为什么在关系代数中将并，交和差运算称为布尔运算？

Question

Why are union, intersection and difference operations of relational algebra called boolean operations?

I found them called that in the first line in section 5.4.1 Boolean operations (Section 5.4 is Relational Algebra and Datalog) in a book named A First Course in DATABASE SYSTEMS by Ullman & Widom.

Answer 1

The statement that you are citing is from the Chapter 5 (Algebraic and Logical Query Languages) of the book “First Course in Database Systems” by Ullman and Widom (Pearson 2013).

In particular, the sections 5.3 and 5.4 of that Chapter treat the language Datalog, which can be used to work on a Relational Data Base, in which a Relation is seen as predicate , not a set (see Section 5.3.1).

In other words, a tuple (x1, x2, ..., xn) of R is seen as the fact that the relation R is true for the arguments specified (x1, x2, ..., xn). In this way one could transform the set operators discussed in the context of Relation Algebra (union, difference, intersection) as rules of Datalog expressed through the use of boolean operators, like AND, NOT, etc.

In fact, you can see that in the same book, in section 2.4.4, they are called Set Operators (as they actually are), so I think the naming of “boolean operators” is due to the fact that they are strictly related to boolean operators and are discussed in the context of a logical view of a database.

Answer 2

The notion of "boolean" relational operations is an idiosyncratic ad hoc distinction/categorization in that book and chapter. The authors identify 3 common relation operators that "can each be expressed simply in Datalog". The next sections at the same level go on to each express some other relational operator. ("Selections can be somewhat more difficult to express in Datalog.") The 3 operators' treatments are similar to each other and different from others' in a certain context-specific "boolean" way. (So it's not a particularly helpful or deep distinction.)

5.4.1 Boolean Operations

The boolean operations of relational algebra--union, intersection, and set difference--can each be expressed simply in Datalog.

• To take the union R ∪ S, [...] As a result, each tuple from R and each tuple of S is put into the answer relation.
• To take the intersection R ∩ S, [...] Then, a tuple is in the answer relation if and only if it is in both R and S.
• To take the difference R - S, [...] Then, a tuple is in the answer relation if and only if it is in R but not in S.

So the "boolean" is because each relation operator corresponds to a certain boolean operator in that context:

UNION returns tuples that are in one operand OR the other
INTERSECTION returns tuples that are in one operand AND the other
DIFFERENCE returns tuples that are in one operand AND NOT the other

(Datalog also literally uses AND and AND NOT, but OR is implicit.)

More importantly, this can be put another way: If every base relation holds the tuples that make a true proposition (statement) from some associated predicate (sentence template parameterized by attribute names) then every result has a predicate made from its operand predicates, and its value holds the tuples that make a true proposition from that predicate:

U holds tuples where U
V holds the tuples where V
U UNION V holds the tuples where U OR V
U INTERSECTION V holds the tuples where U AND V
U DIFFERENCE V holds the tuples where U AND NOT V

But more than that it's not just that those operators correspond to some boolean propositional logic connectives/nonterminals, but that every relational operator corresponds to a predicate logic connective/nonterminal, every query expression has an associated predicate, and every query result value holds the tuples that make a true proposition from that predicate:

U JOIN V holds the tuples where U AND V
R RESTRICT condition holds the tuples where U AND condition
U PROJECT A holds the tuples where FORSOME values for all attributes but A, U
U RENAME A A' holds the tuples where U with A replaced by A'

So the book ties those three operators to the syntax & semantics of Datalog, but really all operators can be tied to the syntax & semantics of predicate logic . Which is the language/notation of precision in mathematics, science (including computer science) & engineering (including software engineering.) In fact, that's how we know what a query means . So it's not really that those three operators are in some sense boolean so much as that all relational operators are logical : every relation expression corresponds to a predicate logic expression.

(The three operators are also frequently called the "set operators", since the set that is the body of the result relation arises from the sets that are the bodies of the operand relations according to the set operators by the same names. And that might be a helpful way to group them in your mind or remember them or their names, and no doubt inspired their names. But in light of the correspondence between relation operators and predicate logic connectives/nonterminals and the correspondence between relation expressions and predicate logic expressions and the correspondence between predicates and relation values, the fact that some relation operators are reminiscent of some set operators is totally irrelevant .) (Just like that some operators are reminiscent of boolean operators. Just like that you can make an algebra with "relations" whose bodies are bags instead of sets.)