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What is correct notation for the following, I have seen both in some books.
To show an empty set, is it Φ(phi) or Ø(slash O) or both.
To show an Universal set, is it ε(epsilon) or U or both. I am little bit confused.

enter image description here

enter image description here We use to denote the universal set, which is all of the items which can appear in any set. This is usually represented by the outside rectangle on the venn diagram.

http://www.cimt.plymouth.ac.uk/projects/mepres/book7/bk7i1/bk7_1i4.htm

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  • $\begingroup$ Where did you see $\Phi$ denote $\varnothing$? $\endgroup$
    – Git Gud
    Jun 30, 2014 at 21:46
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    $\begingroup$ I have never seen $\phi$ personally. You might note that there is no "correct," notation, only an accepted one. If you clearly label your symbols you can be correct, just have very sad readers. $\endgroup$
    – Thoth19
    Jun 30, 2014 at 21:47
  • $\begingroup$ I've seen all of $\phi$, $\emptyset$, $\square$, and $\varnothing$. I dislike all but the last ($\varnothing$). I find the third ($\square$) most peculiar. I have to agree with @Thoth19; you can use any symbol to stand for any notion, but following modern convention makes it more likely the reader will bother finishing what you have written :) $\endgroup$
    – MPW
    Jun 30, 2014 at 21:54
  • $\begingroup$ My book, Passage to Abstract Mathematics, uses $\emptyset$ for an empty set and $U$ for the universal set. For me, that's convenient and less confusing because $\phi$ is already being used in Calculus IV problems for calculating in spherical coordinates. That fancy e means something else in my book...kind of forgot. $\endgroup$
    – usukidoll
    Jul 1, 2014 at 8:18

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The use of $\phi$ or $Ø$ is either from papers and books (and sometimes people) predating modern typesetting standards, or wrong (in term of standard notation). Or, of course, there is a confusion between the symbols because of the striking similarities between all of them.

The symbol $\varnothing$ and its variant $\emptyset$ are derived from the Norwegian-Danish letter Ø:

Common notations for the empty set include "{}", "Ø", and "\emptyset". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Norwegian and Danish alphabet (and not related in any way to the Greek letter Φ).[1] Other notations for the empty set include "Λ" and "0".[2]

The empty-set symbol ∅ is found at Unicode point U+2205.[3] In TeX, it is coded as \emptyset or \varnothing.

(Wikipedia, Empty set)

As for universal sets, those are practically non-existent in modern set theory. I have seen them denoted by $U$ or sometimes $V$, but it might as well be any other letter.

The important thing is that the reader understand what each symbol means. Of course, if there are standard symbols it is better to use them. In the case of the empty set there is. In the case of a universal set, there isn't really a standard notation for it.

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  • $\begingroup$ Note, in his quotation, they are not talking about "the" universal set, but rather "a" universal set: the set of all objects to be discussed. Represented by the outside rectangle in the Venn diagram. So: If we are doing number theory, we may consider a universal set $\{1,2,3,\dots\}$. $\endgroup$
    – GEdgar
    Jun 30, 2014 at 22:37
  • $\begingroup$ GEdgar, yes, this is true. But then anything is a universal set (at least if you accept a purely set theoretical foundation). Note that the generality of basic set theoretical operations like unions and intersection make it so that a universal set in this context is really some arbitrary set. Also note that I said that in set theory there's practically no universal set. It is less common to do number theory when you do set theory, but then again if you take a strict set theoretical foundation you might beg to differ. $\endgroup$
    – Asaf Karagila ♦
    Jun 30, 2014 at 22:42

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