I'm doing discrete math problems in relation to the predicate logic and I'm having problems when figuring out what the antecedent and subsequent implication is.
Most of them if I take them out the first time, but then I start that if I could be this way or the other and I messed up. I have
- help: $h (X, Y)$
- they live in the house of: $v (X)$
- work: $w (X, Y)$
- friends: $f (X, Y)$
All those who help Juan live in Manolo's house. $(\forall X) h(X,j) \to v(X) $
Antonio helps everyone he works with. $(\forall X) w(a,X) \to h(a,X) $
All of Carlos' friends work with Juan. $(\forall X) f(X,j) \to w(X,c) $
Antonio is Carlos's friend. $f(a,c)$
What I sometimes confuse is the way in which the phrase is given. I confuse in this case the second and the third, that I do not know if the meaning of the implication is well done or wrong
ADDITIONAL QUESTION: does there have to be some direct mechanism to detect which is the antecedent and the consequent directly? I always messed up. I already made a response a while ago but I still understand it very well, I do not know if someone has some trick to recognize it better.
First of all, the 'lives in the house of' needs to be 2-place relation:
$v(X,Y): X$ lives in the house of $Y$
Second, use parentheses to indicate the scope of the quantifiers. For example, for the first one, if you do:
$$(\forall X) h(X,j) \rightarrow v(X,m)$$
then the $X$ in $v(X,m)$ is no within the scope of the quantifier. So, instead do:
$$(\forall X) (h(X,j) \rightarrow v(X,m))$$
Similarly, for the second one, you should do:
$$(\forall X) (w(a,X) \rightarrow h(a,X))$$
OK, for the third one you did indeed mix things up: we're talking about friends of Carlos, i.e. you need to use $f(X,c)$, and we're talking about working woith Juan, so you need $w(X,j)$. So, you get:
$$(\forall X) (f(X,c) \rightarrow w(X,j))$$