in R, AND for any b in B, there is at most one a in A such that a, b> is in R.
That is, both the relation and its inverse from B to A are functional. (But they don’t need to be total.)
To put it another way: it is functional and different members of A map to (= are related to) different members of B.
Or again: Different members of A map to different members of B and different members of B map to different members of A.