Welcome to our community

Be a part of something great, join today!

Venn Diagram II.

bergausstein

Active member
Jul 30, 2013
191
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
vance 1.3.4.jpg
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
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

Active member
Jul 30, 2013
191
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

Well-known member
MHB Math Scholar
Mar 10, 2012
834

bergausstein

Active member
Jul 30, 2013
191

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
This is how I would draw them...I will leave the other cases for part c) for you to draw. :D

venn2.jpg

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

bergausstein

Active member
Jul 30, 2013
191
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

Administrator
Staff member
Feb 24, 2012
13,775
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

Active member
Jul 30, 2013
191
this is how would i draw the last case for c.

Venn2.1.jpg
am I correct?
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
The other two cases involve $B$ and $A$ having partial intersection and no intersection. You have drawn the same case I drew.
 

bergausstein

Active member
Jul 30, 2013
191
here's my second attempt for C.
Venn2.2.png

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

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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):

venn3.jpg

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