conditional equality symbol - MathOverflow

conditional equality symbol

2011-04-07T02:42:12Z

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.

Answer by Finn Lawler for conditional equality symbol

2011-04-07T16:36:46Z

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.

Answer by user1837 for conditional equality symbol

2012-10-24T18:55:31Z

If you're like me and use $\rightarrow$ fro implication, you can write $A \Rightarrow B$ to mean "If $A$ is well-defined, then so too is $B$, in which case $A=B$." Similarly, you can write $A \Leftrightarrow B$ to mean, "$A$ is well-defined if and only if $B$ is well-defined, in which case $A=B$." The good thing about those symbols is they actually sort of look like equality.