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Cartesian Product

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Last updated date: 27th Feb 2024
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What is a 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 x B = {(a, b)|a ∈ A and b ∈ B}

The Cartesian product of a 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 generalised 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 x 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 x B = {(a, b)|a ∈ A and b ∈ 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 x A = A2

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 x R = R2.


Cartesian Product Example

Example 1:  Let A = {1, 2} and B = {1, 2, 3, 4, 5, 6}.

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,

  1. X × Y

  2. Y × X

  3. X2

  4. 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.


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).

FAQs on Cartesian Product

1. What is the Cartesian Product of Sets?

Answer: The Cartesian product of sets is defined as the ordered product of two non-empty sets. Or, to put it another way, the set of all ordered pairs is obtained by multiplying two non-empty sets. An ordered pair is when two elements from each set are selected.

2. What is the Cartesian Product Meaning?

Answer: The Cartesian product of two sets P and Q, denoted by P × Q, is defined as the set consisting of all ordered pairs (a, b) for which a ∊ P and b ∊ Q.

3. Who Invented the Cartesian Product?

Answer: The Cartesian product was created by René Descartes. Its name is derived from the same person. René proposed analytic geometry, which helped in the creation of this concept, which we broaden in terms of direct product.

4. What is the Cartesian Product Used For?

Answer: In computing, a Cartesian product is now almost similar to a Cartesian product in mathematics. It will be useful in matrix applications. In SQL, it describes a fault in which you join two tables improperly and end up with numerous records from one table connected to each of the entries from the other, rather than the expected one.

5. What is a Cartesian Product in the Database?

Answer: The Cartesian product, commonly known as a cross-join, retrieves all results from all of the tables in the query. Every row in the first table corresponds to every row in the second table. A Cartesian join can be used to generate a large number of rows for performance reasons.