# Venn Diagram for set operations

## Homework Statement

Hi, the problem states to draw a Venn Diagram for $A\cap(B-C)$

## Homework Equations

$(B - C)$ means include all elements in the set $B$ that are not in $C$.
Definition from my book: Let A and B be sets. The difference of $A$ and $B$, denoted by $A - B$, is the set containing those elements that are in A but not in B. The difference of $A$ and $B$ is also called the complement of $B$ with respect to $A$.

$A\cap(B-C)$ means that after finding $(B - C)$, find where $A$ intersects $(B - C)$.

## The Attempt at a Solution

Step 1:
$(B - C)$
(See png file called "step1")

Step 2:
$A\cap(B-C)$
(see png file called "step2")

My question is, do i shade in region 5, even though $(B - C)$ means "no elements in $C$" but when I have to show $A$ and $B$ intersect, I shade in region 4 obviously, but does that include region 5 because that's also where $A$ and $B$ intersect, but $C$ is also intersecting there as well.

So, in case you didn't follow because I may or may not be good at explaining things, should I include region 5 or exempt it from being shaded in?

#### Attachments

• step1.png
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• step2.png
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## Answers and Replies

PeroK
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So, in case you didn't follow because I may or may not be good at explaining things, should I include region 5 or exempt it from being shaded in?

Region 5 is part of C, isn't it?

FritoTaco
Region 5 is part of C, isn't it?
Yeah, that's why I'm not sure to shade it in. Because originally I shaded all of B and NOT C. The final part is to see where A and B intersect, so I was wondering if I now include region 5 because A and B intersect there. The problem is, that's the region where C is as well (region 5), but originally I did not want to include C.

PeroK
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Yeah, that's why I'm not sure to shade it in. Because originally I shaded all of B and NOT C. The final part is to see where A and B intersect, so I was wondering if I now include region 5 because A and B intersect there. The problem is, that's the region where C is as well (region 5), but originally I did not want to include C.

You are trying to draw ##A \cap (B-C)##. Not ##A \cap B##

Note that you can also write the difference of ##B## and ##C## as:

##B-C = B \cap C^c##, where ##C^c## is the complement of ##C##.

That might makes things even clearer.

Last edited:
FritoTaco
You are trying to draw A∩(B−C)A∩(B−C)A \cap (B-C). Not A∩BA∩BA \cap B
So that means that I should only have region 4 shaded in because it must be the case that $A\cap B$ is not the same as $A\cap (B-C)$ as you stated. Is this correct?

B−C=A∩BcB−C=A∩BcB-C = A \cap B^c, where BcBcB^c is the complement of BBB.
My book never showed this notation, thanks.

PeroK
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So that means that I should only have region 4 shaded in because it must be the case that $A\cap B$ is not the same as $A\cap (B-C)$ as you stated. Is this correct?

Yes, it's only region 4. Note that I mixed up the sets above, which I've now corrected.

If ##B - C## is everything in ##B## that's not in ##C##, then by definition this is ##B \cap C^c##.

FritoTaco
Okay, thank you very much.