Cartesian Product of Sets

Mathematicians use the term ‘product’ to describe a result obtained from multiplying two or more values. Using our example, 45 is obtained by multiplying 9 and 5. You should be familiar with methods for composing sets of two or more elements, such as union and intersection. A third example is a Cartesian product, which yields an ordered pair by joining two sets. Using ordered sets as a reference, Cartesian products can be computed. There is an unlimited number of combinations that contain a member from each of those sets. It is necessary to order two elements, such as x and y, in order to create cartesian products. 

What is an Ordered Pair?

An ordered pair is a pair of objects in which one element is assigned first, while the second element is assigned second; the pair is represented by (a,b). This ordered set has two components, a and b. Therefore a is the first component, and b is the second component.

Define Cartesian Product of Sets

In a cartesian product, the first component is from A, and the second component is from B. One would then get the set of all possible ordered pairs where the first component is from A and the second component is from B. In this instance, we have a set of ordered pairs denoted by A×B.

A × B = {(a, b) : a ∈ A and b ∈ B}

The Cartesian Product of Sets is the Product of Two Sets

A non-empty set P and a non-empty set Q are given. It includes all ordered pairs in which the first component is a member of P and the second component is a member of Q. i.e., P × Q = { (p,q) : p ∈ P, q ∈ Q }

 As long as either P or Q is empty set, then P × Q will also be empty set, i.e., P × Q = φ

Properties of Cartesian Product

1. The Cartesian Product is non-commutative: A × B ≠ B × A

Example: 

A = {3, 4} , B = {a, b}

A × B = {(3, a), (3, b), (4, a), (4, b)}

B × A = {(a, 3), (b, 3), (b, 4), (b, 4)}

Therefore as A ≠ B we have A × B ≠ B × A

2. A × B = B × A, only if A = B

Example:

A={1,2} and B={1,2}

Hence A×B= B×A={(1,1),(1,2),(2,1),(2,2)}

3. The cardinality of the Cartesian Product is defined as the number of elements in A × B and is equal to the product of cardinality of both sets: |A × B| = |A| * |B|

Example: 

A = {3, 4} , B = {a, b}

A × B = {(3, a), (3, b), (4, a), (4, b)}

|A × B| = |A| * |B|

4. A × B = {∅}, if either A = {∅} or B = {∅}

Cartesian Product Calculator

If and only if the corresponding first element and the second element in two ordered pairs are equal, they will be equal.

A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

A x {infinite set} = {infinite set} where A is a non-empty set.

n(A x B)  = n(A) * n(B)

n(A x B x C)  = n(A) * n(B) * n(C)

AxB ≠ BxA

Numerical: If (x + 1, y – 2) = (4,2), find the values of x and y.

Solution: x+1 = 4 or x=3

y-2 =2  or y=4

Numerical: If X = {x, y, z} and P = {q}, form the sets X × P and P × X.

Solution:

X × P = {  (x,q),  (y,q),   (z,q)  }

P X X =  {  (q,x),   (q,y),   (q,z)  }

Numerical: Let A = {1,2,5}, B = {3,4} and C = {4,8,9}. Find A × (B ∩ C) 

Solution: Let’s first find (B ∩ C)  

(B ∩ C)  = {4}

A × (B ∩ C)   = {1,2,5} X {4} =  {(1,4),  (2,4),  5,4)}

Relation: A Subset of Cartesian Product

Set A is a subset of set B if the relation R between the two sets is described. Subsets are derived by describing relationships between elements of A & B. 

• Domain: The set of all the first elements of each pair in an ordered relationship

• Range: The set of all second elements in a relation R 

• Codomain: Whole set B. Note that range ⊆ codomain 

A relation may be represented by:

• Roster form
• Set-builder method
• An arrow diagram

Numerical: Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x +2 }. This relationship can be depicted using an arrow diagram and a set builder’s form. Indicate the domain, the codomain, and the range of R.

Functions: In Terms of Relation

When every element of set A has only one image or output in set B, the relation is said to be a function. In other words, more than one element of set A can be represented in set B by the same image/output. If we compare two elements of set A to two elements of set B, then it still represents a function.

There is no function between sets A and B where the first letter of an element in B is also the first letter of an element in A, as element B in set A has two different images/outputs in set B. The function can be considered a subset of Relation.

Conclusion

We learned about Cartesian products in this article, their properties, and examples. In our previous discussion, we learned that a set is the Cartesian product of two sets. An element in a set is the collection of elements that comprise it. Ordered pairs are the Cartesian product’s elements in this case. It is appropriate to think of an ordered pair as an object composed of two other objects in a specific order. Cartesian products are used to determine what sets of ordered pairs are possible given a given set.