Where to make a conjunction instead of an implication?

Question

We have this statement in natural language:

Benjamin hates all politicians.

I suggested this formula:

$$ \forall X ( politician(X) \Rightarrow hate(benjamin,X)) $$

But our teacher has written this as a solution:

$$ \forall X ( politician(X) \land hate(benjamin,X)) $$

And I quite can't understand the difference between conjunction and implication in cases like this.

Maybe I need more clarification on implication in the real world to get this straight.

Answer 1

Your teacher is right if and only if $X$ is iterating over a set of politicians (i.e. $X\in \rm{Politicians}$). The mathematical formula $$\forall X \quad\rm{politician}(X) \implies \rm{hate}(B,X)$$

is translated into

For all $X$, if $X$ is a politician, $B$ hates it.

Which is true. The second formula $$\forall X \quad\rm{politician}(X) \land \rm{hate}(B,X)$$

is translated into

For all $X$, $X$ is politician and $B$ hates it.

It means not only all $X$ are politicians but also that $B$ hates everyone (from the set $X$ is iterating over).