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More precisely, is there a difference between {p,s,r,q,t} and {{p,s},r,q,t} and if so, how would you show it using a Venn diagram?

I have in my notes a Venn diagram where p, s, r, q and t are all obvious elements of set A. C is a subset of A, fully enclosed. Is C also an element of A? Can I state {C, r, q, t} is a subset of A?

The question on the next page is: {{p, s}, r, q, t} is a subset of A. The answer given is false, but I thought it was true.

Venn Diagram

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There is a difference between $\{p,s,r,q,t\}$ and $\{\{p,s\},r,q,t\}$; the first set has 5 elements, the second set has 4 elements. But I don't know how to show this in a Venn diagram. – Joel Reyes Noche Jun 4 '13 at 13:09
up vote 3 down vote accepted

In terms of this style of diagram, you should think of $\{p,q\}$ as being a dot completely separate from the dots for $p$ and $q$; this is a little counterintuitive, but as the set $\{p,q\}$ is different from both the element $p$ and the element $q$, it needs to be drawn as a separate element. Then you can draw circles to represent the sets $\{p,s,r,q,t\}$ and $\{\{p,s\},r,q,t\}$ so that both contain the dots for $r$, $q$ and $t$, but only one contains the dots for $p$ and $s$, and only the other contains the dot for $\{p,s\}$.

Of course, you might also want to draw a circle around $p$ and $q$ to represent $\{p,q\}$, but this has to be somehow unrelated to the dot - thinking of $\{p,q\}$ as a set is different to thinking of it as an element of other sets for the purposes of this diagram. (For this reason, I don't think Venn diagram notation is particularly good at dealing with these kinds of questions, because you're forced to think of the same thing in two very different ways simultaneously).

This should also answer your other question - $C$ being a subset of $A$ is not the same thing as $C$ being an element of $A$.

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So if there was a dot next to the C we could assume C is an element of A, therefore {C} is a subset of A and C is no longer a subset of A? – Mats Jun 4 '13 at 13:17
Pretty close - I mean that if there was a $C$ with a dot next to it in the picture (maybe just below the $C$ that's there so that it only lies in the set $A$), then $C$ would be an element of $A$. However, if we also keep the $C$ labelling a circle in the picture, then $C$ is still also a subset of $A$; $A$ would then be the set $\{p,q,r,s,t,C\}$, which both contains $C$ as an element, and has the subset $C=\{p,q\}$. (I reiterate that this is confusing, and you should be careful about using Venn diagrams to describe sets being members of, rather than subsets of, other sets). – Matthew Pressland Jun 4 '13 at 13:29

I think Matt covers this quite well mathematically but I thought I'd add an analogy. Imagine I'm seeing what ingredients people have in their pantries i.e. A, B, C, etc. would be people's pantries and then p, q, r, etc. would be the ingredients. Now p might be raspberries and s might be chocolate and then {p, s} could be chocolate coated raspberries, which contains raspberries and chocolate but isn't itself the same as raspberries or chocolate. Now a pantry could have raspberries or chocolate or chocolate coated raspberries or any combination of the three and while they're related, they're each individual items.

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This makes it very clear! I accepted the other answer because it leans closer to the Venn diagrams, but this makes the difference between a subset and a set as an element very clear. – Mats Jun 4 '13 at 14:09
no problem (and Matt's answer was clearly the answer that better addressed the question). – john Jun 4 '13 at 14:17
This is a very neat analogy that I shall try to remember for the next time I get asked a question like this. – Matthew Pressland Jun 4 '13 at 14:21

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