Venn Diagram II: Check Answers & Understand Relationships

In summary, Venn diagrams are useful tools to illustrate set relationships. When using Venn diagrams to illustrate set operations, it is important to consider all possible cases to demonstrate the truth or falsehood of a given condition. In this particular conversation, the use of Venn diagrams was used to illustrate set relationships and operations for different cases, such as subsets, unions, intersections, and complements. This allows for a more thorough understanding and verification of the given conditions.
  • #1
bergausstein
191
0
Another Venn diagram problem. just want to check if my answer is correct since i don't have a solutions manual of the book I'm using.

Use Venn diagrams to illustrate the following.

a. $\displaystyle A\cup B\,=\,A$ if and only if $\displaystyle B\subset A$
b. $\displaystyle A\cap B\,=\,B$ if and only if $\displaystyle B\subset A$
c. $\displaystyle B\subset A$ if and only if $\displaystyle A'\subset B'$
d. $\displaystyle \left(A'\right)'\,=\,A$

my answers
View attachment 1110
 

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  • #2
bergausstein said:
Another Venn diagram problem. just want to check if my answer is correct since i don't have a solutions manual of the book I'm using.

Use Venn diagrams to illustrate the following.

a. $\displaystyle A\cup B\,=\,A$ if and only if $\displaystyle B\subset A$
b. $\displaystyle A\cap B\,=\,B$ if and only if $\displaystyle B\subset A$
c. $\displaystyle B\subset A$ if and only if $\displaystyle A'\subset B'$
d. $\displaystyle \left(A'\right)'\,=\,A$

my answers
View attachment 1110
I think you should have drawn the universal set too, especially while showing diagrams relating to complements.
 
  • #3
caffeinemachine said:
I think you should have drawn the universal set too, especially while showing diagrams relating to complements.

i have drawn the universal set. w/c is set A. right?
 
  • #4
bergausstein said:
i have drawn the universal set. w/c is set A. right?
What is 'w/c'??
 
  • #5
caffeinemachine said:
What is 'w/c'??

i mean i have drawn the universal set. it's set A.
 
  • #6
This is how I would draw them...I will leave the other cases for part c) for you to draw. :D

View attachment 1112

edit: caffeinemachine is correct, the universal set (the rectangles in my sketches) is needed for complementation.
 

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  • #7
MarkFL said:
This is how I would draw them...I will leave the other cases for part c) for you to draw. :D

View attachment 1111

edit: caffeinemachine is correct, the universal set (the rectangles in my sketches) is needed for complementation.

why $\displaystyle A\subset B$ ? shouldn't it be $\displaystyle B\subset A$ since B is inside A?
 
  • #8
bergausstein said:
why $\displaystyle A\subset B$ ?

Darn it...after all that work, I manage to foul it up...of course those should read $B\subset A$. I amaze myself sometimes...(Tongueout) Thank you for catching this error! (Yes)

edit: I edited the drawing, and reattached it. :D
 
  • #9

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  • #10
The other two cases involve $B$ and $A$ having partial intersection and no intersection. You have drawn the same case I drew.
 
  • #11
here's my second attempt for C.
View attachment 1116

i just want to ask why do we have to show different cases for each problems?
 

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  • #12
Demonstrating the truth for the given condition is not enough. You must also show it is false for the other cases to be thorough. This is how I would demonstrate the second case is not true for c):

View attachment 1117

The area in blue is within $A'$ but not in $B'$, hence $A'\not\subset B'$
 

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FAQ: Venn Diagram II: Check Answers & Understand Relationships

What is a Venn diagram and how is it used in science?

A Venn diagram is a graphical representation of relationships between different sets of data. It is commonly used in science to visually show the intersection and differences between different groups or categories of data.

How do I create a Venn diagram?

To create a Venn diagram, you will need to first determine the sets of data you want to compare. Then, draw two or more overlapping circles on a piece of paper or using a software program. Label each circle with the name of the set it represents. Next, add the data points or elements that belong to each set within their respective circles. Finally, add any data points that overlap between the sets in the overlapping sections of the circles.

What does it mean when two sets in a Venn diagram overlap?

When two sets in a Venn diagram overlap, it means that there are data points or elements that are shared between the two sets. This can indicate a relationship or similarity between the two sets.

How do I interpret a Venn diagram?

To interpret a Venn diagram, you should first look at the data points or elements within each circle and their relationship to the circle's label. Then, examine the data points in the overlapping sections and determine what they have in common between the two sets. You can also use the size of each circle and the size of the overlapping sections to determine the relative size or importance of each set or relationship.

Can Venn diagrams be used for more than two sets of data?

Yes, Venn diagrams can be used for any number of sets of data. However, as the number of sets increases, the complexity of the diagram also increases. To create a Venn diagram with more than three sets, you will need to use more circles and overlapping sections, making it difficult to interpret visually. In these cases, it may be more useful to use other types of graphs or charts to represent the data.

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