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The term ‘product’ refers to multiplying two or more values together in mathematics. For instance, 45 is the result of multiplying 9 and 5. Basic set operations such as Union and Intersection must be understood, which are done on two or more sets. The Cartesian Product is another operation that takes two sets and returns a collection of ordered pairs. In set theory, the Property of Cartesian Product is most typically used. Furthermore, many real-life objects, such as a deck of cards, chess boards, computer graphics, and so on, maybe represented using cartesian products. Computers display most digital images as pixels, which are graphical representations of cartesian products.
What is the Property of Cartesian Product?
The Property of Cartesian Product is the product of any two sets that are ordered, which means that the resultant module includes all possible and ordered pairs in which the first element belongs to the first set and the second element belongs to the second set. Because the sequence in which they appear is significant, we refer to them as the first and second elements, respectively. To create a new set from two given sets, A and B, we employ ordered pairs.
Example of a Cartesian Product
Assume A is a collection of two colors and B is a collection of three things, with A=red, blue and B=b,c,s.
A belt, coat, and shirt are represented by b, c, and s, respectively.
From these two sets, how many pairs of objects can be made?
We can see that there will be separate pairs if we proceed in a fairly ordered manner, as seen below: (red,b),(red,c),(red,s)(blue,b),(blue,c),(blue,s)
As a result, we have six separate items. First, a cartesian product is a combination of two words: Cartesian and product. The name Cartesian was coined by René Descartes, a French mathematician, and philosopher.
Cartesian Product of Empty Set
After learning about the Property of Cartesian Product, the empty set is a one-of-a-kind set that has no elements. Its size and cardinality, or the total number of elements in a set, will both remain 0. A void set is another name for an empty set. The empty set ∅ is the Cartesian product of C and the empty set ∅.
Let C × ∅ = {(a,b)| a ∈ C, b∈ ∅}. There is no element in ∅. C × D =∅ if and only if C = ∅ or D = ∅. If and only if one of the sets is empty, the cartesian product of the two sets will result in an empty set.
Cartesian Product Of Relations
The Property of the Cartesian Product of two sets of relations is the same as the relation between two sets of relations. The cartesian product is usually used to express a set rather than a relation. Furthermore, because the universal relation connects every element of one set to an element of another, it can be expressed as the cartesian product of relations.
Properties of Cartesian Product
- The Cartesian product is non-commutative in the following ways:
A×B≠B×A
It means that the multiplication sequence is crucial in determining the cartesian product.
- A×B=B×A, if only A=B
- A×B=ϕ, if either A=ϕ or B=ϕ
- The Cartesian product is associative:
It signifies that the three-cartesian set’s product is the same, which means that it doesn’t matter which bracket is multiplied first because the final result will be the same.
- Over a set intersection, there is a distributive property:
A×(BC) =(A×B) ∩(A×C)
- Set the union’s distributive property:
A×(BC) =(A×B) ∪(A×C)
- If A⊆B, then A×C⊆B×C for any set C.
Important Notes on Cartesian Product
Here’s a rundown of some key concepts to keep in mind while studying the property of Cartesian products:
- Ordered pairs are sometimes referred to as 2-tuples.
- The cross-product of the product set of C and D is the cartesian product of two sets, C and D.
- The collection of all ordered pairs created by the product of these two non-empty sets will be the final cartesian product of the two sets.
Cartesian Product in Relational Algebra
In relational algebra, the Cartesian product is a binary operator. As a result, to identify the Cartesian product, the two relations involved must have disjoint headers, which means there should be no common attribute name. The Cartesian Product is defined as two relations or two sets of tuples in relational algebra. It will couple each tuple from the left set (relation) with all the tuples in the right set one by one (relation).
Conclusion
We learned about the Property of Cartesian Product and several instances in this lesson. The Cartesian product of two sets is a set, as we understood. A collection of elements makes up a set. The ordered pairs are the elements of a Cartesian product in this example. An ordered pair should be treated as a single item made up of two other things in a specific order. A cartesian product is used to find the set of all possible ordered pairings from a collection of numbers.
