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So, you’ve got an XML
document containing a bunch of integers: <Test> <x>2</x> <x>4</x> <x>5</x> <x>7</x> <x>10</x> </Test> You want to know: Is
it true that not all the integers are even? To find the answer,
you create this XPath expression:
not (every $x in /Test/* satisfies $x mod 2 eq 0) “Is
it not the case that every x in Test is even?” Alternatively,
you could have written the XPath expression this way:
some $x in /Test/* satisfies not ($x mod 2 eq 0) “Are
there some x in Test that are not even?” The
two XPath expressions are equivalent – they produce the same results. Let’s
denote the test for evenness ($x mod 2 eq 0) with p(x), where p represents the evenness test on x.
Let’s
denote the set of values being evaluated ( /Test/* ) with seq. Then
the two XPath expressions can be written:
not (every $x in seq satisfies p(x))
some $x in seq satisfies not p(x) Using
a slightly more abstract (non-XPath) notation, we can write the expressions as:
not (for all x) p(x)
(for some x) [not p(x)] The
two expressions are equivalent. “for
all” and “for some” are called quantifiers. In particular, “for all” is called the
universal quantifier and “for some” is called the existential quantifier. The
equivalence of these two expressions:
not (for all x) p(x)
(for some x) [not p(x)] is
called the “duality of quantifiers”. Here
are two more equivalent expressions, exhibiting the duality of quantifiers:
not (for some x) p(x)
(for all x) [not p(x)] Converting
them to XPath, we have:
not (some $x in /Test/* satisfies $x mod 2 eq 0)
every $x in /Test/* satisfies not ($x mod 2 eq 0) Both express this: “Is it the case that none of the x values are even?” Neat! /Roger |