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Cartesian product
Sunil Kumar Singh
This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †
We have seen that set operations convey the notion of arithmetic operations. One such similar operation is product of two sets called Cartesian product. Since sets are collection not a single quantity, the product operation here involves combining or pairing each of the elements of one set with that of another set. We use symbol X to denote product operation. The Cartesian product of two sets A and B is symbolically represented as :
A × B It is important to understand that we do not multiply elements as we do in arithmetic instead we pair elements together. This is the meaning of product for the sets. We denote one such pair within a pair of small brackets like :
(a, b) where a ∈ A and b ∈ B. Note that elements from two sets are separated by comma.
1 Ordered pair
The order of pairing is important. The pair (a,b) and (b,a) are dierent. This ordering is required as there are real time situations, where order makes a dierence. Consider for example, we are required to nd the integers which can be formed from two integer subsets like {1,2,3} and {3,4,5}. Clearly, 13 and 31 represent dierent integers. We need to distinguish them. All pairs formed from two sets should be distinct. Keeping this restriction in mind, let us work out an example to nd ordered pairs formed from elements of two sets.
A = set of rst letter of the names of cities = {N, D, H}
B = set of numbers denoting ight numbers = { 001 , 002 , 003 } All possible ordered pairs formed from two sets are :
(N, 001) , (N, 002) , (N, 003) , (D, 001) , (D, 002) , (D, 003) , (H, 001) , (H, 002) , (H, 003) There are all together 9 ordered pairs. From this example, we can deduce a method for writing ordered pairs from two sets. We begin with the rst elements of two sets. Progressively, we change the elements from the second set till it is exhausted, while keeping the elements from the rst set unchanged. Then, we
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switch to next element from rst set and start with rst from the second set. Again, we change the elements from the second set progressively till it is exhausted, while keeping the elements from the rst set unchanged. We continue in this manner till all elements from the rst set is also exhausted. From this discussion, it is also evident that two ordered pairs are equal if and only if the corresponding rst and second elements are equal.
2 Cartesian product
The Cartesian product of two sets is de ned in terms of ordered pairs.
De nition 1: Cartesian product The Cartesian product of two non-empty sets A and B is the set of all ordered pairs of the elements from two sets. We should emphasize the use of word non-empty, The Cartesian product of a non-empty set with an empty set is equal to empty set.
A × φ = φ On the other hand, if one of the sets is in nite, then resulting Cartesian product is also in nite. We express the Cartesian product set in set building form as :
A × B = {(x, y) : x ∈ A, y ∈ B} Here, use of "," in the set builder form is equivalent to "and". Therefore, we can write Cartesian product of two sets also as :
A × B = {(x, y) : x ∈ A and y ∈ B} Further, we can emphasize two ways validity of the conditional statements as in the case of other set operators :
If (x, y) ∈ A × B ⇔ x ∈ A and y ∈ B
2.1 Graphical representation
The ordered pairs can be represented in the form of tabular cells or points of intersection of perpendicular lines. The elements of one set are represented as rows, whereas elements of other set are represented as columns. Look at the representation of ordered pairs by points in the gure for the example given earlier.
(A ∩ B) × (B ∩ C) = {(6, 1) , (5, 1)}
Note that the elements in the given set are not ordered. It is purposely given this way to emphasize that order is requirement of ordered pair not that of a set.
2.3 Numbers of elements
We have seen that ordered pairs are represented graphically by the points of intersection. The numbers of intersections equal to the product of numbers of rows and columns. Thus, if there are p elements in the set A and q elements in the set B, then total numbers of ordered pairs are pq. In symbolic notation,
n (A × B) = pq
3 Multiple products
Like other set operations, the product operation can also be applied to a series of sets in sequence. If A 1 ,A 2 ,...... .., An is a nite family of sets, then their Cartesian product, one after another, is symbolically represented as :
A 1 × A 2 ×................ × An This product is set of group of ordered elements. Each group of ordered elements comprises of n elements. This is stated as :
A 1 × A 2 ×... × An = {(x 1 ,x 2 ,... , xn) : x 1 ∈ A 1 ,x 2 ∈ A 2 ,... , xn ∈ An}
3.1 Ordered triplets
The Cartesian product A × A × A is set of triplets. This product is de ned as :
A × A × A = {(x, y, z) : x, y, z ∈ A} We can also represent Cartesian product of a given set with itself in terms of Cartesian power. In general,
⇒ An^ = A × A ×...... × A where n is the Cartesian power. If n = 2, then
⇒ A^2 = A × A This Cartesian product is also called Cartesian square.
3.1.1 Example
Problem 3 : If A = {-1,1}, then nd Cartesian cube of set A. Solution : Following the method of writing ordered sequence of numbers, the product can be written as :
A × A × A = {(− 1 , − 1 , −1) , (− 1 , − 1 , 1) , (− 1 , 1 , −1) ,
The total numbers of elements are 2x2x2 = 8.
3.2 Cartesian Coordinate system
The Cartesian product, consisting of ordered triplets of real numbers, represents Cartesian three dimensional space.
R × R × R = {(x, y, z) : x, y, z ∈ R} Each of the elements in the ordered triplet is a coordinate along an axis and each ordered triplet denotes a point in three dimensional coordinate space.
Cartesian coordinate system
Figure 2: The coordinate of a point is an ordered tripplet.
Similarly, the Cartesian product " R × R " consisting of ordered pairs de nes a Cartesian plane or Cartesian coordinates of two dimensions. It is for this reason that we call three dimensional rectangular coordinate system as Cartesian coordinate system.
4 Commutative property of Cartesian product
The Cartesian product is set of ordered pair. Now, the order of elements in the ordered pair depends on the position of sets across product sign. If sets "A" and "B" are unequal and non-empty sets, then :
A × B 6 = B × A In general, any operation involving Cartesian product that changes the "order" in the "ordered pair" will yield dierent result.
2: For distribution over intersection operator
⇒ LHS = A × (B ∩ C) = {a, b} × { 2 }
⇒ LHS = {(a, 2) , (b, 2)} Similarly,
⇒ RHS = (A × B) ∩ (A × C) = {(a, 1) , (a, 2) , (b, 1) , (b, 2)} ∩ {(a, 2) , (a, 3) , (b, 2) , (b, 3)}
⇒ RHS = {(a, 2) , (b, 2)} Hence,
⇒ A × (B ∩ C) = (A × B) ∩ (A × C) 3: For distribution over dierence operator
⇒ LHS = A × (B − C) = {a, b} × { 1 }
⇒ LHS = {(a, 1) , (b, 1)} Similarly,
⇒ RHS = (A × B) − (A × C) = {(a, 1) , (a, 2) , (b, 1) , (b, 2)} − {(a, 2) , (a, 3) , (b, 2) , (b, 3)}
⇒ RHS = {(a, 1) , (b, 1)} Hence,
⇒ A × (B − C) = (A × B) − (A × C)
5.1 Analytical proof
Let us consider an arbitrary ordered pair (x,y), which belongs to Cartesian product set A × (B ∪ C) . Then,
⇒ (x, y) ∈ A × (B ∪ C) By the de nition of product of two sets,
⇒ x ∈ A and y ∈ (B ∪ C) By the de nition of union of two sets,
⇒ x ∈ A and (y ∈ B or y ∈ C)
⇒ (x ∈ A and y ∈ B) or (x ∈ A and y ∈ C)
⇒ (x, y) ∈ A × B or (x, y) ∈ A × C By the de nition of union of two sets,
⇒ (x, y) ∈ (A × B) ∪ (A × C) But, we had started with " A × (B ∪ C) " and used de nitions to show that ordered pair (x,y) belongs to another set. It means that the other set consists of the elements of the rst set at the least. Thus,
⇒ A × (B ∪ C) ⊂ (A × B) ∪ (A × C) Similarly, we can start with " (A × B) ∪ (A × C) " and reach the conclusion that :
⇒ (A × B) ∪ (A × C) ⊂ A × (B ∪ C) If sets are subsets of each other, then they are equal. Hence,
⇒ A × (B ∪ C) = (A × B) ∪ (A × C) Proceeding in the same manner, we can also prove distribution of product operator over intersection and dierence operators,
A × (B ∩ C) = (A × B) ∩ (A × C)
A × (B − C) = (A × B) − (A × C)
