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Give an example, and find the Venn diagram, for $$A\subseteq B,\quad B\subseteq C, \quad C\subseteq A$$

Our solution: We proved it to conclude that $A=B=C$.

  • so, the example is $A=\{1,2,3\}$, $B=\{1,2,3\}$, $C=\{1,2,3\}$
  • but in the Venn diagram, my friend and I disagreed. He drew three intersecting circles and put the elements in Intersection. But I think there is a mistake, I'm not sure about my solution. I drew a circle for three groups and put the elements $\{1,2,3\}$ inside.

So, the question is:

What is the right solution? Three circles or one circle?

Blue
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hind
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    Both are correct. Venn diagram does not require an area to be a non-empty set. – cr001 Aug 10 '20 at 13:40
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    Both are correct! As long as there are no elements outside each of the three regions, they are valid. You can even draw the same circle for A and B, and another circle C which intersects the earlier circle(s), and put the three elements in the common intersection. – abcd123 Aug 10 '20 at 13:41
  • @cr001 Both may be correct but what is the most accurate and why? – hind Aug 10 '20 at 13:51
  • @SarGe i think that cuz the relate is equal between the sets A, B, C not intersection – hind Aug 10 '20 at 13:55
  • It depends on context. In case of solving a problem, I would say your friend is the most accurate, but in a research paper, I would say yours is the most accurate. – cr001 Aug 10 '20 at 13:58
  • @abcd123 Does this express that the related between the three seta is equality? I don't think – hind Aug 10 '20 at 13:59
  • @cr001 thank you – hind Aug 10 '20 at 14:04
  • @hind As long as the region outside the intersection is empty, yes. – abcd123 Aug 10 '20 at 14:10

2 Answers2

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The usual approach would be your friend's. You want a circle for each set, then to locate the elements to show which set(s) they belong to. You are correct that in this problem you really have three names for the same set. Note there could be elements in the universe that are outside all three sets. Nothing in the problem prohibits that. I might put $4$ in the outside region to show that possibility.

Ross Millikan
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2

Both are Venn diagrams according to the usual definition:

sets are depicted by means of arbitrarily shaped (but bounded by simple closed curves) regions subject to a combinatorial requirement that there be exactly $2^N$ connected subregions (Cut-the-Knot)

Yours is a scaled Venn diagram:

A Venn diagram in which the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram. (Wikipedia)

JMP
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