venn diagram of powerset!!!!!!!!!!!!!!!!!!!!!!!!

[pendem.venkat kumar]

is there any procedure to draw venn diagram for the power set of a given set...................as it is a set.......?
                   

[Dylan Faullin]

I think so, but I haven't tried it out for anything more than a set of 3 elements. But the procedure should go something like this:

Write each element of the set down.

Draw a circle around each element.

Draw a circle around every possible combination of elements (this would be the tricky part).

Draw one big circle around all the elements.

Draw one circle off to the side for the empty set.

Shouldn't that Venn Diagram every element of the powerset, overlapping the sets that share elements in common? I'm not sure if this is what you meant.

I just tried it for 4 elements. It gets messy and hard to follow. Maybe if you use different colors...

Hope this helps!

[pendem.venkat kumar]

thanks for your valuable reply sir and i didn't get about why you have written null set separately?????

Dylan Faullin said:

I think so, but I haven't tried it out for anything more than a set of 3 elements. But the procedure should go something like this:

Write each element of the set down.

Draw a circle around each element.

Draw a circle around every possible combination of elements (this would be the tricky part).

Draw one big circle around all the elements.

Draw one circle off to the side for the empty set.

Shouldn't that Venn Diagram every element of the powerset, overlapping the sets that share elements in common? I'm not sure if this is what you meant.

I just tried it for 4 elements. It gets messy and hard to follow. Maybe if you use different colors...

Hope this helps!

[Dylan Faullin]

I honestly wasn't sure what to do with the null set. In fact, I wasn't even sure if what I was proposing was exactly what you were looking for. I was trying to draw a venn diagram that represented each element of the power set of a set. So for my drawing here, I took the set S={a,b,c,d}. The powerset would contain the following elements:
{a}
{b}
{c}
{d}
{a,b}
{a,c}
{a,d}
{b,c}
{b,d}
{c,d}
{a,b,c}
{a,c,d}
{a,b,d}
{b,c,d}
{a,b,c,d}
null set

I then tried to draw a venn diagram such that there would be a region to correspond to each of the above sets. Was this what you intended?

As for the null set, I wasn't sure where to put it. Seeing as it doesn't belong as an element to any of the other sets, I thought it should be off to the side by itself.
pendem.venkat kumar said:

thanks for your valuable reply sir and i didn't get about why you have written null set separately?????

Dylan Faullin said:

I think so, but I haven't tried it out for anything more than a set of 3 elements. But the procedure should go something like this:

Write each element of the set down.

Draw a circle around each element.

Draw a circle around every possible combination of elements (this would be the tricky part).

Draw one big circle around all the elements.

Draw one circle off to the side for the empty set.

Shouldn't that Venn Diagram every element of the powerset, overlapping the sets that share elements in common? I'm not sure if this is what you meant.

I just tried it for 4 elements. It gets messy and hard to follow. Maybe if you use different colors...

Hope this helps!

[Joshua Zucker]

You should draw a circle for each as in the regular Venn diagram. Only instead of circles for sets A, B, C, and intersections being the overlap, now it will be elements a, b, c, and intersections meaning sets of two or three of those elements.

[Christian Baune]

Should be self explanatory :-)

[Dylan Faullin]

Very nice illustration. It seems as though I had it inverted. Thank you for the correction, though I think you could do it either way. It depends how you want to define your Venn Diagram regions. You are defining a region as containing the sets within a powerset whereas I defined my regions as representing the sets within a powerset. For example, the region that contains A, B, and C is representing the set {a,b,c}. The set {a,b} is a subset of {a,b,c} and is therefore represented by a region (containing A and B) that is contained within the region representing {a,b,c}. Seems logical to me. Am I wrong in thinking of it this way?

Should any mention be made of the null set in your venn diagram? After all, the null set is part of the power set... Where would you put it in the diagram? Or is it just 'understood'?

Thanks again for taking the time to create/share that diagram.

Christian Baune said:

Should be self explanatory :-)

[Christian Baune]

The empty set is here but can't be seen :-)
(intersection of none, part of outside)

I think that nested sets are not convenient because it became quickly messy.

My representation does not give the feeling that "{A,B}'" is contained in "{A,B,C}" but it has the advantage to be clear.(Even if I had to put coloureds dots to help reader figure it out)
But I couldn't get it work for 3 items or more.
And it's quite normal.
If you had 1 set, each set would contain 1 group of dot.(1 possibility)
If you had 2 sets, each set would contain 2 groups of dots.(3 possibilities)
If you had 3 sets, each set would contain 4 groups of dots.(7 possibilities)
If you had 4 sets, each set would contain 8 groups of dots.(15 possibilities)
As you can see in my diagram, each set contain 7 groups of dots. My diagram is wrong :-) (missing a triplet)



The diagram you did is, at my sense, better matching reality because you've the feeling that these are included sets. But it's messy.

[Dylan Faullin]

The fact that you couldn't quite get it to work for more than 3 elements (and the nature of the issue in getting it to work) gave me the feeling that it might be related to the 4-color theorem...It was strange. The thought just jumped into my head. I didn't take the time to see if this is true. Very interesting though...So perhaps it may not even be possible for power sets derived from more than 3 elements.

Also, good job on catching the missing set. I didn't notice that anything was amiss in your diagram. You have a sharp eye.

Christian Baune said:

The empty set is here but can't be seen :-)
(intersection of none, part of outside)

I think that nested sets are not convenient because it became quickly messy.

My representation does not give the feeling that "{A,B}'" is contained in "{A,B,C}" but it has the advantage to be clear.(Even if I had to put coloureds dots to help reader figure it out)
But I couldn't get it work for 3 items or more.
And it's quite normal.
If you had 1 set, each set would contain 1 group of dot.(1 possibility)
If you had 2 sets, each set would contain 2 groups of dots.(3 possibilities)
If you had 3 sets, each set would contain 4 groups of dots.(7 possibilities)
If you had 4 sets, each set would contain 8 groups of dots.(15 possibilities)
As you can see in my diagram, each set contain 7 groups of dots. My diagram is wrong :-) (missing a triplet)



The diagram you did is, at my sense, better matching reality because you've the feeling that these are included sets. But it's messy.

[Joshua Zucker]

You need to move your rectangles around a bit to get a 4-set Venn diagram, like if you scroll down a bit at
http://math.gmu.edu/~eobrien/Venn4.html
to find the rectangle-based option.

A nice five-set drawing is at http://www.maa.org/editorial/mathgames/mathgames_11_01_04.html

I still think the common-sense approach here is to have one region (oval or rectangle in these examples) for each element, and then the sets containing more elements are in the overlap.

If you want to go the other route, try having each region stand for a subset NOT containing the given element. That might work out quite nicely. Then the place in the middle where everything overlaps is the empty set, and the whole page is your original set, and so on. I think that might fit the intuition you're trying to communicate here.