Is It Possible to Represent Group Products Using Venn Diagrams in Algebra?

[soopo]

Homework Statement

Let G be a group, and let H and K be two subgroups of G.
Then the union of H and K is a subgroup of G such that the intersection of H and K is an empty set.

Can you visualize the product hk in a venn diagram for all h
which belongs to H and for all k which belongs to K?

The Attempt at a Solution

You cannot because:

Take a point h at the top of H group where derivate of the "circle" is zero.
Take similarly a point k at the top of K.

Let k=(-1, 10) and h=(1, 10).
The product kh is
10 * 10 = 100
in y-direction, while
-1 * 1 = -1
in x-direction.

Thus, we get a point kh = (-1,100)
which does NOT belong to H nor K.

This suggests me that the set of the union H and K is greater than than
the high-school presentation of the union.
It seems that we cannot use venn diagrams in such problems.

 

[HallsofIvy]

Homework Statement

Let G be a group, and let H and K be two subgroups of G.
Then the union of H and K is a subgroup of G such that the intersection of H and K is an empty set.

Are you stating this as a general proposition? If so it is certainly not true! The intersection of two subgroups of group G cannot be empty. They must both include at least the group identity. Additionally, a group, G, of order 8 may have a subgroup of order 4 which itself contains a subgroup of G of order 2. It is true that if two subgroups have non-trivial intersection (that is, not just the group identity) then one must be a subgroup of the other.
(I finally looked at your attachment. That's exactly what it says!)

Can you visualize the product hk in a venn diagram for all h
which belongs to H and for all k which belongs to K?

The Attempt at a Solution

You cannot because:

Take a point h at the top of H group where derivate of the "circle" is zero.
Take similarly a point k at the top of K.

I have absolutely no idea what you are talking about here. What do you mean by the "top" of a subgroup and what "circle" are you talking about?

Let k=(-1, 10) and h=(1, 10).
The product kh is
10 * 10 = 100
in y-direction, while
-1 * 1 = -1
in x-direction.

Thus, we get a point kh = (-1,100)
which does NOT belong to H nor K.

This suggests me that the set of the union H and K is greater than than
the high-school presentation of the union.
It seems that we cannot use venn diagrams in such problems.

What group G, and subgroups H and K are you talking about in your example?

 

[soopo]

I have absolutely no idea what you are talking about here. What do you mean by the "top" of a subgroup and what "circle" are you talking about?

Please, see the new attachment. I think the groups as 2D circles. Groups contain all points inside the circles.

What group G, and subgroups H and K are you talking about in your example?

I am talking about the groups G, H and K in the first attachment. I am trying to show that you cannot use Venn diagrams to visualize groups, since in the 2D plane, the union of H and K seems to be greater than the size of G.

 

[Mark44]

How can you say that H U K seems to be larger than G? Your drawing doesn't show G, that I can see.

Also, you are imposing a 2D structure on your Venn diagram that groups generally don't have. A group is a collection of things, together with some operation. You are assuming the things are points (ordered pairs), and apparently are assuming that the operation is multiplication.

It is completely incorrect to talk about the "top" of a group, or the shape of a group, as well as the derivative of a tangent line on the circle. If k = (-1, 10) and h = (1, 10), as you wrote, what justification do you have for saying that kh = 100 "in one direction" and -1 in the other direction. You are making all sorts of assumptions about the members of arbitrary groups, and the operation on group members that are not justified by the given conditions of your problem.

 

[soopo]

How can you say that H U K seems to be larger than G? Your drawing doesn't show G, that I can see.

Also, you are imposing a 2D structure on your Venn diagram that groups generally don't have. A group is a collection of things, together with some operation. You are assuming the things are points (ordered pairs), and apparently are assuming that the operation is multiplication.

It is completely incorrect to talk about the "top" of a group, or the shape of a group, as well as the derivative of a tangent line on the circle. If k = (-1, 10) and h = (1, 10), as you wrote, what justification do you have for saying that kh = 100 "in one direction" and -1 in the other direction. You are making all sorts of assumptions about the members of arbitrary groups, and the operation on group members that are not justified by the given conditions of your problem.

Thank you for pointing out the assumptions which I made!

The http://mathstat.helsinki.fi/~fluch/algebra_1-au06/algebra1.pdf" says that if we talk about groups we often
make an assumption that the binary operator is multiplication for
non-abelian groups, while plus for abelian groups.
I therefore assume that the binary operator is multiplication.

It clear my confusion:
I now know that more what the "collection" of sets mean.
It seems that we use naive set theory to handle the problem.

It is now clear that we cannot make such an assumption that the things are ordered pairs.

Your answer raised a question.

How often do we need to use Venn diagram and ordered pairs in Algebra?

In other words, how do you visualize the sets in Algebra?
I know little about lattice diagrams, Cayley's tables and mappings between sets.
Is there any other way to visualize the relations in sets?

 

[Mark44]

How often do we need to use Venn diagram and ordered pairs in Algebra?

38% of the time.

Seriously, though, I don't know how often, but Venn diagrams might be useful to get an understanding of set union, set intersection, and complements.

How often to we need to use ordered pairs in Algebra? I guess it would depend on the sets you're working with. In the set of the problem you posted, ordered pairs don't enter into it at all. If you're working with the direct product of two sets, then you are working with ordered pairs.

 

[soopo]

38% of the time.

Seriously, though, I don't know how often, but Venn diagrams might be useful to get an understanding of set union, set intersection, and complements.

How often to we need to use ordered pairs in Algebra? I guess it would depend on the sets you're working with. In the set of the problem you posted, ordered pairs don't enter into it at all. If you're working with the direct product of two sets, then you are working with ordered pairs.

Thank you for your answers!

I will be ready for the next time when I meet such a problem.