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Is there a standard notation (perhaps $A \stackrel{\leftarrow}{=} B$) meaning "in all situations where $B$ is defined, $A$ is defined and equals $B$"?

The kind of situation in which such a notation would be useful is the teaching of formulas like $$\lim_{x \rightarrow a} (f(x)-g(x)) = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x).$$ When I teach such formulas I take pains to teach them as theorems, with hypotheses that must be satisfied (in this case, the existence of $\lim_{x \rightarrow a} f(x)$ and $\lim_{x \rightarrow a} g(x)$) before the truth of the formula can be concluded, and I call to the students' attention the asymmetry of the situation (whenever the RHS is defined the LHS is defined and must be equal to it, but it is emphatically NOT always the case that when the LHS is defined the RHS must be defined and must be equal to it). I feel that one way to help students remember what the theorem says would be to use a variant of the equals sign when summarizing the theorem by a formula.

Has anyone introduced such a symbol? I think it would be at least as useful as the ":=" ("is defined as") symbol.

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It might be useful. But if you're talking about teaching calculus, I'm afraid that a lot of students would just write down "=" for such a symbol and it wouldn't be worth the trouble. –  Michael Lugo Apr 7 '11 at 19:41

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up vote 5 down vote accepted

Freyd and Scedrov, in their book Categories, Allegories, use for this 'directed equality' a peculiar symbol that they call a Venturi tube and that looks a bit like $\mathrel{>=}$, so that $x \mathrel{>=} y$ means if $x$ is defined then so is $y$ and $x=y$. You can find some discussion at the nLab page on Kleene equality (the symmetric version of this) and in this nForum thread.

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