Cartesian Product

The cartesian product of two or more sets is the set of all ordered pairs/n-tuples of the sets. It is most commonly implemented in set theory. In addition to this, many real-life objects can be represented by using cartesian products such as a deck of cards, chess boards, computer images, etc. Most of the digital images displayed by computers are represented as pixels which are graphical representations of cartesian products.

In this article, let's learn about the cartesian product, its properties, and the product of sets with solved examples for a better understanding. We will discuss the cartesian product of two or more sets and relations.

Cartesian product is the product of any two sets, but this product is actually ordered i.e, the resultant set contains all possible and ordered pairs such that the first element of the pair belongs to the first set and the second element belongs to the second set. Since their order of appearance is important, we call them first and second elements, respectively. We use ordered pairs to obtain a new set from two given sets A and B.

• An ordered pair (p, q) consists of two values p and q. Example: (1, 3) and (- 4, 10) are ordered pairs where these pairs of numbers are in a specific order.
• Consequently, (p, q) ≠ (q,p) unless p = q. In general, (p, q) = (s, t) if and only if p = s and q = t. Example: (1, 3) is not equivalent to (3, 1) i.e., (1, 3) ≠ (3, 1).

An ordered pair is a pair of numbers in a specific order. For example, (1, 2) and (- 4, 12) are ordered pairs. The order of the two numbers is important: (1, 2) is not equivalent to (2, 1) -- (1, 2)≠(2, 1).

Cartesian Product Definition

If C and D are two non-empty sets, then the cartesian product, C × D is the set of all ordered pairs (a, b) with the first element from C and the second element from D. Similar to the other product operations, we use the same multiplication sign × to represent the cartesian product between two sets. Here, we use the notation C × D for the Cartesian product of C and D.

By using the set-builder notation, we can write the cartesian product as:

C × D = {(a,b): a ∈ C, b ∈ D}. Here a belongs to set C and b belongs to set D.

If both the sets are the same i.e, if C = D then C × D is called the cartesian square of the set C and it is denoted by C²

C² = C × C = {(a,b): a ∈ C, b ∈ C}

The cartesian products of sets can be considered as the product of two non-empty sets in an ordered way. The final product of the sets will be a collection of all ordered pairs obtained by the product of the two non-empty sets. In an ordered pair, two elements are taken from each of the two sets.

Finding Cartesian Product

Consider two non-empty sets C = {x, y, z} and D = {1, 2, 3} as shown in the below image:

The cartesian product, also known as the cross-product or the product set of C and D is obtained by following the below-mentioned steps:

• The first element x is taken from the set C {x, y, z} and the second element 1 is taken from the second set D {1, 2, 3}
• Both these elements are multiplied to form the first ordered pair (x,1)
• The same step is repeated for all the other pairs too until all the possible combinations are chosen
• The entire collection of all such ordered pairs gives us a cartesian product C x D = {(x,1), (x,2), (x,3), (y,1), (y,2), (y,3),(z,1), (z,2), (z,3)}.
• Similarly, we can find the cartesian product of D x C.

Let us find the cartesian product of the two sets C and D, where C = {11,12,13} and D = {7, 8}. After following the steps mentioned above:

• The resultant product C x D will be: {(11,7), (11,8),(12,7),(12,8),(13,7),(13,8)}.
• Similarly, we can find the cartesian product of D and C as D × C = { (7,11),(7,12),(7,13),(8,11),(8,12),(8,13)}.
• The cartesian products C × D and D × C do not contain exactly the same ordered pairs. Hence, in general, C × D ≠ D × C.

Cartesian Product of Several Sets

We can extend or define the cartesian product to more than two sets. The cartesian product of several input sets is a larger set that contains every ordered combinations of all the input set elements. The cartesian product of three sets P, Q, and R can be written as:

P × Q × R = { (a,b,c): a ∈ P, b ∈ Q, c ∈ R }

Let us consider the example of three sets A, B and C, where A = {2,3} , B = {x,y}, and C = {5,6}. In order to find the cartesian product of A × B × C, let us find the cartesian product of A × B first.

A × B = {(2,x), (2,y),(3,x),(3,y)}.

A × B × C = {(2,x,5), (2,x,6), (2,y,5), (2,y,6), (3,x,5), (3,x,6), (3,y,5), (3,y,6)}

The empty set is a unique set with no elements. Both its size or cardinality i.e, the total count of elements in a set will remain zero. An empty set is also referred to as a void set. The Cartesian product of C and the empty set ∅ is the empty set ∅. Let C × ∅ = {(a,b)| a ∈ C, b∈ ∅}. There is no element in ∅. C × D =∅ if and only if C = ∅ or D = ∅. Here, the cartesian product of two sets will result in an empty set if and only if, either of the sets is an empty set.

Consider the example: If C = {1, 2} and D = ϕ. Then, C × D = ϕ and D × C = ϕ.

These are the properties of the empty set:

• Empty set's subset is the empty set itself: ∀ C:C ⊆ ∅ ⇒ C = ∅
• The empty set's power set is the set containing only the empty set: 2ⁿ = 2⁰ = 1.
• The cardinality of the empty set i.e., the number of elements of the set is zero: n(∅) = 0

Cartesian Product of Countable Sets

The cartesian product of two countable sets is countable. Let us take these two cases to understand this:

• Consider an integer b in such a way that b > 1. Then the cartesian product of b countable sets is countable.
• Consider the two countable sets A = {a₀, a₁, a₂, ...} and B = {b₀, b₁, b₂, ...}. If both the sets A and B are countable, then the resulting set will also be coutable.

The cartesian product of relations is the same as the relation across two sets. Generally, the cartesian product is represented for a set and not for a relation. Further, the universal relation relates every element of one set to an element of another set, and hence it can be represented as the cartesian product of relations.

Here are some important properties that are to be followed while determining the cartesian product. The properties are as follows:

Cardinality of a Cartesian Product

The cardinality of a set is the total number of elements present in the set. The cardinal number of A is n(A) = number of all the elements in set A. Example: The cardinal number of a set of English alphabets A = (a, b, c .....x. y. z) is n(A) = 26. The сardinality of a cartesian product of two sets C and D is equal to the product of the cardinalities of these two sets: n(C × D) = n(D × C) = n(C) × n(D). Similarly, n(C₁×…× C_n) = n(C₁) ×…× n(C_n).

Consider two sets C and D, where C = {2,3} and n(C) = 2, D = {5,4,7} and n(D) = 3. So, n(C × D) = n(C) × n(D) = 2 × 3 = 6. Here, we can see that the cardinality of the output set C × D is equal to the product of the cardinalities of all the input sets C and D. That is, 6.

Related Articles

• Cartesian Coordinates
• Operations on Sets
• Venn Diagrams
• Roster Notation
• Universal Set
• Intersection of Sets

Important Notes on Cartesian Product

• Sometimes, ordered pairs are also referred to as 2−tuples.
• The cartesian product of two sets C and D is also known as the cross-product or the product set of C and D
• The final cartesian product of two sets will be a collection of all ordered pairs obtained by the product of these two non-empty sets.

What Is Cartesian Product of Sets?

The cartesian products of sets can be considered as the product of two non-empty sets in an ordered way. The final product of the sets will be a collection of all ordered pairs obtained by the product of the two non-empty sets. In an ordered pair, two elements are taken from each of the two sets. Example: Cartesian product of two sets A = {2,3} and B = {x,y} will be: A × B = {(2,x), (2,y),(3,x),(3,y)}.

What Is a Cartesian Product Used For?

Cartesian product finds its uses in many real-life objects such as a deck of cards, chess boards, computer images, etc. The digital images that we see on our computers and mobile phones are represented as pixels which are graphical representations of cartesian products. In mathematics, the cartesian product is most commonly implemented in set theory.

How To Find Cartesian Product of Three Sets?

Let's find the cartesian product of three sets by using an example. Consider three sets C, D, and E where C = {1,2}, D = {0,1}, and E = {0,4}.

• First find the cartesian product of C and D, C × D = {(1,0),(1,1),(2,0),(2,1)}.
• Now, let's find the product of (C × D) and E, ((C × D) × E) = {((1,0),0),((1,1),0),((2,0),0),((2,1),0),((1,0),4),((1,1),4),((2,0),4),((2,1),4)}.
• We can now remove the parenthesis in the first ordered pair c x D x E = {(1,0,0),(1,1,0),(2,0,0),(2,1,0),(1,0,4),(1,1,4),(2,0,4),(2,1,4)}

Is Cartesian Product Commutative?

No, the cartesian product is not commutative. As per the properties of the cartesian product, the result depends on the order of the sets. Consider the two sets C and D: C × D ≠ D × C. But, C × D = D × C, if and only if C = D. If C = {11,12,13} and D = {7, 8}, then:

C x D will be: {(11,7), (11,8),(12,7),(12,8),(13,7),(13,8)}. D × C = { (7,11),(7,12),(7,13),(8,11),(8,12),(8,13)}.

What Is The Cartesian Product Of Relations?

The cartesian product of relations does not exists because the relations itself is an ordered pair, which relates the elements of one set with the elements of another set. Relations in maths is another form of connecting the elements of two sets, similar to the cartesian product of sets.

Does Order Matter in Cartesian Product?

Yes, the order in which the sets are multiplied in a cartesian product matters as the cartesian product is not commutative. The cartesian product does not satisfy the commutative property. Two sets A and B are such that, the cartesian product A × B will not be equal to the cartesian product B × A.
A x B ≠ B x A

What Is the Cartesian Product of Two Empty Sets?

The cartesian product of two empty sets will also be an empty set. As per the properties of the cartesian product, consider two sets A and D, such that A × D = ∅ if either A = ∅ or D =∅. Also, if both the sets are empty sets, then the resulting cartesian product will also be empty.

How Do You Find the Cardinality of a Cartesian Product?

The cardinality of a set is the total number of elements in the set. The сardinality of a cartesian product of two sets C and D is equal to the product of the cardinalities of these two sets: n(C × D) = n(D × C) = n(C) × n(D). Consider two sets A = {2,5} and C = {4,1}. The cardinality of A and C are 2 and 2. Hence, the cardinality of their product will be n(A × C) = n(A) × n(C) = 2 × 2 = 4.

Is Cartesian Product an Equivalence Relation?

A cartesian product is an equivalence relation if and only if the cartesian product is a product of a set with itself. An equivalence relation is a relation that is reflexive, symmetric, and transitive.