Predicate Logic

• Feb 6th 2010, 06:04 AM
jds80021
Predicate Logic
I had Discrete math in my undergrad CompSci program, I am now in grad school and having to use it again and I am a bit rusty: Need help with verifying some predicate logic translations.

English: Notevery airport has a runway for large jets.
My translation:
hasrunwayfor(x,j,l): true if airport x has a runway for jet j, that is large l, false otherwise
My Formula: ∃x:Airport|∀j:Jet|Runway(j,l)
Do my translation and formula look correct?
Help is greatly appreciated. Thanks in advance
• Feb 6th 2010, 06:40 AM
Plato
Quote:

Originally Posted by jds80021
English: [FONT=&quot]Notevery airport has a runway for large jets.

This may be a surprise, but that statement is logically equivalent to:
Some airport does not have a runway for large jets.
• Feb 6th 2010, 06:57 AM
bmp05
Plato,
is it possible to use a conjunction in this case. Such as:
$
(\forall A)(\exists R_{Large})[A \wedge \neg R_{Large}]
$

Although this would be true for airports without large runways - something different.
• Feb 6th 2010, 07:01 AM
jds80021
Logical equivalent
Thanks Plato, Yes, I can see the logical equivalent , but what is the predicate logic statement and formula?
• Feb 6th 2010, 07:04 AM
Plato
Quote:

Originally Posted by bmp05
Plato,
is it possible to use a conjunction in this case. Such as:
$
(\forall A)(\exists R_{Large})[A \wedge \neg R_{Large}]
$

Although this would be true for airports without large runways - something different.

$
(\forall A)(\exists R_{Large})[A \wedge \neg R_{Large}]
$

That tranlates as "All airports have a runway not for large jets."
Is that logically equivalent to “Not every airport has a runway for large jets.”?
• Feb 6th 2010, 07:12 AM
bmp05
Plato. There should be a computer program for this- and there is isn't there? Prolog or something?

Anyway, I was way off the mark... so what about: "Some airports have a runway not for large jets?"
$(\exists A)(\forall R_{Large})[A \wedge \neg R_{Large}]$

$Runway(A, B)$ can this type of statement (?) always be replaced by a simpler statement?
• Feb 6th 2010, 07:23 AM
Plato
If $L(x)$ is the predicate “x has a runway for large jets” and
$A(x)$ is the predicate “x is an airport”.
Then the translation is $\left( {\exists x} \right)\left[ {A(x) \wedge \neg L(x)} \right]$ .