| slovo | definícia |
constructive proof
(foldoc) | constructive proof
A proof that something exists that provides an
example or a method for actually constructing it.
For example, for any pair of finite real numbers n < 0 and p > 0,
there exists a real number 0 < k < 1 such that
f(k) = (1-k)*n + k*p = 0.
A constructive proof would proceed by rearranging the above to
derive an equation for k:
k = 1/(1-n/p)
From this and the constraints on n and p, we can show that 0 < k <
1.
A few mathematicians actually reject *all* non-constructive
arguments as invalid; this means, for instance, that the law of
the excluded middle (either P or not-P must hold, whatever P is)
has to go; this makes proof by contradiction invalid. See
intuitionistic logic.
Constructive proofs are popular in theoretical computer science,
both because computer scientists are less given to abstraction
than mathematicians and because intuitionistic logic turns out
to be an appropriate theoretical treatment of the foundations of
computer science.
(2014-08-24)
|
| | podobné slovo | definícia |
non-constructive proof
(foldoc) | non-constructive proof
existence proof
(Or "existence proof") A proof that something exists
that does not provide an example of that thing or a method for
finding an example. (A constructive proof does provide such an
example or method).
For example, for any pair of finite real numbers n < 0 and p > 0
there exists a real number 0 < k < 1 such that
f(k) = (1-k)*n + k*p = 0.
A non-constructive proof might proceed by observing that as k
changes continuously from 0 to 1, f(k) changes continuously from n
to p and, since they lie either side of zero, f(k) must pass
through zero for some intermediate value of k. This proof does
not tell us what that value of k is, only that it exists.
Cantor's proof that the real numbers are uncountable can be
thought of as a non-constructive proof that irrational numbers
exist. There are existence theorems with no known constructive
proof.
(2014-08-23)
|
|