[1] Let U,V be non-empty sets. The Cartesian product UxV consists of the set of ordered pairs (u,v) for all u in U and for all v in V. u is called the first coordinate and v is the second coordinate of the ordered pair (u,v).
Notation: UxV = {(u,v) | u in U and v in V}.
Example [1.1]
Let U = {1,2,3,4} and V = {a,b,c}.
Then UxV = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c), (4,a), (4,b), (4,c)}
and VxU = {(a,1), (b,1), (c,1), (a,2), (b,2), (c,2), (a,3), (b,3), (c,3), (a,4), (b,4), (c,4)}.
The ordered pairs in VxU are the reverse of the ordered pairs in UxV.
Example [1.2]
It is impossible to list all the ordered pairs of natural numbers in NxN. The following gives an idea:
NxN = {(1,1), (1,2), ..., (1,n), ..., (2,1), (2,2), ..., (2,n), ..., ..., (m,1), (m,2), ..., (m,n), ... }
[2]Two ordered pairs are equal if and only if their first coordinates are equal and their second coordinates are equal.
Notation: (u1, v1) = (u2, v2) if and only if u1 = u2 and v1 = v2.
None of the ordered pairs in VxU in example [1.1] are equal to any ordered pair in UxV. But reversing all of the ordered pairs in NxN simply produces all of the ordered pairs in NxN.
[3] (Symmetry) A set of ordered pairs is symmetric if reversing all of the ordered pairs in that set reproduces all of that same set.
The only way that a set of ordered pairs be symmetric is that the set be equal to the Cartesian product of some set with itself.