Cartesian Product and Ordered Pairs

Cartesian Product:

Cartesian product comprises of two words – Cartesian and product. The word Cartesian is named after the French mathematician and philosopher René Descartes (1596-1650) Cartesian product of two non-empty sets A and B is denoted by \( A × B \) and defined as the collection of all the ordered pairs (a,b) such that \( a \in A \) and \( b \in B \) .

\(~~~~~~~~~~~~~~~\) \( A × B \) = { \( ( a,b):a \in A, b\in B \) }

It is also called the cross product, set direct product or the product set of A and B . One very important thing to note here is that it is the collection of ordered pairs. By ordered pair, it is meant that two elements taken from each set are written in particular order. So, if a ≠ b , ordered pairs (a,b) and (b,a) are distinct.

To take an example, let us take P as the set of grades in a school from set Q as the sections for the grades. So, we have P and Q as:

\(~~~~~~~~~~~~~~~~~~~\) P = {8,9,10}

\(~~~~~~~~~~~~~~~~~~~\) Q = {A,B,C,D}

So,\( P × Q \) , according to the definition will be equal to,

\( P × Q \) = { (8,A) , (8,B) , (8,C) , (8,D) ,(9,A) , (9,B) ,(9,C),(9,D),(10,A),(10,B),(10,c),(10,D)}

There are total of 12 ordered pairs. If n(P) and n(Q) represent number of elements in the sets P and Q respectively, then n(P) = 3 and n(Q) =4. So, n(P×Q) = 3 × 4 = 12. Refer figure 1 for the depiction of the same. In the figure, we can clearly observe how \( P × Q \) forms a plane, also referred to be as Cartesian plane. Each point represents an ordered pair which has first element from set P and second element from set Q. If number of elements in set A and B is p and q respectively, then number of elements in the Cartesian product of sets will be pq i.e.

If n(A) = p and n(B) = q and , then n( A × B) = pq.

From this property, we can draw two conclusions:

  • When one or both the sets are empty, A × B = \( \phi\) .
  • If anyone of the sets are infinite, even A × B is an infinite set.

Figure 1: Depiction of all possible ordered pairs for \( P × Q \)

For two ordered pairs to be equal, their corresponding elements must be equal. E.g. If ordered pairs (9,13) and (x+3 , y+6) are equal,

\(~~~~~~~~~~~~~~~~~~~~\) x + 3 = 9 ⇒ x = 6

\(~~~~~~~~~~~~~~~~~~~~\) y + 6 = 13 ⇒ y = 7

Cartesian product of sets is not limited to only two sets. It also holds for more than two sets. But the complexity increases as we increase the number of sets. For three sets A, B and C, an element of A ×B × C is represented as (a,b,c) and it is called an ordered triplet. If we take Cartesian product of two sets as , R × R where R is the set of real numbers, that represents the entire two dimensional Cartesian plane. Similarly, R × R × R represents three dimensional Cartesian space.

It is interesting to know what is Cartesian product and what are ordered pairs. But what is even more interesting is how Descartes got this idea. He was lying on his bed when he saw a fly. After a lot of buzzing from the fly, he noticed something very simple yet outstanding. He could mark the position of the fly using three parameters, distance from the two adjacent walls and distance from the floor. And each time the fly moved, there was a new set of values for the new position. This gave Descartes an idea and he invented Coordinate systems.