Set Theory begins with a fundamental binary relation between and object $o$ and a set $A$. If $o$ is a member of $A$, write $o \in A $.
I thought that a binary relation is a collection of ordered pairs of elements of $A$.
Why is relating one element of a set to the set a binary relation?
Thanks.
I thought that a binary relation is a collection of ordered pairs of elements of A.
Well, no: strictly speaking, "is" is the wrong word here. A binary relation is not a set of ordered pairs -- rather, in the context of set theory, (some) relations are implemented or modelled by sets. As the more careful authors say clearly.
For the record, here are three pretty familiar (or ought-to-be familiar) reasons for not actually identifying binary relations with sets of ordered pairs.
Note first that some binary relations are "too big" to have corresponding sets as graphs. For an obvious but pertinent example, take the relation of set membership itself. The ordered pairs $(x, y)$ where $x \in y$ are too many to form a set on standard set theories. If a relation it too big to have a corresponding set as its graph, it can't be that set. So it is immediate that not all relations are sets.
Can we at least identify "small" relations with sets? Well, it might be conventional to model a binary function $R$ as corresponding to [say] the ordered pairs $(x, y)$ such that $Rxy$, and then treat the ordered pairs by the Weiner-Kuratowski construction. But note that at both steps we are making arbitrary choices from a range of possibilities. By making compensating adjustments in the modelling scheme, you could use the set of ordered pairs $(y, x)$ or the ordered pairs $((x, \emptyset),(y, \{\emptyset\}))$, etc.: and similarly, you could choose a different set-theoretic representation of ordered pairs. Since the association of the binary relation with a set involves arbitrary choices, there isn't a unique right way of doing it. None, then, can be reasonably said to reveal what the relation really is. We are, to repeat, in the business of modelling or implementing or representing (relative to some chosen scheme of representation).
Most importantly, it is a type confusion to identify a relation with an object like a set. A binary relation holds, or fails to hold, of two objects. In terms of Frege's nice metaphor, a relation like e.g. the numerical less-than relation is "unsaturated", comes with two slots waiting to be filled (filling the slots in the binary less-than relation yields a numerical truth or falsehood). In modern terminology, relations in general, and binary relations in particular, have an intrinsic arity. By contrast, objects aren't unsaturated, don't have slots waiting to be filled, don't intrinsically have arities, don't hold of objects. And what applies to objects in general applies to those objects which are sets in particular. So binary relations -- like other relations -- aren't sets.
For corresponding points about functions, see In what manner are functions sets?
When you "start with" set theory, you do not know that a relation is a set of ordered pairs.
You start from first-order language, where you have variables ($x$, $y$, ...) and prediates ($P, Q$, ...).
Thus, you will use a binary predicate $e(x,y)$ such that $e(o,A)$ holds when $o$ belongs to $A$; in this case, we abbreviate it with :
$o \in A$.
In first-order language, $n$-ary predicates, when $n >1$ are called : relations.
Here the term binary relation has a different meaning, you should interpret it as one would colloquially interpret the term binary relation: a relation between two things. It's not a mathematical concept, but rather one belonging to the natural language.
Restating, binary relation as a mathematical term is a set of ordered pairs, as a natural language term it is a 'relation' between two things and you can't expect to define 'relation' here.
Actually, a binary relation is a relationship between two things ( $ a $ belongs to $ A $ , for example, so as $ A $ is included in $ B $ ). Then you could write an ordered pair $ (a, A) $ in the first case and $ (A, B) $ for the second.
The distinction between a ‘binary relation’ and ‘an ordered pair’ is particularly germane when considering the set membership relation, ‘$\in$’, because it clearly illustrates a case of the chicken vs. the egg.
As the Wikipedia article states, a set theory is some logic (say first order with identity for arguments sake) with the addition of the $\in$-relation and axioms dictating its use.
Now while not necessary, ordered pairs are often defined in terms of a set theory as a later development. When defined in such a way, an ordered pair is essentially reduced to a logical statement involving the $\in$-relation, as are the sets which aggregate the ordered pairs.
That last bit is important, for while we can certainly describe ‘binary relations’ in terms of sets of ‘ordered pairs’, consider what happens when you try to define the $\in$-relation in terms of such set theoretical ‘ordered pairs’. Without some additional machinery our definitions turn circular: ordered pairs in terms of the $\in$-relation, and the $\in$-relation in terms of (sets of) ordered pairs.
The take away here, I believe, is that while it can be useful to talk about relations in general in terms of sets of ordered sequences, particularly when relations are the object of study, in practice relations are properly a part of the underlying logical language used to make assertions about objects, not objects about which we make assertions.
