Cartesian Product of Sets: Definition, Formula and Properties

A cartesian product is a set of all ordered pairs and tuplets. It is most commonly implemented in set theory. Various real-life examples can be used to explain the cartesian product of sets like chess boards, computer images, and decks of cards. The digital images displayed by the computers are represented as pixels, the graphical representation of cartesian products.

While plotting a graph paper one must note the axes like the x-axis and y-axis. Example (7,6) depicts the value on the x-axis as 7 and that on the y-axis as 6. This is not the same as (6,7). The value of the x-coordinate always comes before that of the y-coordinate. A Cartesian product is the product of x and y in an ordered fashion.

Read more: Coordinate geometry

KeyTerms: Cartesian products, Non-empty sets, Coordinates, Ordered pairs, Distributive properties, Equivalence sets

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Cartesian Product of Sets Definition

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The product of two non-empty sets in an ordered fashion is called the cartesian product of sets. The product of two non-empty sets gives a collection of all ordered pairs and it is the final product of the sets. Two elements are taken from each set for an ordered pair.

To understand better the cartesian product of sets, here is an example:

• Let A and B be the two sets such that A is a set of three colors of curtains and B is a set of three colors of the sofa, i.e.

A = {brown, green, yellow}

B = {red, blue, purple}

• The number of pairs of colored objects that we can make from a set of curtains and sofa in different combinations can be paired as given below:

(brown, red), (brown, blue), (brown, purple), (green, red), (green, blue), (green, purple), (yellow, red), (yellow, blue), (yellow, purple)

• The Cartesian product has nine such pairs since each of the defined sets A and B contains three members.The above-ordered pairs represent the definition for the Cartesian product of the sets given. This product is denoted by “A × B”.

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Cartesian Product of Sets Formulas

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• Given: Two non-empty sets P and Q. The set of all ordered pairings of elements from P and Q make up the Cartesian product P x Q, i.e.,

P × Q = {(p,q): p ∈ P, q ∈ Q}

• If either P or Q is the null set, then P × Q will also be an empty set, i.e., P × Q = φ.

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Finding Cartesian Product

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• Consider two non-empty sets C = {x, y, z} and D = {1, 2, 3} 
• The cartesian product, also known as the cross-product or the product set of P and Q is obtained as explained:
• The first element x is taken from the set P {x, y, z} and the second element 1 is taken from the second set Q {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 P x Q = {(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 Q x P.

Finding Cartesian Product

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Cartesian Product of Two sets

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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 Two sets

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Cartesian Products of Three Sets

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Suppose P is a non-empty set and the Cartesian product P × P × P represents the set P × P × P ={(x, y, z): x, y, z ∈ P} which means the coordinates of all the points in three-dimensional space. This forms the basis for the Cartesian product of three sets. 

Example: Find the Cartesian product of three sets P = {p, q}, Q = {1, 2} and R = {x, y}.

Solution: Given,

P = {p, q}

Q = {1, 2}

R = {x, y}

The ordered pairs of P × Q × R can be formed as given below:

1st pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (p, 1, x)

2nd pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (p, 1, y)

3rd pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (p, 2, x)

4th pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (p, 2, y)

5th pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (q, 1, x)

6th pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (q, 1, y)

7th pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (q, 2, x)

8th pair ⇒ {p, q} × {1, 2} × {x, y} ⇒ (q, 2, y)

Thus, the ordered pairs of P × Q × R can be written as:

P × Q × R = {(p, 1, x), (p, 1, y), (p, 2, x), (p, 2, y), (q, 1, x), (q, 1, y), (q, 2, x), (q, 2, y)}

Cartesian Product of Three sets

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Properties of Cartesian Product

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Things to Remember

• The set of all ordered pairs and tuplets is called a cartesian product. 
• It is the product of x and y in an ordered fashion.
• Ordered pairs are also referred to as 2−tuples.
• The cartesian product of two sets C and D are 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.
• Cartesian products are non-commutative and non-associative.
• Cartesian products have distributive properties over the intersection of sets and over the union of sets.

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