Cartesian Product of Sets

The cartesian product of two or more sets is a set of all ordered pairs in which the first element comes from the first set and the next from the second.  

Symbol

If A and B are two sets, their cartesian product is written as A × B. It is also called the cross-product.

Formula

In the set builder notation, A  × B is written as: 

A × B = {(a,b) | a ∈ A, b ∈ B}

Here, a belongs to set A, and b belongs to set B. 

For Two Sets

If A = {p, q, r} and B = {7, 8, 9} are two non-empty sets, then their cartesian product is:

A × B = {(p, 7), (p, 8), (p, 9), (q, 7), (q, 8), (q, 9)}

Similarly, we can find the cartesian product of B × A.

B × A = {(7, p), (7, q), (7, r), (8, p), (8, q), (8, r), (9, p), (9, q), (9, r)}

Thus, in general, A × B ≠ B × A

Set With Itself

If both sets are the same, A = B, then A × B is called the cartesian square of set A, and it is written as A²

A² = A × A = {(a,b): a ∈ A, b ∈ A}

Here, we also conclude that if A = B, then A × B = B × A

Empty Set

The cartesian product of any set and an empty set produces the empty set. If A is a set and ɸ is an empty set, then A × ɸ = {(a, b) | a ∈ A, b ∈ ɸ}

In general, the cartesian product of two sets results in an empty set if and only if either of the sets is null.

Symbolically, if A = ɸ or B = ɸ, then A × B = ɸ

For Three Sets

The cartesian product can be defined and extended to more than two sets. The cartesian product of more than two sets is a larger set containing every ordered pair of all the given set elements. 

In the set builder notation, A × B × C is written as: 

A × B × C = {(a,b,c) | a ∈ A, b ∈ B, c ∈ C}

Example

If A = {5, 6}, B = {s, t}, and C = {g, h}, to find the cartesian product of A × B × C, we first find the cartesian product of A × B followed by the (A × B) × C

A × B = {(5, s), (5, t), (6, s), (6, t)}

Now, A × B × C = {(5, s), (5, t), (6, s), (6, t)} × {g, h}

⇒ A × B × C = {(5, s, g), (5, s, h), (5, t, g), (5, t, h), (6, s, g), (6, s, h), (6, t, g), (6, t, h)}

Properties

Commutative

The cartesian product depends on the order of sets, and it is thus non-commutative.

If A and B are two sets, then A × B ≠ B × A

Associative

The cartesian product is not associative. If A, B, and C are three sets, then (A × B) × C ≠ A × (B × C)

Distributive

Intersection of Sets

If A, B, and C are three sets, then according to the distributive property of the intersection of sets  A × (B ∩ C) = (A × B) ∩ (A × C)

Union of Sets

If A, B, and C are three sets, then according to the distributive property of the union of sets A × (B ∪ C) = (A × B) ∪ (A × C)

Cardinality

The cardinality of the cartesian product of sets A and B is the total number of ordered pairs in A × B. 

If x is the number of elements in set A, and y is the number of elements in set B, then the cardinality of the Cartesian product of sets A and B is xy.

Mathematically, it is written as if n(A) = x, n(B) = y, then n(A × B) = n(A) × n(B) = xy.

Similarly, for n number of sets, n(A₁×…× A_n) = n(A₁) × … × n(A_n)

Solved Examples