Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

When I'm told to represent a Venn diagram for $(A-B)\cup (B\cap C)$ are these two valid?

The first one:

enter image description here

The second one:

enter image description here

I don't seem to understand clearly whether an union implies having both sets "touch" each other in the diagram or if it doesn't matter at all as long as I color them red as I did in the second one.

Also, note that the exercise doesn't actually tell me if $A,B,C$ are really intersecting each other (only $A$ with $B$ and $B$ with $C$ but never $A$ with $C$), is that supposed to make a difference in the way I display the diagrams?

share|improve this question
add comment

3 Answers

up vote 4 down vote accepted

While both diagrams represent the set $(A \setminus B ) \cup ( B \cap C )$, the second is done under the additional assumption that $A \cap C = \emptyset$. Generally Venn diagrams are supposed to represent all of the possible interactions between the sets they represent, and if you do not know beforehand that $A \cap C = \emptyset$, then the second diagram loses information (in my opinion, anyway).

As such, I would be hesitant to provide the second as an answer to the question, as you could have equally done the following: represent the sets $A , B , C$ as discs which do not overlap at all (this is the situation $A \cap B = \emptyset$, $A \cap C = \emptyset$ and $B \cap C = \emptyset$. Then $( A \setminus B ) \cup ( B \cap C )$ would be represented by filling in the $A$ circle, and leaving the rest blank.

share|improve this answer
add comment

They are both valid. The sets need not "touch" each other or be connected in any manner.

share|improve this answer
add comment

Both diagrams are correct. The set of $(A-B)\cup (B\cap C)$ is, by definition, all of the elements that are either in $A-B$ or in $B$ AND $C$ (not an exclusive or, the element could be in both). Hence, whether the sets "touch" each other or not is actually irrelevant. I hope you find this somewhat enlightening!

share|improve this answer
    
Wait, if the element could be in both at the same time, the second drawing would make no sense, or would it? I mean, since the two red areas are visually separated, it would be impossible for an element to be in both, no? –  Omega Dec 3 '12 at 6:05
    
In fact, since we're considering $A-B$ and $B\cap C$, it couldn't be in both, but in general, a union is not an exclusive "or". To clarify, in this setting, if it could be in both, then $A\cap B^c\cap B\cap C$ would have to be nonempty ($A-B=A\cap B^c$), but in this case, it couldn't be nonempty. –  Clayton Dec 3 '12 at 6:09
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.