Compare relations

We can compare relations. Since the restrict operation returns a relation, we can compare two filtered results.

$S[STATUS=20]

$S[CITY="London"]

The equality comparison, using = will return true for this case.

$S[STATUS=20] = $S[CITY="London"]

true

We can also test whether one relation is a subset of another, using <=. This symbol is meant to resemble the subset symbol, ⊆ (which you can use instead, if your keyboard can create it).

$S[STATUS=20] <= $S[CITY="London"]

true

$S[STATUS=20] <= $S

true

$S[STATUS=20] <= $S[STATUS=30]

false

There is also a superset operator: >= (or ⊇). And there are pure subset and pure superset operators (i.e. excluding the cases when both are equal): < and > (or ⊂ and ⊃)

Heading

We can use the heading function to get the set of attributes for a relation.

heading($S)

And because this returns a relation we can do further operations on it. For example, to find the common attributes in $S and $SP:

heading($S) & heading($SP)

Cardinality

We can count how many tuples are in a relation using count().

count($S[STATUS=20])

2

We can check if a relation has no tuples using empty().

empty(dee)

false

empty(dum)

true

Degree

We can count how many attributes are in a relation using count and heading.

count(heading($S))

4

Quantification

We can add a boolean attribute to a relation (or use an existing one). Then we can use the all or exists (or ∀ or ∃ if your keyboard can create them) operators to check if all or any tuples have the attribute set true.

exists($S{*, test:=(STATUS=20)}, test)

true

all($S{test:=(STATUS=20)})

false

all($S{test:=(STATUS<100)})

true

exists and all should be more efficient than counting filtered matches since they can short-circuit and finish early without necessarily checking all the tuples.