Reading a first order logic definition, with for all

Question

I'm trying to read the following FOL for a functional binary relation:

$R \subset X \times Y$

$\forall x \in X, \forall y \in Y, \forall z \in Y, (x,y) \in R \land (x,z) \in R \implies y = z$

I'm confused on how to read this. It's my understanding "∀ __ is True if the value of __ is True for all values of x" and that "∀ __ is False if the value of __ is False for any value of x.

So above, $\forall z \in Y$ will bind $z$ to values where $y \ne z$ and the entire statement will be false. Am I reading this wrong?

Answer 1

Maybe it's worth looking at the innermost part first and understanding what it means:

$(x,y) \in R \land (x,z) \in R \implies y = z$

Remember, $P \implies Q$ is a logical statement that is true if either:

a) $P$ is false; or
b) $P$ and $Q$ are both true

i.e. if you know that the left-hand side is true, you can conclude that the right-hand side is also true; but if the left-hand side is false you have no information about the right-hand side.

So in this case, $(x,y) \in R \land (x,z) \in R \implies y = z$ means "If we know that $xRy$ and $xRz$ both hold, then in fact we also know that $y$ and $z$ are equal. However, if either $xRy$ or $xRz$ doesn't hold, then who knows?"

Answer 2

You are misunderstanding the bracketing here: $\forall x \in X, \forall y \in Y, \forall z \in Y, (x,y) \in R \land (x,z) \in R \implies y = z$

Should be: $\forall x \in X, \forall y \in Y, \forall z \in Y, ((x,y) \in R \land (x,z) \in R) \implies y = z$