MHB Venn Diagram II: Check Answers & Understand Relationships

[bergausstein]

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

 

[caffeinemachine]

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.

 

[bergausstein]

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?

 

[caffeinemachine]

i have drawn the universal set. w/c is set A. right?

What is 'w/c'??

 

[bergausstein]

What is 'w/c'??

i mean i have drawn the universal set. it's set A.

 

[MarkFL]

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.

 

[bergausstein]

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?

 

[MarkFL]

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

 

[bergausstein]

this is how would i draw the last case for c.

View attachment 1115
am I correct?

 

[MarkFL]

The other two cases involve $B$ and $A$ having partial intersection and no intersection. You have drawn the same case I drew.

 

[bergausstein]

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?

 

[MarkFL]

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 $A'\not\subset B'$