Confusion on equivalence class $[a]=${$(x_a,z);∀z∈C$} meaning

Question

For an equivalence class $[a] = \{ (x_a,z) : z \in C \}$ means $[a] = \{x_a\} \times C$ where $z∈C$? Am I defining it correctly?

Answer 1

Yes, this is a perfectly valid characterization: all these are equivalent:

$$\{(x_a,z)\mid \forall z \in C \} = \{x_a \} \times C = \{(x_a,z)\mid z \in C\}$$

That said, I would prefer one of the latter two notations (which I'd use depending on context). The latter seems a more "proper" setbuilder notation for the set you're describing.

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Mostly posting this so this question can be finally considered to have an answer and thus be removed from the unanswered queue. Made it Community Wiki since I have nothing much of substance to add.