Ordered Pair
An ordered pair of a and b, with first co-ordinate a and second co-ordinate b is defined by,
(a,b) = {{a}, {a,b}}
The proof for the above ordered pair is given by,
If (a,b) and (x,y) are two ordered pairs, then (a,b) = (x,y) if and only if a=x and b=y.
Proof:
If in case, a = b, then, (a,b) => the singleton set of {{a}} or the singleton set of {{b}}.
Also, conversely, if (a,b) itself is a singleton set, results in {a} = {a,b}, so {b} ϵ {a,b} so which leads to a=b
On the other hand if we assume , (a,b) = (x,y) then both of them have to be singletons which can lead to
Singletons on both the sides, like {a} = {x}, so that a = x.
Also both the pairs should have atleast have one element which is not a singleton({a,b} and {x,y}), it results to {a,b} ={x,y}.
Now since we already know that b ϵ {a,b}, b is also a subset of {x,y}.
Since a=x, b also cannot be equal to x and hence, b = y.
T his completes the proof that, (a,b) = (x,y) If and only if a=x and b=y.
Cartesian product:
When A and B are two sets, we can define a set which contains the ordered pair (a,b) such that a is in A and b is in B.
Assuming above, we can conclude that {a} ϲ A and {b} c B, implies that {a,b} c A U B. Also, {a} C (A U B) and {b} C {AUB}, which implies, (a,b) C {A U B}.
There is an important observation from this, {a}, {b}, {a,b} are actually some of the members of the power set of the elements of the set {a,b}. So we can conclude that, (a,b) ϵ ‽(a,b). Its also obvious that ‽(a,b) has more elements than the ordered pair hence we can define a new set ‽(‽(a,b)) when a ϵ A and b ϵ B.
Applying the axiom of specification and axiom of extension we can obtain the set A X B which is the Cartesian product such that
A X B = {x:x=(a,b) for some a in A and some b in B}.
We can take forward the equation in a converse order for any set of ordered pairs there exists a set R such that, for sets A and B, R C (A X B).
Now, ({a},{a,b}) ϵ U R. Now, since {a,b} is one of the subsets of the ordered pair, its safe to assume that, {a,b} ϵ U R. This also assumes that a and b also belongs to UUR.
Projections
Its also often desirable to construct projections (like projection in relation algebra for example). By using the definitions,
A = { a: for some b in ((a,b) ϵ R)}
B = { b: for some a in ((a,b) ϵ R)}
