FOL: Two friends have different friends

Question

I am new to FOL, and I have some difficulties understanding it. Here are two things:

1. Two PEOPLE have different friends
2. Two friends have the same other friend

A relation for "a is a friend of b" is given by E(a,b)

My solution would be

1. $\forall$ a,b $\rightarrow$ $\neg$(E(a,v) $\wedge$ E(b,v))
2. $\forall$ a,b E(a,b) $\rightarrow$ E(a,v) $\wedge$ E(b,v)

where a,b are Humans v is a Var. If this is correct, can somebody explain why we use $\rightarrow$ instead of $\leftrightarrow$ ? e.g. I saw this:

Brother(John, Paul) $\rightarrow$ Brother(Paul, John)

but wouldn't it be more

Brother(John, Paul) $\leftrightarrow$ Brother(Paul, John)?

Answer 1

For the second one: two friends have other friends, I would suggest: $\forall a,b: E(a,b) \rightarrow [ \exists c :E(a,c) \land E(b,c)] $ and I assume it is implied that if we use the letter $c$ , we have that $a \neq c , b \neq c$.

Answer 2

Assuming that the two sentences meaning it that for every pair of people the specific property holds (you could also interpret them with at least one or exactly one) this would be my suggestion:

1. $\forall a,b: \neg(a=b) \land E(a,c) \rightarrow \neg E(b,c)$
2. I think your solution is ok, without any other constraint, this because the sentence doesn't specify other constraints (but still I'm not sure about the necessity of the existentialistic operator for c)

I also don't understand you comment about the notation: your $a,b,v$ MUST all be variables, then the possibility of them to not represent necessary a human element of your domain should be managed or via the definition of the $E$ function or via another function (like $Human(x)$). However like this it's ok if you have only humans in your domain.

The use of implication instead of equivalence (and I'm talking about the solution 2, because the implication in 1 doesn't make sense to me) is because you are saying that the property holds given the fact that a and b are friends, otherwise you cannot say anything.
With equivalence (or double implication) you would also state that two people that have the same friend MUST also be friend with each other. Sentence 2 only specify the first relation, not the second one.