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The problem I am working on is:

Use a Venn diagram to illustrate the relationship $A⊆B$ and $B⊆C$.

From my understanding, I should be drawing several Venn diagrams, corresponding to the different situations that are possible.

Since $A \subseteq B$ does not specify whether $A$ is a proper subset or not, we have two situations:

$(1)$: $A=B$

$(2)$: $A \subset B$

The same two situations apply to $B \subseteq C$


(This part is just a side note, but I'd appreciate a corroboration of my reasoning) Also, by the transitive law, since for x we find in A, it is true that we can find it in B ($A \subseteq B$); and since for every x in B, it is true that we can find it in C ($B \subseteq C$), we can say that $A \subseteq C$. This is reasonable, because all of A is contained in B, and all of B is contained in C, so all of A must be contained in C.


Now, the different situations we have are as follows:

$(1)$: $A=B$ and $B \subset C$

$(2)$: $A \subset C$ and $B \subset C$

$(3)$: $A=B$ and $B=C$

$(4)$: $A \subset C$ and $B=C$

So, I would have to make a Venn diagram for each situation?

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I think you just have to draw three concentric circles. The cases where an inclusion is mutual (equality) are implicit I think. Also, your case #2 is wrong; nothing prevents A and B to be disjoint with your statements. –  Sh3ljohn Feb 25 '13 at 15:34
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1 Answer

up vote 3 down vote accepted

Here's the list I came up with, with the "$\subsetneq$" symbol meaning "is a proper subset of":

$(1): A = B\;$ and $\;B = C.\;$ (And hence $A = C$ by transitivity)

$(2): A = B\;$ and $\;B \subsetneq C.\;$ (And hence $A \subsetneq C$, necessarily).

$(3): A \subsetneq B\;$ and $\; B = C.\;$ (And hence, $A \subsetneq C$, necessarily).

$(4): A \subsetneq B\;$ and $\; B \subsetneq C.\;$ (And hence, $A \subsetneq C$, necessarily).

Yes, the Venn diagram for each of the four scenarios would necessarily be different, if we are distinguishing between equality of sets and "being a proper subset of" a set, so you can cover all possible relationships by drawing a Venn diagram for each case, separately. For example, case $(1)$ would be a single circle, labeled A = B = C. Case two would be a circle A = B, contained within the larger circle C. etc...

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So, in short, my analysis is correct? –  Mack Feb 25 '13 at 16:09
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Where you have $A \subset C$, it should be $A \subset B$, since $A\subset C$ will then follow. And your use of $\subseteq$ should be $\subset$ in your cases (1) and (2) ($B\subset C$), since you're considering equality separately. Does that make sense? –  amWhy Feb 25 '13 at 16:13
    
Yes, thank you! –  Mack Feb 25 '13 at 17:07
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