Any total relation is extensional

Question

In An Introduction to Set Theory by William A. R. Weiss, Chapter $6$ it says: "Any total relation is extensional." The book defines $R$ on a set $C$ to be extensional when the following is true:

$$∀~x∈C~~ ∀~y∈C, ~~[x=y↔∀z∈C (⟨z,x⟩∈R↔⟨z,y⟩∈R)]$$

And total if the following is true:

$$∀~x∈C~~ ∀~y∈C,~~ [⟨x,y⟩∈R ∨ ⟨y,x⟩∈R ∨ x=y]$$

But it seems that not all total relations are extensional. For example taking $C = \{ a,b \}$ (where $a \neq b$) and $R = C \times C$ one would have a total relation that is not extensional. Is the book wrong or am I overlooking something?

Answer 1

$\def\r{\mathrel{\rm R}}$As far as I can see you are correct.

Here is a conjectural resolution of the confusion. Suppose that the book intended to say that "total" means

for all $x,y\in C$, exactly one of $x\r y$ and $y\r x$ and $x=y$ is true.

This property implies extensionality. Proof. First, it is clear that $x\r x$ is never true. So if $x,y\in C$ and $x\ne y$ then we have either

• $x\r y$ and $x\not\r x$; or
• $y\r x$ and $y\not\r y$.

So $\r$ is extensional.