Peter states the following facts about the football premiership:
If St Kilda beat Hawthorn, Franklin is included next week
Hawthorn beating St Kilda means that Geelong will top the ladder
Hawthorn will make the top eight only if Carlton miss out on the finals
If Geelong lose to Essendon then Geelong can't top the ladder
If Hawthorn fail to make the top eight, Franklin will be dropped next week
a) Formulate these five facts as statements in propositional logic.
I did this:
p: St Kilda beat Hawthorn
q: Franklin included next week
x: Geelong top the ladder
y: Hawthorn will make the top 8
z: Carlton miss out on finals
a: Geelong lose to Essendon
b) Is it the case that if Geelong don't manage to top the ladder, Carlton miss out
on the finals? Give your reasoning.
So is this part asking whether is true or false?
Do I use rules of inference from my 5 propositions from part a) to figure out b)? I can't seem to link them together...
Any help would be much appreciated!
Ok I 'think' I got it after many trials, can anyone confirm if my working is correct/wrong. Thanks very much!
From 1. name this 1'.
Assuming 5 and 1' are true and then using Hypothetical Syllogism we yield to be true.
From 3. name this 3'. Assuming 3' is true then using Simplification yields to be true.
Looking at and using Hypothetical Syllogism we yield to be true.
From 2. we have name this 2'.
Using Hypothetical Syllogism with 2' and yields .
Thus is true!
Thanks for the help guys!
Ackbeet, thanks for that pickup, that was a stupid mistake by me - however I was under the impression that "A if B" means B implies A, however "A only if B" means A implies B?
edit - ah just as I posted this, plato confirmed it haha
Hmmm, it seems I am stuck with a further extension of this problem.
The extension is that "can one deduce that either Geelong lose to Essendon or Carlton make the finals?"
So my interpretation of this question is that is true or false?
Now to prove this, if is true or is true then is true, however if and are both false, then is false.
But no matter how hard I try, I can not seem to find any way of manipulating the information we have to prove what I described above