What is the difference between these two ordered pairs I am studying Linear Algebra and often I come across things that belong the abstract algebra but still peak my interrest. I was reading up on metric and metric spaces (on wikipedia)
Wikipedia says that a metric space is an Ordered pair pair (M,d) where M is a set and d a metric
d: MxM---)$\mathbb{R}$. But how come a coordinate in the x-y plain is also a ordered pair (a,b) without a function b:AxA--) $\mathbb{R}$ (thats stupid I know but it try to make a point.)
And when I was learning about vectorspaces I came across a quadruple A vector space is defined as a quadruple $(\mathbf{V},\mathbb{K},\oplus,\odot)$ where $\mathbf{V}$ is a set of elements called vectors, $\mathbb{K}$ is a field $(\mathbb{K},+,\cdot)$ , $\oplus$ is a binary operation (called sum) on $\mathbf{V}$ such that $(\mathbf{V},\oplus)$ is an Abelian group and $a\odot\mathbf{v}:\mathbb{K}\times\mathbf{V} \rightarrow \mathbf{V}$ is a scalar multiplication. But how do you know the underling relation between these objects in the quadruple (ordererd pair(metric space)) Do all quadruples have the same formath of relations I mean does the first symbol and the third symbol always have a relation or does this need to be fined , another example is the second symbol always a field or can It be any object? Are the last 2 always "binaire operations" (yeah wikipedia says that scalar multplication can sometimes be called a binaire operations altough it clashes with the definition) Summary: I want to know what can be in these quadruples  and How you can know the underling "relation" (probaly not the right word but I hope you get it) between these different things in the quadruple. I did some wikipedia surfing myself but didnt come to a clear answer and I dont realy know where to look.
Thanks in Advance
 A: When we say $(A,\square,\triangle)$ is a field, we are saying that the base set of the field is $A$, that $\square$ and $\triangle$ are binary operations on $A$ satisfying certain properties.
We use an ordered triple because we need to distinguish between the first operation and second operation, they don't satisfy the same axioms. In the case of that field, for example, we know that $\triangle$ distributes over $\square$, but no the other way around, or that the every element of $A$ has an inverse with respect to the $\square$ operation, and that's not true for $\triangle$. If we write "let $A$ a field with operations $\#$ and $\&$" it's not clear from the statement which operation corresponds to the "addition" of field and which to the "multiplication". Writing them in a triple get rid of the ambiguity.
Also, I think you have a misconception, because you ask "how do you know the underling relation between these objects in the quadruple "(in this case, triple). The point is that the notation "$(A,\square,\triangle)$" alone is incomplete. The correct notation would be "$(A,\square,\triangle)$ is a field".
From $(x,y)$ we obviously can't know if that denotes a group or a metric space (or a completely different thing). Examples of COMPLETE notations are is "$(x,y)$ is a group" or "the metric space $(x,y)$".
A: In these definitions, we are talking about specific structures. And, the idea that governs these structures is what's important.
When you are talking about the ordered pair $(M,d)$, you are basically talking about a space with a metric (or distance). When you are considering the point $(a,b)$, you are talking about a point on the plane. These two ideas being completely different, I don't understand why you are even trying to match them together.
When you are talking about the quadruple $(\mathbf{V},\mathbb{K},\oplus,\odot)$, you are basically referring to a vector space. This space comprises of a set of vectors $\mathbf{V}$, a field $\mathbb{K}$, and a couple of operations that we call addition and scaling. Again, a quadruple $(a,b,c,d)$ is just a four dimensional point. So, it doesn't need all these relations- it's just a simple point!
The reason behind using the words "ordered pair" or "quadruple" in these definitions is to just be rigorous. To understand that, just think of how else to put the same idea without using loose layman terms.
Also, if you think a little more rigorously, then an ordered pair $(a,b)$ can simply be thought of as the very innocent set $\{\{a\},\{a,b\}\}$. A metric space or a vector space comes with much more structure embedded in it. They involve operations, functions, sets and so on. So, of course, you should not expect any ordered pair to come with such structures.
Hope that helps.
Edit: A question in the comments ask whether the order is important. Well, according to me, not really, but yes.
First thing is the convention that we follow. We really think of the pair or the quadruple as (Space,Metric)$=(M,d)$ or (Set,Field,Addition,Scaling)$=(\mathbf{V},\mathbb{K},\oplus,\odot)$. So, you can sense an importance of an ordered pair.
Secondly, just think of the notation $(d,M)$. Notice that $d$ is a function from $M$ to $\mathbb R$. So, to define $d$, we first need to define $M$. So, using $(d,M)$ (which talks of $d$ first, and then $M$) makes much less sense than $(M,d)$. Similarly, to define scaling or addition, you need the definitions of $\mathbf{V}$ and $\mathbb{K}$ first.
I hope that answers the question.
A: An ordered pair is just that.  It is two things that are associated and we want to keep track of which is first and which is second.  They may be the same type of thing or not.  In your metric space example the first is a space and the second is a metric.  They are not the same type of thing at all, but we need both to make a metric space.  In your $\Bbb R^2$ example they are two coordinates that together define a point in the space.  Triples and quadruples are similar but with more things that need to be in the list.  You shouldn't think of ordered pairs as a class because they are just a nice way of organizing two things that go together.  When you see an ordered pair in a text there should have been a definition of the type of ordered pairs that are being considered.  Your example of a metric space is typical.  Similarly,  a group is a set of elements and an operation on those elements (that satisfies certain axioms).  The space in your metric space may already be a vector space, so it may be a quadruple itself.  There is nothing special inherent to the positions in an ordered pair, triple, ...  It is a convenient structure to keep things together that belong together.  For a specific use there should be more definition.
