Relations & Cartesian product explanation I'm confused about this notation:
R c A x B

(R is a subset of the Cartesian product of A and B).
Cartesian product of AxB looks like this, right?
    1       2        3
x (x,1)   (x,2)    (x,3)
y (y,1)   (y,2)    (y,3)
z (z,1)   (z,2)    (z,3)

Where 123 index is B, and xyz index is A.
And R in the notation means 'Relation' (I guess).
But how can a relation be a subset of AxB?
Or is it just a way of writing that R is defined in the context of AxB?
For example: R is defined like this:
R = {(x,y) e Z x Z | y = x^2 + 2x - 3}

And that this definition of R occurs for the Cartesian product only?
I'm lost here, can someone explain this to me?
I'm also confused about the following notations:
If R c AxB is a relation, and (a,b) is an element of that relation, we note this: aRb instead of this: (a,b) e R.
And what is the notation 

UcA

?
And, what exactly is a relation?
Thanks a lot!
 A: First of all, I wish to point out that your example definition of a relation:
$$R = \{(x,y)\in\mathbb Z\times\mathbb Z\mid y=x^2+2x-3\}$$
IS a subset of $\mathbb Z\times\mathbb Z$.
All of its elements are members of $\mathbb Z\times\mathbb Z$ according to your own definition.
Note that each element of $\mathbb Z$ is a scalar whereas each element of $\mathbb Z\times\mathbb Z$ is a tuple.

If the above is confusing in any way, I suggest consulting a good dictionary and thoroughly clearing up the terms "tuple," "scalar," "relation."

However, a brief explanation, not intended to be rigorous:
A scalar is something that's considered as a whole; it's not considered as having parts.
(Note: Anything can always be considered as having parts, but the parts don't need to be considered for every application.  Outside the realm of math, consider a car dealer's inventory: each car is a single entry.  A car parts dealer considers car parts, but even then each part is ultimately comprised of molecules and atoms, but this fact is irrelevant.  In much the same way, a scalar is considered to be indivisible at the level of abstraction we're dealing with, whether it is or isn't indivisible in other abstractions.)
You know what a set is, by your question.  (I hope.)  A tuple is an ordered set.  Usually we are only concerned with tuples that all have the same number of elements.  "Ordered pair" is another word for "tuple."  Bigger tuples might be called "3-tuples," "4-tuples," etc.

Since you are probably used to considering relations as either "true" or "false"—such as the relation "is greater than"—consider set-theoretic relations the same way.
Let's say $R$ is the "is greater than" relation.  Then we have, in your notation:
$$R = \{(x,y)\in\mathbb Z\times\mathbb Z\mid x>y\}$$
Note that this consists of a set of ordered pairs taken from the Cartesian product $\mathbb Z\times\mathbb Z$ according to a certain rule.  In other words, if the "greater than" relation evaluates to "true," we include the tuple in our set $R$; if it evaluates to "false," we exclude the tuple.
So $R$ already IS a subset of the Cartesian product.  Set theory is just a way to formalize this.
And thus you're saying:
$$x>y \iff (x,y)\in R$$
Which is to say:
$$xRy\iff (x,y)\in R$$
And in both of these definitions, $x\in\mathbb Z\land y\in\mathbb Z$, which is to say $(x,y)\in\mathbb Z\times\mathbb Z$, so $R\subseteq\mathbb Z$.

In other words, your question is just about notation.  I hope that helps.

I will add that a Cartesian product has no indices in set theory.  Sets are unordered.  A Cartesian product of two sets is also a set, so it's unordered.  The elements of the set (the elements of the Cartesian product) are themselves ordered pairs, but there is no order between pairs.  In other words, the Cartesian product is a set of tuples; it's NOT a matrix.
Matrix multiplication is a different thing.  Matrices ARE ordered; sets (including Cartesian products) are not ordered.
A: First I would like to clarify something when in mathmatics we say something has a definition it means that "Whenever we use this word it is short for this longer term".
The definition of relationship is an example of this.  "A relationship between sets A and B is defined as a subset of the cross product between A and B".  You can substitute the names of the sets (A and B) for the sets you are considering.
Pairs of sets can have many diffrent relationships between and diffrent sets have diffrent relationships.
Then you are given an example of a perticular relationship between $\mathbb{Z}$ and $\mathbb{Z}$.  This relationship is all the pairs where that formular is true.
A: Your indexing is confusing but seems to get the point.  
If A = { a,b }, B = { 0,1 }, then AxB = { (a,0), (a,1), (b,0), (b,1) }.  
Relations are defined to be subsets of a product.  It is the definition.
"aRb" is another way of writing "(a,b) in R" just
like "isn't" is another way writing "is not".
R is a relation of ZxZ;  a binary relation about integers.
The notation UcA is meaningless except for your unusual use of c to
mean subset;  so it could mean U subset A.
