# Can someone check my logic (sentential logic)

• Jul 12th 2010, 04:18 PM
swtdelicaterose
Can someone check my logic (sentential logic)
D: Dan is Married
G: George is Married
F: Fred is Married

At most one of them is married.

I figured you have just have to take the negation of the cases where two or more people are married (three cases), and the case when all of them are married. So I did this:

$\displaystyle \neg\left[\left(\left(G\&\left(F\&D\right)\right)\vee(G\&D)) \vee((G\&F)\vee(F\&D))\right]$
• Jul 12th 2010, 06:03 PM
Ackbeet
That's one valid approach. The other valid approach would be to OR the admissible cases together:

$\displaystyle D\vee F\vee G\vee\neg(D\vee F\vee G).$

[EDIT]: See Plato's post below for a correction to this approach.

I think you'll find this approach more scalable, if you throw more guys into the mix!

Incidentally, because disjunction is associative, you probably don't need that many parentheses, unless your teacher is being a stickler about wff's.
• Jul 13th 2010, 03:08 AM
Plato
The following is a true statement if all three were married.
$\displaystyle D\vee F\vee G\vee\neg(D\vee F\vee G).$

We want at most one is married.
$\displaystyle (\neg D \wedge \neg F \wedge \neg G) \vee (D \wedge \neg F \wedge \neg G) \vee (\neg D \wedge F \wedge \neg G) \vee (\neg D \wedge \neg F \wedge G)$.
• Jul 13th 2010, 03:21 AM
avonm
$\displaystyle \neg ( ( D \wedge F ) \vee (D \wedge G) \vee (F \wedge G) )$
• Jul 13th 2010, 03:30 AM
avonm
remember

$\displaystyle (D \wedge F \wedge G) \to (D \wedge F)$
$\displaystyle (D \wedge F \wedge G) \to (D \wedge G)$
$\displaystyle (D \wedge F \wedge G) \to (F \wedge G)$

so you don't need to care of

$\displaystyle (D \wedge F \wedge G)$