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this post gives some explanation about the definition and operation of sets.

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the paragraph above Axiom 1b uses this symbol $\Leftrightarrow$ to indicate that p and q ­are equivalent.

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the paragraph below Axiom 1b uses this symbol $\Longleftrightarrow$ to give the Definition of "Union of two sets"

so, is the difference between $\Longleftrightarrow$ and $\Leftrightarrow$ is as follow?

$\Leftrightarrow$ represents equivalent

$\Longleftrightarrow$ represents definition

JJJohn
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    One is longer than the other. Which one to use is really a matter of taste. And its meaning depends on context. I think that's all. –  Jul 30 '19 at 14:29
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    They're the same symbol and have the same "if ond only if" meaning. The one that is shorter is probably done just to keep the text from wrapping to a new line. – Joe Wells Jul 30 '19 at 14:32
  • @Mauro: Not quite. Note that the text states that $p\Leftrightarrow q$ is a sentence. Namely, the shorter arrow is a connective. This is typographically unsound, though, and using $\leftrightarrow$ is much more reasonable. – Asaf Karagila Jul 30 '19 at 14:35
  • In short, $\Leftrightarrow$ is a connective in the symbols of the logic. So $p\Leftrightarrow q$ is a sentence. On the other hand $\iff$ is a statement about sentences. This is an awful notation, and often the short arrow is written as $\leftrightarrow$. – Asaf Karagila Jul 30 '19 at 14:36
  • @Mauro: It's not a matter of opinion. It is literally stated in the text. So I am honestly quite confused by your disagreement. Also note that the statement of the Axiom of Extensionality, (Ax. 1b) is using the connective form of a double-implication, which again is in line of what is written in the text and what I point out in the comments. – Asaf Karagila Jul 30 '19 at 14:40
  • @AsafKaragila: I am confused why you closed this post as a duplicate of the linked one. Clearly they are related, but this one (not very interesting though) specifically asks two symbols ($\Longleftrightarrow$ and $\Leftrightarrow$) that are not the ones ($\equiv$ and $\leftrightarrow$) in the linked post. –  Jul 30 '19 at 14:46
  • @Jack: True, the symbols are different. But $\Leftrightarrow$ should be $\leftrightarrow$ and $\iff$ would be $\equiv$. Should I have copy-pasted my answer with these changes? – Asaf Karagila Jul 30 '19 at 14:48
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    @AsafKaragila would please provide any Authoritative text book of topology that uses your notation? – JJJohn Jul 30 '19 at 14:55
  • @yaojp: I don't know. I've never read a topology book, as that kind of approach is uncommon where I studied. I've never heard complaints like this about Munkers or Engelking. So I'd reckon you can start with them. – Asaf Karagila Jul 30 '19 at 15:02
  • @AsafKaragila are you recommending a book you have not read? – JJJohn Jul 30 '19 at 15:06
  • @yaojp: No. I am telling you what are the two standard textbooks that I've seen people use often, and not once heard a complaint about the logical connectives there. If you read my comment carefully, you'll see it begins with "I don't know.", meaning I don't know what book to recommend you for this purpose. – Asaf Karagila Jul 30 '19 at 15:07
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    Compare with : van Dalen, Logic and Structure (e.g. page 18) where $\to$ and $\leftrightarrow$ are used as propositional connectives while $\Rightarrow$ and $\Leftrightarrow$ are used as abbreviations in the text for "if..., then..." and "iff". – Mauro ALLEGRANZA Jul 30 '19 at 15:09
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    Having said that, the choice of the short version and the long one of the same symbol is at minimum a bad choice. In addition, it seems that the author uses "not" and "and" in formulas (instead of the usual symbols : $\lnot, \land$) and uses symbols in the text (see page 4 : (p ⇒ q) and (q ⇒ p) ⟺ (p ⇔ q), which is IMO terrible...). – Mauro ALLEGRANZA Jul 30 '19 at 15:12
  • @AsafKaragila would you please give the complete titles of those 2 books? – JJJohn Jul 30 '19 at 15:19
  • https://www.google.com/search?q=Munkers+topology and https://www.google.com/search?q=Engelking+topology, I guess. – Asaf Karagila Jul 30 '19 at 15:21
  • Whoever voted to reopen, please let me know when the question is open. I would be happy to copy-paste my answer from the duplicate if you endorse this sort of blatant copy-paste behavior. – Asaf Karagila Jul 30 '19 at 15:21

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