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A set is a well-defined group of elements. In mathematics, there are many types of sets based on the number of elements they have. Cartesian product in set relations function implies the product of two elements of Cartesian in an arranged manner.
The Cartesian product in set relations functions formula for set A and set B (none of these two sets are empty sets) is expressed below:
A×B={(a,b) |a ∈ A and b∈ B}
The set of all achievable ordered pair (a,b) is called the Cartesian product denoted by AB, where aA and bB.
Cartesian Product
Cartesian represents the two-dimensional coordinate plane formed by the intersection of two axes. The two axes of Cartesian are x-axis and y-axis. Cartesian product in set relations function implies the product of two elements of Cartesian in an arranged manner. The Cartesian product of set A and set B (none of these two sets are empty sets) is denoted by the following expression:
A×B
This product is also called the cross product or the product of set A and set B.
The Cartesian product of set A{a1,a2} and set B{b1,b2} (none of these two sets are empty sets) is expressed in the following expression:
A×B={(a1b1) ,(a1b2) ,(a2,b1) (a2b2)}
If in the Cartesian product any set is a null set, then the result of the Cartesian product is also a null set.
Cartesian Product in Set Relations Functions Formula
The Cartesian product in set relations functions formula for set A and set B (none of these two sets are empty sets) is expressed below:
A×B={(a,b) |a A and bB}
The set of all achievable ordered pair (a,b) is known as the Cartesian product denoted by AB, where aA and bB.
The Cartesian product of set A and set B is not equal to the Cartesian product of set B and set A except A=B.
Cartesian Product of Functions
Suppose there are two functions:
The first function is f, which exists from X to A
The second function is g, which exists from Y to B.
The Cartesian product of these two functions can be represented as:
(f×g)(x,y)={f(x), g(y)}
Properties of Cartesian Product in Set Relations Functions Formula
In set relations functions formula there are different properties of the Cartesian product. These properties are very helpful in the calculation of different sets while performing mathematical operations for set theory.
The properties of the Cartesian product In set relations functions formula are listed below.
Non-commutative Property: The Cartesian product in set relations functions formula is non-commutative. Suppose sets A and B are two non-empty sets; then the Cartesian product of set A and set B is not equal to the Cartesian product of set B and set A.
A×B ≠ B×A
Non-Associative Property: The Cartesian product in set relations functions formula is non-associative. Suppose sets A, B and C are three non-empty sets; then for the Cartesian product of set A and set B the non-associative property can be expressed as:
(A×B )×C≠A×(B×A)
Distributive Property of Intersection: The Cartesian product of set A and intersection of set B and set C is equal to the intersection of the Cartesian product of set A and B and Cartesian product of set A and C.
A×(B∩C )=(A×B)∩(A×C)
Distributive Property of Union: 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 union of the Cartesian product of set A and C.
A×(B∪C )=(A×B)∪(A×C)
Distributive Property of Difference: Distributive property of difference of set A, set B and set C for Cartesian product is expressed as:
A×(B-C )=(A×B)-(AC)
Examples
1. Two set is given A{1,3,5} and B{2,4}. Find the Cartesian product of set A and set B.
Solution:
The first element of set A is 1, the second element of set A is 3 and the third element of set A is 5. For the second set B, the first and second element is 2 and 4, respectively.
The ordered pair for first element is {(1,2), (1,4)}
The ordered pair for second element is {(3,2), (3,4)}
The ordered pair for third element is {(5,2), (5,4)}
The Cartesian product of set A and set B is,
A×B={(1,2), (1,4),(3,2), (3,4),(5,2), (5,4) }
Conclusion
The product of elements of Cartesian of various sets in a systematic way is called the Cartesian product in set relations function. In set relation functions, the Cartesian coordinate helps to obtain all possible ordered pairs of different sets. To find the Cartesian coordinate, the order of multiplication plays a significant role.
