Everyone has exactly 1 favorite book (z)

Question

I am learning discrete math and I fail to understand this:

Everyone has exactly 1 favorite book (z):

∀x(∃yF(x, y)∧∀z(F(x, y)→(y=z)))

I mean, this part I don't understand: ∀z(F(x, y)→(y=z))

Are there some good resources and examples to practice and learn it?

Answer 1

There is a typo in the formula …

It must be :

$∀x∃y(F(x,y) ∧ ∀z(F(x,z) → (y=z)))$.

The part $∀x∃yF(x, y)$ means : "for every (person) $x$ there is (at least) one (book) $y$ such that $F(x,y)$".

At least one means : "one or more".

Thus, to specify "exactly one" we have to add a formula that exludes the cases : two, three,etc.

This can be done saying that : "for every (book) $z$, if $F(x,y)$ then $z$ must be the same as $y$", i.e. :

$∀z(F(x,z) → (y=z))$.