Third edition of Artificial Intelligence: foundations of computational agents, Cambridge University Press, 2023 is now available (including the full text).

## 16.3 Relations and the Relational Algebra

Relations are common in AI and database systems.
The
**relational algebra** defines operations on relations and is
the basis of relational databases.

A **scope** *S* is a set of variables.
A **tuple** *t* on
scope *S*, has a value on each variable in its scope. We write *val(t,X)* to be the
value of tuple *t* on variable *X*. The value of *val(t,X)* must be in
*dom(X)*. This is like the mathematical notion of tuple,
except the index is given by a variable, not by
an integer.

A **relation** is a set of tuples, all with the same scope. A
relation is often given a name. The
scope of the tuples is often called the relation **scheme**. A
**relational database** is a set of relations. A scheme of a
relational database is the set of pairs of relation names and relation schemes.

A relation with scope *X _{1},...,X_{n}* can be seen as a Boolean factor on

*X*, where the true elements are represented as tuples.

_{1},...,X_{n}Often a relation is written as a table.

**Example 16.2:**The following is a tabular depiction of a relation,

*enrolled*:

Course | Year | Student | Grade |

cs322 | 2008 | fran | 77 |

cs111 | 2009 | billie | 88 |

cs111 | 2009 | jess | 78 |

cs444 | 2008 | fran | 83 |

cs322 | 2009 | jordan | 92 |

The heading gives the scheme, namely *{Course, Year, Student, Grade}*, and every other
row is a tuple. The first tuple, call it *t _{1}* is defined by

*val(t*,

_{1},Course)=cs322*val(t*,

_{1},Year)=2008*val(t*,

_{1},Student)=fran*val(t*.

_{1},Grade)=77The order of the columns and the order of the rows is not significant.

If *r* a relation with scheme *S*, and *c* is a condition on the
variables in *S*, the **selection** of *c* in *r*, written
*σ _{c}(r)*, is the set of tuples in

*r*for which

*c*holds. The selection has the same scheme as

*r*.

If *r* is a relation with scheme *S*, and *S _{0} ⊆S*, the

**projection**of

*r*onto

*S*, written

_{0}*π*, is the set of tuples of

_{S0}(r)*r*where the scope is restricted to

*S*.

_{0}**Example 16.3:**Suppose

*enrolled*is the relation given in Example 16.2.

The relation *σ _{Grade > 79} (enrolled)* selects those tuples in

*enrolled*where the grade is over 79. This is the relation:

Course | Year | Student | Grade |

cs111 | 2009 | billie | 88 |

cs444 | 2008 | fran | 83 |

cs322 | 2009 | jordan | 92 |

The
relation *π _{{Student,Year}}(enrolled)* specifies what years students
were enrolled:

Student | Year |

fran | 2008 |

billie | 2009 |

jess | 2009 |

jordan | 2009 |

Notice how the first and the fourth tuple of *enrolled* become the same tuple
in the projection; they represent the same function on *{Student,Year}*.

If two relations on the same scheme, the **union**,
**intersection** and **set difference** of these are defined as the
corresponding operations on the set of tuples.

If *r _{1}* and

*r*are two relations, the natural

_{2}**join**of

*r*and

_{1}*r*, written

_{2}*r*is a relation where

_{1}r_{2}- the scheme of the join is the union of the scheme of
*r*and the scheme of_{1}*r*,_{2} - a tuple is in the join, if the tuple restricted to the scope of
*r*is in the relation_{1}*r*and the tuple restricted to the scope of_{1}*r*is in the relation_{2}*r*._{2}

**Example 16.4:**Consider the relation

*assisted*:

Course | Year | TA |

cs322 | 2008 | yuki |

cs111 | 2009 | sam |

cs111 | 2009 | chris |

cs322 | 2009 | yuki |

The join of *enrolled* and *assisted*, written *enrolled assisted* is the relation:

Course | Year | Student | Grade | TA |

cs322 | 2008 | fran | 77 | yuki |

cs111 | 2009 | billie | 88 | sam |

cs111 | 2009 | jess | 78 | sam |

cs111 | 2009 | billie | 88 | chris |

cs111 | 2009 | jess | 78 | chris |

cs322 | 2009 | jordan | 92 | yuki |

Note how in the join, the information about *cs444* was lost, as there
was no TA in that course.