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Additional Material



Products of Sets

Discussions here involve another useful way of "joining" sets, quite unlike any of the operations on sets discussed in Volume A Chapter 2. They are generalizations of the coordinate plane and coordinate space of analytic geometry which involve ordered pairs (x,y) and ordered triples (x,y,z), respectively. Introductory examples provide the basic idea.

Let sets U = {a,b,c} and V = {1,2,3,4}. From these sets form 12 ordered pairs:
     (a,1),   (a,2),   (a,3),   (a,4)
     (b,1),   (b,2),   (b,3),   (b,4)
     (c,1),   (c,2),   (c,3),   (c,4)
This table shows that the notation for an ordered pair is ( , ) or more exactly (an object taken from the first set, an object taken from the second set), where U is the first set, and V is the second set. The objects in the notation are called coordinates: in (b,3), b is the first coordinate and 3 is the second coordiante. The collection of all ordered pairs given here is

UxV = {(a,1), (a,2), (a,3), (a,4), (b,1), (b,2), (b,3), (b,4), (c,1), (c,2), (c,3), (c,4)}
The set UxV is called the product of U and V.
Because of the 'x' in the notation UxV it is may be called the "cross product of two sets."
(Technical definition) (x,y) is in UxV if and only if x is in U and y is in V.
The product of two sets of m objects and n objects respectively has (the product) mn ordered pairs.

If W = {β,γ} then UxVxW consists of 24 ordered triples. Among them are (a,4,β) and (c,2,γ).In the last ordered triple, c is the first coordinate, 2 is the second coordinate and γ is the third coordinate. It would require a three-dimensional table to show properly all of the ordered triples in UxVxW.
UxVxW is the product" of U,V,W or the "cross product of the three sets."
The product of three sets of k objects, m objects and n objects respectively has (the product) kmn ordered triples.
In future discussions products of more than three sets will not be needed. But it is easy to extend the above discussions to such products. UxVxW is the product" of U,V,W or the "cross product of the three sets." It is important to determine if ordered pairs are equal and ordered triples are equal.

Two ordered pairs are equal if and only if their first coordinates are equal and their second coordinates are equal.
Technical definition) (x,y) = (x',y') if and only if   x = x' and y = y'.
Notice that a single equality between ordered pairs involves two equations between pairs of corresponding coordinates.

Two ordered triples are equal if and only if their first coordinates are equal, their second coordinates are equal, and their third coordinates are equal.
Technical definition) (x,y,z) = (x',y',z') if and only if   x = x' and y = y' and z = z'.
A single equality between ordered triples involves three equations between pairs of corresponding coordinates.

Notice that the ordered pairs in UxV and VxU are different. In VxU the first coordinates are numbers, and the second coordinates are letters. This means that the product is not commutative.

Most useful is the product of the (real) number line with itself RxR. It is the familiar set of all ordered pairs (x,y) where x and y are real numbers. They form the coordinate plane of analytic geometry. Often the symbol R2 = RxR is used.
Similarly R3 = RxRxR is coordinate space of solid analytic geometry with coordinates (x,y,z).

(Technical definition) A relation between two sets is a subset of their product.
Notation: A relation between a set U and a set V is a subset of the product UxV.

It is important which set is mentioned first: a relation between U and V or a relation between V and U. They are different if U and V are different.
Many, many years ago a relation was called a multi-valued function, but this term is seldom used currently.