Problem in set theory where elements of a set are geometrical figures. I am facing a problem in analyzing sets having their elements as geometrical figures like triangles, rectangles or circles. May be it is because I have not studied advanced geometry but still I want to know how shall I approach questions like these.
My book asks me to find whether set of all triangles in a plane is a subset of set of all rectangles in that plane. First I thought that it is not. But then I thought that both the sets would be infinite ( there can be infinite rectangles or triangles in a plane. Also all such triangles could be a part of rectangles and vice versa. So I am stuck whether it is a subset or not.
Another question which is brief but vital for me is that what universal set would we choose for the set of all equilateral triangles in a plane. I thought about this and concluded that as triangles are merely a collection of points, then the universal set we should chose should be set containing all the points in a Euclidean plane. But I thought it may be also wise to choose set of all triangles as universal set. So what it would be ?
It also leads me to another question that whether a set can have more than one universal sets and if not how are they chosen ?
I'm sorry that I am putting all these questions together here but I found them related and relevant to be put together. Also I may be wrong somewhere and I want you let me know it as I am only a beginner now.
Thanks for your help . 
 A: You talk about very different sets. Let's take the questions one at a time.
Is the set of all triangles a subset of the set of all rectangles?
No. Let us call $T$ the set of all triangles and let us call $R$, the set of all rectangles.  The elements of $T$ are triangles whereas the elements of $R$ are rectangles. A set $A$ is a subset of a set $B$ if all elements of $A$ are also elements of $B$. But no triangle is a rectangle so no elements of $T$ are elements of $R$. 
To dispel a misunderstanding. You say "triangles could be a part of rectangles". If indeed you consider your shapes to be sets of points in the plane then you can have a specific triangle $t$ be part of a specific rectangle $r$. But this just means that the $t$ is a subset of $r$, that is an element of $T$ is a subset of an element of $R$ and this has no bearing on $T$ being a subset of $R$.
To give an easy example consider the sets $A=\{\{1,2\},\{2,3\},\{1,2,3\}\}$ and $B=\{\{1,3\},\{1\}\}$. So the elements of $A$ are the set, $\{1,2\}$ the set $\{2,3\}$ and the set $\{1,2,3\}$ and the elements of $B$ are the sets $\{1,3\}$ and $\{1\}$. You can see that no elements of $A$ are equal to any elements of $B$ and vice versa so the sets are disjoint. In particular $A$ is not a subset of $B$ and $B$ is not a subset of $A$. At the same time though $\{1\}$ is a subset of both $\{1,2\}$ and $\{1,2,3\}$ and $\{1,3\}$ is a subset of $\{1,2,3\}$. So each element of $B$ is a subset of some element of $A$.
What universal set should I choose?
What universal set you should choose is hard to say. This depends on what you are doing. In general in mathematics we rarely choose universal sets (these usually wouldn't be sets anyway) since they make little sense. You can't choose the set of all points on the plane since that set does not contain the members of either the set $T$ or the set $R$. The members of $T$ and $R$ are subsets of the set of all points on the plane. What you could choose if you wanted a set of which $T$ and $R$ would both be subsets would be the power set of the set of all points in the plane. That seems hardly useful though. Such a set will contain many sets you probably don't know how to deal with in any reasonable sense.
You could choose the set of all triangles as a universal set which would probably make a little more sense than choosing the power set. But it wouldn't contain any rectangles so $T$ wouldn't be a subset of it. A more reasonable choice would be the set of all triangles and rectangles but it's rather arbitrary.
This takes us to your last question.
Can a set have more than one universal sets and if so how are they chosen ?
The answer to this is probably yes depending on what you mean by universal set. I can't find any rigorous definition of a universal set, as I said we don't use them in mathematics in general because they make little sense. There is the universe which contains all sets but it is not a set itself (at least from inside), but that is way too big for anything you might want to do with it.
I think that some teachers use the idea of a universal set to allow for drawing things like Venn diagrams and defining the complement. We don't usually call these universal sets though they are just the places we're currently working. So these might be a set containing all triangles in the plane if we are looking at equilateral and non equilateral triangles, or they may be the set of all convex polygons with four or less sides if we are looking at triangles, rectangles, trapezoids, parallelograms and squares. Which we choose to be the "universal set" depends on what we want to do.
