Venn Diagram for set operations

In summary, the conversation is about drawing a Venn Diagram for the set A\cap(B-C). The problem is clarified and it is determined that region 4 is the only region that should be shaded, as it represents the intersection of A and (B-C). The conversation also discusses the notation for the difference of two sets, B-C, and how it can also be written as B \cap C^c.
  • #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.
 
Last edited:
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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.
 

1. What is a Venn Diagram for set operations?

A Venn Diagram is a graphical representation of sets and their relationships. It consists of overlapping circles or ovals, with each circle representing a set and the overlapping regions representing the intersection of the sets.

2. What are the basic set operations represented in a Venn Diagram?

The basic set operations represented in a Venn Diagram are union, intersection, and complement. Union is represented by the overlapping regions of the circles, intersection is represented by the common area between the circles, and complement is represented by the areas outside of the circles.

3. How do you read a Venn Diagram for set operations?

To read a Venn Diagram, start with the circles representing the sets. The areas outside of the circles represent the complement of the sets, while the overlapping regions represent the intersection of the sets. The entire area within the circles represents the union of the sets.

4. How is a Venn Diagram for set operations useful in real-life situations?

A Venn Diagram for set operations can be useful in many real-life situations, such as organizing data, identifying commonalities and differences between groups, and solving logic problems. It can also be used in decision-making, as it can help visualize the relationships between different options.

5. Are there any limitations to using a Venn Diagram for set operations?

While Venn Diagrams are useful for visualizing set operations, they can become complicated and difficult to read when there are more than three sets involved. Additionally, they do not show the exact elements of each set, so they may not be suitable for representing large or complex sets.

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