Venn Diagram for set operations

  • #1
FritoTaco
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Homework Statement



Hi, the problem states to draw a Venn Diagram for [itex]A\cap(B-C)[/itex]

Homework Equations



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

[itex]A\cap(B-C)[/itex] means that after finding [itex](B - C)[/itex], find where [itex]A[/itex] intersects [itex](B - C)[/itex].

The Attempt at a Solution


Step 1:
[itex](B - C)[/itex]
(See png file called "step1")

Step 2:
[itex]A\cap(B-C)[/itex]
(see png file called "step2")

My question is, do i shade in region 5, even though [itex](B - C)[/itex] means "no elements in [itex]C[/itex]" but when I have to show [itex]A[/itex] and [itex]B[/itex] intersect, I shade in region 4 obviously, but does that include region 5 because that's also where [itex]A[/itex] and [itex]B[/itex] intersect, but [itex]C[/itex] 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?
 

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  • #2
FritoTaco said:
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?
 
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  • #3
PeroK said:
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.
 
  • #4
FritoTaco said:
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.
 
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  • #5
PeroK said:
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 [itex]A\cap B[/itex] is not the same as [itex]A\cap (B-C)[/itex] as you stated. Is this correct?

PeroK said:
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.
 
  • #6
FritoTaco said:
So that means that I should only have region 4 shaded in because it must be the case that [itex]A\cap B[/itex] is not the same as [itex]A\cap (B-C)[/itex] 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##.
 
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  • #7
Okay, thank you very much.
 
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