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In my book there are questions where there are two intersecting sets $A$ and $B$ and the questions begin by saying something like "In the given Venn diagram ... ". But there is no Universal set surrounding the two sets $A$ and $B$. Just the two sets themselves. Strictly speaking is this kind of diagram an Euler diagram or a Venn diagram ? Does a Venn diagram need a Universal set ?

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    $\begingroup$ There's an implied, unspecified universal set, but for all you know, it's just $A \cup B$. In any case, if you're not interested in points outside of $A \cup B$, there's no need to show an enclosing box. $\endgroup$ – quasi Aug 27 '17 at 14:52
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    $\begingroup$ take the book page as Universal set (ha ha) $\endgroup$ – MAN-MADE Aug 27 '17 at 14:54
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    $\begingroup$ It's implied if not present. A Venn diagram is more general than an Euler diagram because the presence of an overlap between set representations does not implies the existence of elements in that overlap. $\endgroup$ – Joffan Aug 27 '17 at 14:54
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    $\begingroup$ As a somewhat different way of looking at it, some relations between sets (union, intersection, difference) do not require a universal set, while others (compliment) do. You can use Venn/Euler diagrams to represent the former, which doesn't require you to specify a universal set. $\endgroup$ – Kajelad Aug 28 '17 at 4:41
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The distinction between an Euler diagram and a Venn diagram is not the presence or absence of a universal set, but in the relations that are represented. In a Venn diagram, every possible intersection is shown, even if one or more of those intersections is empty. In an Euler diagram, only "interesting" (i.e. nonempty) intersections are shown. For example (blatantly stolen from Wikipedia):

A Venn diagram:

enter image description here

An Euler diagram showing the same information:

enter image description here

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  • $\begingroup$ But if the Universal set isn't specified there is one possibility not represented, i.e. the complement of the union of the sets that are specified. $\endgroup$ – Kantura Aug 27 '17 at 19:14
  • $\begingroup$ In that case the universal set is the union of the sets shown. In a Venn diagram, this follows from the fact that every possible intersection is shown. In an Euler diagram, this follows from the fact that only non-empty sets are shown, hence the complement is empty. $\endgroup$ – Xander Henderson Aug 27 '17 at 22:45

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