What is a cartesian product?

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The cartesian product M1×M2 of two sets M1 and M2 is the set of all pairs (x1,x2) with x1∈M1 and x2∈M2. More generally, the cartesian product M1×M2×…×Mn of n sets M1,M2, …,Mn ist the set of all n-tuples (x1,x2,…,xn) with x1∈M1, x2∈M2, …, xn∈Mn. We may write

M_1\times ... \times M_n = \{ (x_1,...,x_n) \left| x_i \in M_i \textrm{ for } i=1,...,n \right. \} .

Before we turn to some examples, let us recall the notion of n-tuples. A tuple is a list of finitely many objects which do not have to be different from each other. An n-tuple contains n objects. Like a set, a tuple is essentially a collection of objects, but it is important to notice two main differences to sets:

  • In a set, we would not write an element more than once. The reason is that an object either is an element of a given set or not. It can’t belong to the set several times. In tuples, it is perfectly fine to have the same object in several positions of the tuple. For example there is nothing wrong about the 3-tuple (0,1,1). On the other hand, in the set {0,1}, writing the element 1 several times would not have any meaning.
  • When we write down a set, the order of the elements doesn’t matter. The set {0,1} can also be written as {1,0}. However, for tuples, the order matters. The 2-tuples (or pairs) (0,1) and (1,0) are not the same. Also (0,1,1) and (1,0,1) are two different tuples. Two tuples (x1,x2,…) and (y1,y2,…) are only the same, if they have the same number of entries and if x1=y1, x2=y2, and so on.

Therefore it makes sense to speak of the first, second or j-th entry of a tuple and we can say that if a tuple belongs to the cartesian product M1×M2×…×Mn , the first entry belongs to M1, the second one belongs to to M2 and so on.

Examples of cartesian products

As a simple example, let us consider the two sets of letters M1={a,b,c,d} and M2={x,y,z}. The cartesian product M1×M2 is the set of all 2-tuples with one of the letters a,b,c,d in the first position and one of the letters x,y,z in the second position. This set consists of the following twelve elements:

abcd
x(a,x)(b,x)(c,x)(d,x)
y(a,y)(b,y)(c,y)(d,y)
z(a,z)(b,z)(c,z)(d,z)
All elements of the cartesian product {a,b,c,d}×{x,y,z}

As a next example we consider the cartesian product {0,1}×{0,1}×{0,1}. It is the set of all 3-tuples with either 0 or 1 in all three positions. We obtain the following set of eight elements:

{0,1}×{0,1}×{0,1}={(0,0,0), (0,0,1), (0,1,0), (1,0,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1)}.

These sets are finite. If infinite sets are involved in the cartesian product, we obtain an infinite set as a result. For example the cartesian product

\{0,1\} \times \mathbb{Z}\times \mathbb{Q} = \{(x_1,x_2,x_3) \left| x_1\in \{0,1\}, x_2\in \mathbb{Z}, x_3 \in \mathbb{Q} \right. \}

is the set of all 3-tuples with either 0 or 1 in the first position, an integer in the second position and a rational number in the third position.

Cartesian products and real spaces

Cartesian coordinates of a point in the plane

Cartesian products of the set of the real numbers with itself play an important role in physics as they represent continuous spaces. For example the cartesian product

\mathbb{R}^2=\mathbb{R}\times \mathbb{R}= \{(x_1,x_2)\left| x_1,x_2\in \mathbb{R} \right. \}

is the set of all pairs of two real numbers x1 and x2. These two numbers can be interpreted as the two coordinates of a point with respect to two perpendicular coordinate-axes (i.e. two continuous rays of real numbers). These are called cartesian coordinates. As the cartesian product is the set of all of these points described by two real coordinates, it represents the entire plane. In the same sense does

\mathbb{R}^3=\mathbb{R}\times \mathbb{R}\times \mathbb{R}= \{(x_1,x_2,x_3)\left| x_1,x_2,x_3\in \mathbb{R} \right. \}

describe the three-dimensional space. For higher n, the cartesian products

\mathbb{R}^n=\{(x_1,...,x_n)\left| x_1,...,x_n\in \mathbb{R} \right. \}

may be used to describe higher dimensional spaces, such as four-dimensional space-time in the theory of relativity.

To each point of such a space we can associate a vector which we usually denote by the tuple of coordinates. Because of this correspondence, there is sometimes a confusion between the notion of a tuple and a vector. A vector in the sense of linear algebra is an element of a vector space. In other words, it belongs to a set on which there is a sensible notion of adding two elements and of the multiplication of elements of this set with scalars. In particular, the result of these operations has to belong to the same set. In the physics literature, there is sometimes the more narrow notion of a vector as an object which has a length and a direction. For the above real spaces, both of these notions are in perfect agreement. If we add two tuples component-wise, we obtain another tuple of the same space and the same is true if we multiply a tuple (also component-wise) by a real number. So these cartesian products are indeed vector spaces and their tuples are naturally used to represent vectors.

However, as one becomes used to this correspondence already at school, it is important to keep in mind that not every tuple is a vector and not every cartesian product is a vector space. In most of the above examples we would have trouble to interpret tuples as vectors. Let us consider another example: For each bank account of a certain bank, we construct a 2-tuple (a,b) where a is the number of the account and b is the amount of money on this account. It makes no sense to take the sum (a1,b1)+(a2,b2)=(a1+a2,b1+b2) of such tuples, or to multiply them with a number k, because a1+a2 or k·a1 will most likely not be the numbers of existing accounts, and even if they are, there is no reason why the amount of money on these accounts should be b1+b2 or k·b1. There is also no meaningful way to assign a lenght or a direction to these tuples. So they certainly do not stand for any vectors. The notion of tuples and cartesian products is very general and only in some cases there is additional structure due to which they stand for vectors and vector spaces.

Where does the word cartesian come from?

The adjective “cartesian” comes from the name Cartesius which is a latinized version of the last name of the philosopher and mathematician René Descartes (1596-1650). Descartes lived at a time when it was still very common in the academic world to communicate in latin. It is one of the curiosities in the history of mathematics that apparently Descartes was not actually the inventor of the cartesian coordinates (as it is mentioned in books by Boyer from 1968 and Boyer and Merzbach from 1989). Pierre de Fermat (1601-1665) had used such coordinates already before Descartes’ famous work on analytic geometry appeared. However, as Fermat had never published his work, the perpendicular coordinate axes were associated with Descartes’ famous name thereafter.

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