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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

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    $\begingroup$ Just guessing. Perhaps it's some sort of preference sign? (From the point to the next closing parenthesis or the end). That is: $A\leftrightarrow.B\wedge C$ is the same as $A\leftrightarrow(B\wedge C)$ $\endgroup$ – ajotatxe Dec 15 '17 at 14:53
  • $\begingroup$ Thanks. I was worried it was something more specific! don't understand the down vote thing? My first post. Don't even know how to vote! $\endgroup$ – Drex Dec 15 '17 at 15:02
  • $\begingroup$ Be careful! I was only guessing! $\endgroup$ – ajotatxe Dec 15 '17 at 15:04
  • $\begingroup$ I believe you are correct. Using this information I found another post on this site:- math.stackexchange.com/questions/311871/… $\endgroup$ – Drex Dec 15 '17 at 16:10
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    $\begingroup$ It is only a parentheses. The second one is $∀x ( x ∈ \{ a,b \} ↔ (x=a ∨ x=b) )$. The "dot notation" was due to Peano and used in logic into W&R's Principia Mathematica. $\endgroup$ – Mauro ALLEGRANZA Dec 15 '17 at 17:02
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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).

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  • $\begingroup$ I assumed the quoted material was from a regular maths book, written in that usual semi-formal way where you notationally use what works without worrying too much about formal punctiliousness. But yes ok "informal" wasn't the mot juste, and I've deleted it! $\endgroup$ – Peter Smith Dec 16 '17 at 9:24

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