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At https://proofwiki.org/wiki/Definition:Well-Defined/Operation , a well-defined operation is defined as:

$$x, x' \in [x]_{\mathcal R}, y, y' \in [y]_{\mathcal R} \implies x \circ y = x' \circ y'$$

shouldn't this be written as:

$$x, x' \in [x]_{\mathcal R}, y, y' \in [y]_{\mathcal R} \implies [x \circ y]_{\mathcal R} = [x' \circ y']_{\mathcal R}$$

Since the site is used a lot, I guess I am wrong. Am I ?

vkubicki
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1 Answers1

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I think you are correct. For instance, if $S=\mathbb Z$ and $\mathcal R = 3\mathbb Z$ (i.e. congruence mod 3) then multiplication of the classes is a well defined binary operation, $0,3\in[0]_{\mathcal R}$ and $2,6\in[2]_R$, but $2\cdot 3 \neq 6\cdot 0$. Unless they want to create another definition not equivalent to well definition of maps... It does not seem to be the case.

nelynx
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