What does $\bar{\in}$ symbol mean?

Question

I found in an old Russian math book the following symbol:

$\bar{\in}$ or $\overline{\in}$ used in the given context:

$\eta_\epsilon(x)=1, \ x\in A^\epsilon$

$\eta_\epsilon(x)=0, \ x\overline{\in}A^{3\epsilon}$

The context is that the function $\eta$ vanishes at the point $2\epsilon$ on the x axis , and exists between $0$ and $2\epsilon$.

The meaning of this should be something like $\not\in$, one would think, but in this case $ A^{3\epsilon}$ is already outside the domain of $x$, so it would have sufficed to write

$\eta_\epsilon(x)=0, \ x\not\in A^{2\epsilon}$ .

So my guess is that the symbol $\overline{\in}$ means something else.

Does anyone know what?

Thanks

Answer 1

You can see Hausdorff's Set Theory (original German edition 1937), page 12:

$a \ \bar \varepsilon \ A$ for "$a$ is not an element of $A$".

You can comapre it Compare with Bernays' Axiomatic Set Theory (1958), page 58: "the complement of $A$ is $\bar A = \{ x \mid x∉A \}$.

The use of "bar" for negation is due also to Hilbert & Ackermann (1928), where the negation of proposition $X$ is denoted with $\bar X$.