Double-ended arrow with a dot after. Notation Question Good afternoon.
I have a quick question. What is the meaning of ↔. ?
(as opposed to purely ↔). The context arises in some basic set definitions (below). The first has no 'dot' whereas the 2nd and 3rd do:-
∀x ( x ∈ {a} ↔ x=a )
∀x ( x ∈ {a,b} ↔. x=a ∨ x=b )
∀x ( x ∈ {a,b,c} ↔. x=a ∨ x=b ∨ x=c )
I know that the ↔ by itself means 'if and only if'  and I imagine the addition of the dot still means something similar, but I can't find out what. 
Thanks for your help.
Drex
 A: This sort of use of the dot as punctuation can be just thought of as marking a pause, so that
∀x ( x ∈ {a,b} ↔. x=a ∨ x=b )
is in effect
∀x ( x ∈ {a,b} ↔ [wait for it ...!]  x=a ∨ x=b )
so naturally gathering what follows into a unit. Hence it is a bracketing device. (And "↔." is no more a unit with its own significance than is, say, "↔ ("
The once-common use of dots for bracketing duty seems to have its origins in  Principia Mathematica. I'd forgotten I'd written that answer on official dotty conventions at https://math.stackexchange.com/q/312074, so thanks for the link that saves me writing something similar! :) But as I say, in relaxed contexts like the present example, you needn't worry about official conventions. Just think of it as like a pause to break up a sentence and group it. As we do in ordinary language. Thus compare the familiar example of the party invitation ...

Bring your partner or come alone [pause] and have a good time
Bring your partner [pause] or come alone and have a good time

Without the spoken pause, or the written comma, it would be ambiguous. Likewise without the dot, or bracketing, or an operator-precedence-convention, 
∀x ( x ∈ {a,b} ↔ x=a ∨ x=b )
would be potentially ambiguous (even though one reading is daft).
