

What is The Cartesian Product?
The Cartesian product of two sets A and B, denoted A B, is the set of all ordered pairs (a, b) where an is in A and b is in B in mathematics, specifically set theory. In terms of set-builder notation are as given below,
$ A \times B = {(a,b) | a \in A \; {\text{and}} \; b \in B}$
The Cartesian product of set of rows and a set of columns can be used to make a table. The cells of the table contain ordered pairs of the type if the Cartesian product rows columns are used (row value, column value). The Cartesian product of ‘n’ sets, also called an n-fold Cartesian product, is a similar concept that can be represented by an n-dimensional array with each element being an n-tuple. A 2-tuple or couple is an ordered pair. The Cartesian product of an indexed family of sets can be defined even more generally. The Cartesian product is a development of René Descartes' definition of analytic geometry, which is further generalized in terms of direct product.
For more understanding let’s discuss one cartesian product of sets example,
Let S & R be two sets such that n(S) = 4 and n(R) = 2. If in the Cartesian product we have (a,1), (b,-1), (c,1), (d, -1). Find S and R, where m, n, x, and y are all distinct.
Solution: S = set of first elements = {a, b, c, d} and R = set of second elements = {1, -1}.
Cartesian Product Definition
The Cartesian product A B between two sets A and B is the set of all possible ordered pairs with the first element from A and a second element from B. The cartesian product formula is given below,
$ A \times B = {(a,b) | a \in A \; {\text{and}} \; b \in B}$
The standard Cartesian coordinates of the plane, where A represents the set of points on the x-axis, B represents the set of points on the y-axis, and A × B represents the XY-plane, are an example.
If A = B, we can denote the Cartesian product of A with itself as $ A \times A = A^2$.
For example, since we can represent the x-axis and the y-axis as the set of real numbers (R), then we can write the XY-plane as $ R \times R = R^2$.
What are The Ordered Pairs?
A set of two things with an order associated with them is referred to as an ordered pair. In most cases, ordered pairings are written in parenthesis (as opposed to curly braces, which are used for writing sets). The element ‘p’ is called the first entry or first component of the ordered pair ( p, q ), and the element ‘q’ is called the second entry or second component of the pair.
Two ordered pairs (p, q) and (x, y) are equal if and only if p = x and q = y. In general,
(p, q) ≠ (x, y).
Steps to Find the Cartesian Product
The cartesian product is also called a cross product. Let us consider two non-empty sets say C = {x, y, z} and D = {1, 2, 3}, these two sets can be represented as shown below,
Hence the cross product of C and D can be found by following the steps:
Now take the first element from set C i.e ‘x’ and the first element from set D i.e ‘1’.
These two elements are combined to form an ordered pair (x,1).
Now take first element from C i.e ‘x’ and second element from D i.e ‘2’, hence ordered pair would be (x,2).
This process is repeated until all the possible ordered pairs are formed.
The obtained cartesian product would be sequence of all the ordered pairs.
Cartesian product, CD = {(x,1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3),(z, 1), (z, 2), (z, 3)}
Properties of Cartesian Product
While determining the cross product there are some important properties that are to be followed.
Property 1: The result of cartesian product depends on the order of the pairs, i.e they are non-commutative.
Consider the two sets A and B:
A × B ≠ A × B
A × B = A × B, if and only if A = B.
A × B = ∅, if either A = ∅ or B =∅
Property 2: The rearrangement of the ordered pairs can change the result, hence it doesnot obey associative property. Hence cartesian product is non-associative.
For three sets A, B, and C, (A × B) × C ≠ A × (B × C)
Property 3: The cartesian product is aligned to the distributive property of intersection of the given sets.
For three sets A, B, and C, A × (B ∩ C) = (A × B) ∩ (A × C)
Property 4: The cartesian product is aligned to the distributive property of union of the given sets.
For three sets A, B, and C, A × (B ∪ C) = (A × B) ∪ (A × C)
Cardinality of the Cartesian Product
The total number of elements present in a set is called cardinality of a set. For the set A the cardinal number or cardinality is represented by ‘n(A)’.
Where n(A) = Total number of elements
Whereas the cardinal number of a cartesian product of two sets will be the cross product of cardinal numbers of each set. It can be represented as,
n(A × B) = n(B × A) = n(A) × n(B)
Cartesian Product Solved Example
Example 1: Let A = {1, 2} and B = {1, 2, 3, 4, 5, 6}.
Solution: A × B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6)}
B × A = {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)}
Therefore, in this case, A × B ≠ B × A.
Hence the Cartesian product is not commutative.
Example 2: Given X = {2,3,4,5} and Y = {3,4,5,6}. Find the following sets,
X × Y
Y × X
X2
Y2
Solution:
The given sets are X = {2,3} and Y = {3,4,5,6}.
1. X × Y
By definition, the Cartesian product X × Y contains all feasible ordered pairs ( a, b ) such that a A and b B are the same. As a result, we may write
X × Y = {(2,3), (2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6)}.
2. Y × X
Similarly, by definition, the Cartesian product Y × X contains all feasible ordered pairs (a, b ) such that a A and b B are the same. As a result, we may write
Y × X = {(3,2), (3,3), (4,2), (4,3), (5,2), (5,3), (6,2), (6,3)}.
3. X2
The cartesian square is defined as the X × X, so we can write as,
X × X = {2,3} × {2,3} = {(2,2), (2,3), (3,2), (3,3)}
4. Y2
The cartesian square is defined as the Y × Y, so we can write as,
Y × Y = {3,4,5,6} × {3,4,5,6} = {(3,3), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,3), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}.
Hence it’s solved.
Practise Questions
1. If set A = {3, 5}, then the cardinal number of A × A × A is
4
6
1
8
Ans: Option d
2. If set A = {2, 4, 6}, B = {8, 9} and C = {7, 8}, then find the relation A × (B ∩ C).
{(2,7), (4,7), (6,7)}
{(2,8), (4,8), (6,8)}
{(2,9), (4,9), (6,9)}
None of the above
Ans: Option b
Facts
The given two sets of a cartesian product can be represented in the form of a two-dimensional table. Where the entries will be the elements of each set represented horizontally and vertically.
In Cartesian product, the first element of an ordered pair will always be chosen from the first set of elements.
Conclusion
The Cartesian product is also termed a cross product, and it is usually implemented in the set theory. Hence the final cartesian product will be the set of all the ordered pairs. It is used in daily life to represent the images of a computer, deck of cards, etc.
FAQs on Cartesian product: Introduction, Definition, Formula, and Examples.
1. What is the Cartesian product of two sets in Maths?
The Cartesian product of two non-empty sets, A and B, is the set of all possible ordered pairs (a, b) where the first element 'a' is from set A and the second element 'b' is from set B. It is denoted by A × B. The formula in set-builder notation is: A × B = {(a, b) | a ∈ A and b ∈ B}.
2. How do you find the Cartesian product of two sets? Provide an example.
To find the Cartesian product A × B, you pair every element of set A with every element of set B. For example, if Set A = {1, 2} and Set B = {p, q}:
- Take the first element of A (1) and pair it with each element of B: (1, p), (1, q).
- Take the second element of A (2) and pair it with each element of B: (2, p), (2, q).
The resulting Cartesian product is the set of all these pairs: A × B = {(1, p), (1, q), (2, p), (2, q)}.
3. What is an ordered pair, and why is the order so important in a Cartesian product?
An ordered pair is a pair of elements written in a specific sequence, like (a, b). The order is crucial in a Cartesian product because it indicates which set each element belongs to. The first element always comes from the first set, and the second element from the second set. This means that, in general, the ordered pair (a, b) is not equal to (b, a) unless a = b. This is why the Cartesian product is not commutative.
4. Is the Cartesian product A × B the same as B × A? Explain why or why not.
No, the Cartesian product A × B is generally not the same as B × A. This property is known as being non-commutative. The reason is that the order of elements in the pairs is reversed. For instance, if A = {1} and B = {2}, then A × B = {(1, 2)}, but B × A = {(2, 1)}. Since (1, 2) ≠ (2, 1), the resulting sets are different. A × B is only equal to B × A in the special case where A = B.
5. What is the formula to calculate the number of elements in a Cartesian product?
The number of elements in a Cartesian product, also known as its cardinality, is the product of the number of elements in each individual set. If n(A) is the number of elements in set A and n(B) is the number of elements in set B, the formula is: n(A × B) = n(A) × n(B).
6. How does the Cartesian product relate to the concepts of relations and functions?
The Cartesian product forms the foundation for understanding relations and functions. A relation from a set A to a set B is defined as any subset of their Cartesian product, A × B. A function is an even more specific type of relation where every element in the first set (A) is uniquely paired with exactly one element in the second set (B). Therefore, the Cartesian product can be seen as the universal set of all possible pairings from which relations and functions are derived.
7. Can the Cartesian product be calculated for more than two sets?
Yes, the Cartesian product can be extended to three or more sets. For three sets A, B, and C, the Cartesian product A × B × C is the set of all possible ordered triplets (a, b, c), where a ∈ A, b ∈ B, and c ∈ C. For example, if A = {0, 1}, B = {x, y}, and C = {m}, then A × B × C = {(0, x, m), (0, y, m), (1, x, m), (1, y, m)}.
8. What happens to the Cartesian product if one of the sets is an empty set (∅)?
If either set A or set B (or both) is an empty set (∅), their Cartesian product A × B will also be an empty set. This is because there are no elements in the empty set to form any ordered pairs. So, if A = ∅ or B = ∅, then A × B = ∅.
9. What are some real-world applications or examples of the Cartesian product?
The Cartesian product is a fundamental concept used in various fields:
- Coordinate Geometry: The Cartesian coordinate plane is the Cartesian product of the set of real numbers with itself (ℝ × ℝ), where every point is an ordered pair (x, y).
- Databases: In SQL, a CROSS JOIN operation performs a Cartesian product on two tables, creating all possible combinations of rows.
- Computing: It is used in computer graphics to define pixel coordinates on a screen and in programming to generate all possible combinations of different parameters.
- Daily Life: A deck of 52 playing cards can be described as the Cartesian product of the set of ranks {A, 2, ..., K} and the set of suits {♠, ♥, ♦, ♣}.





