# Venn Diagram II.

#### bergausstein

##### Active member
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

#### caffeinemachine

##### Well-known member
MHB Math Scholar
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
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
i have drawn the universal set. w/c is set A. right?
What is 'w/c'??

#### bergausstein

##### Active member
What is 'w/c'??
i mean i have drawn the universal set. it's set A.

#### MarkFL

##### Administrator
Staff member
This is how I would draw them...I will leave the other cases for part c) for you to draw.

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

#### bergausstein

##### Active member
This is how I would draw them...I will leave the other cases for part c) for you to draw.

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
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... Thank you for catching this error!

edit: I edited the drawing, and reattached it.

#### bergausstein

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

am I correct?

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#### MarkFL

##### Administrator
Staff member
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
here's my second attempt for C.

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

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#### MarkFL

##### Administrator
Staff member
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):

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