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The Cartesian product is the product of any two sets, but it is actually ordered, meaning that the resultant set contains all feasible and ordered pairs in which the first element belongs to the first set and the second element belongs to the second set. We refer to them as the first and second elements, respectively, because the order in which they appear is significant. We use ordered pairs to generate a new set from two given sets A and B.
Cartesian Product Definition
The cartesian product, C D, is the set of all ordered pairs (a, b) with the first element from C and the second element from D if C and D are two non-empty sets. We use the same multiplication sign to express the cartesian product between two sets as we do for the other product operations. For the Cartesian product of C and D, we use the notation C D.
If we use notation,then we can write the cartesian product it as:
C × D = {(a,b): a ∈ C, b ∈ D}. Here a is in set C and b is in set D.
If both the sets are the same i.e, if C = D then C × D is called the cartesian square of the set C and it is denoted by C2
C2 = C × C = {(a,b): a ∈ C, b ∈ C}
Cartesian Product of Sets
The ordered product of two non-empty sets is what Cartesian products of sets are. The end product of multiplying the two non-empty sets together will be a collection of all ordered pairs. In an ordered pair, two elements are picked from each of the two sets.
Cartesian Product of Empty Set
The empty set is a unique set that has no elements. It will have zero size and cardinality (the total number of elements in a set). An empty set is also known as a void set. The Cartesian product of C with the empty set is the empty set. A C and b are represented by C = (a,b)|. There isn’t a single element inside. If and only if C = or D =, C D = is true. The cartesian product of the two sets will result in an empty set if and only if one of the sets is empty.
Cartesian Product of Countable Sets
It is possible to count the cartesian product of two countable sets. Take a look at these two examples to see what I mean:
Consider an integer b that is greater than one. Countable is then the cartesian product of b countable sets.
Consider the two countable sets A = a0, a1, a2,…, and B = b0, b1, b2,…, respectively. If both sets A and B are countable, the resultant set will be countable as well.
Conclusion
The set of all ordered pairs/n-tuples of two or more sets is called the cartesian product. Furthermore, many real-life objects, such as a deck of cards, chess boards, computer graphics, and so on, can be represented using cartesian products.
