I don't really understand the difference between → (implication) and ∧ (conjunction) in propositional logic. As far as I know:
- A ∧ B is only true when both A and B are true.
- A → B is only true when it's not the case that A is true and B is false.
However, when we have to translate English sentences into mathematical expressions with quantifiers I have some problems. For example:
T(x,y): "student x likes cuisine y"
U: All the students at your school and all the cuisines.
∀x∀z∃y( (x≠z) → ¬ (T(x,y) ∧ T(z,y) ))
The solution of this is: "Two different students don't like a cuisine".
I don't understand this, though. Because the expression between the parenthesis ((x≠z) → ¬ (T(x,y) ∧ T(z,y)) could also be true if x≠z was false (Since FALSE → TRUE is TRUE). Therefore, I think the right answer should be as follows:
∀x∀z∃y( (x≠z) ∧ ¬T(x,y) ∧ ¬T(z,y) ))
So, the expression would only be true if x≠z is true and T(x,y) and T(z,y) are not true (Since TRUE ∧ ¬ FALSE ∧ ¬ FALSE is TRUE).
You see what I mean? It's very confusing. Could anybody help me?