Properties of Cartesian Product of Sets

The Cartesian product of sets is the group of all the possible ordered pairs of sets in which the product of sets of elements is obtained in an ordered way. The properties of the Cartesian product of sets help to understand the product of two or more sets. There are different properties that need to be followed while determining the Cartesian product of sets.

Some important properties of the Cartesian product of sets are non-associative property, distributive property of intersection, union and difference, subset property, empty set property, etc. The non-commutative property of Cartesian products describes the significance of the order of sets.

Non-Commutative Property

The non-commutative property of a Cartesian product is an essential property, which describes the significance of the order of sets. According to this property, the Cartesian product of two sets is not equal to the Cartesian product of the same sets when the order of Cartesian is interchanged.

To understand it better, let’s take an example. If A and B are the two non-empty sets. Then, the Cartesian product of A and B is not equal to the Cartesian product of B and A. The following expression represents the non-commutative property for sets A and B.

A × B ≠ B × A

For the non-commutative property of the Cartesian product, all sets must be non-empty sets. If any of the sets is non-empty in the above case, then this proposition will no longer be valid.

Null Set Property

This property is similar to the non-commutative property. According to the null set property, if one of the sets is an empty set, then, the Cartesian product of these sets is always an empty set. It can be expressed in the following way.

A × Φ = Φ

For example, A is a non-empty set, and B is an empty set. Then, the Cartesian product of sets A and B is also an empty set or null set.

A × B = Φ

Non-Associative Property

In the Cartesian product of the non-empty sets, the binary multiplication operation is not presumed to be associative unless one of the involved sets is a null set.

If three sets, A, B, and C, are non-empty sets, the binary Cartesian product of two sets is not assumed to be associative by this property. The following expression represents the non-associative property of the Cartesian product for the three sets, A, B, and C.

(A ×B ) × C ≠ A × (B × A)

Distributive Property

This property can be expressed in terms of the intersection, union, and difference of the sets.

Distributive Property of Intersection

The Cartesian product of a non-empty set and the intersection of another two non-empty sets is equal to the Cartesian product of the first and second set and the first and third set.

Suppose A, B, and C are the three non-empty sets. Then, by this property, the Cartesian product of set A and the intersection of set B and set C is equal to the intersection of the Cartesian product of set A and B and the Cartesian product of A and C. The following expression represents the distributive property of intersection of Cartesian products for three sets, A, B, and C.

A × (B ∩ C ) = (A × B) ∩ (A × C)

Distributive Property of Union

Distributive property of sets is used for the operation of union. The Cartesian product of a non-empty set and union of another two non-empty sets is equal to the Cartesian product of the first and second set and the first and third set.

Let A, B, and C are the 3 non-empty sets. Then, according to this property, the Cartesian product of set A and the union of set B and set C is equal to the union of the Cartesian product of set A and B and the Cartesian product of set A and C. The following expression represents the distributive property of union of Cartesian products for three sets A, B, and C.

A × (B ∪ C ) = (A × B) ∪ (A × C)

Distributive Property of Difference

The Cartesian product of sets in which one set represents the subtraction of two sets is equal to the Cartesian product of difference of the first set with other two sets of the subtraction operation.

Let’s say sets A, B and C are non-empty sets. With the use of the distributive property over set difference the Cartesian product of set A and the difference of set B and set C is equal to the difference of the Cartesian product of set A and B and the Cartesian product of A and C. The following expression represents the distributive property over set difference of Cartesian product for three sets A, B and C.

A × (B-C )=(A × B)-(A × C)

Subset Property

If there is a set from another set, which consists of some elements of the first set, then the second set is called the subset of the first set. The subset property of the Cartesian product gives the relation as a part between the two Cartesian products of these three sets.

Let us suppose, there is a set B which is a subset of set A. For this case, the Cartesian product of two sets, A and another set, say C, is a subset of the Cartesian product of two sets, Set B and Set C. The expression for subset property is given below.

A × C ⊆ B × C

Importance of Cartesian Product Theorems

The Cartesian product of sets is the collection of all the possible ordered pairs available for the sets. There are so many theorems that are helpful while estimating the Cartesian product of different sets. The reasons why the theorems of Cartesian product are important are:

• The Cartesian product theorems of sets used to estimate the result of complex set theory problems.
• The Cartesian product theorems explain and prove the various properties of the Cartesian product. 
• These theorems also help to understand the importance of order of sets in the Cartesian product.

Question Based on Cartesian Product Theorems

• Prove the following theorem for the Cartesian product.

A × B  ≠ B × A

Solution:

Let the set A{3,2} and B{4,5}. The Cartesian product of set A and set B can be given below.

A × B ={(3,4),(3,5),(2,4),(2,5)}

The Cartesian product of sets B and A can be given below.

B × A ={(4,3),(4,2),(5,3),(5,2)}

Here, {(3,4),(3,5),(2,4),(2,5)}{(4,3),(4,2),(5,3),(5,2)}

Therefore, A × B  ≠ B × A,

•  Let A {1,3}, B{2,4}, and C{4,6}. Find

A×(B ∩C)

Solution:

The common element in set B and set C is 4.

(B ∩C)={4} 

Therefore, A × (B ∩C)={(1,4)(3,4)}.

Conclusion

The properties of the Cartesian product are the characteristics of the product of two sets. In this article, we learned that the Cartesian product of sets is non-commutative and non-associative. There are various properties of the Cartesian product which should be kept in mind while estimating the Cartesian product of sets.