Venn Diagram for set operations

[FritoTaco]

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?

 

[PeroK]

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]

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.

 

[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]

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.