Profile
Settings
Refer your friends
Sign out
In set theory, the Cartesian product is often utilized. 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, graphical representations of cartesian products.
What is Cartesian Product?
The Cartesian product is the product of any two sets. Still, it is ordered, meaning that the resultant set comprises all viable and ordered pairings in which the first element belongs to the first set and the second element belongs to the second set. Because the order in which they appear is crucial, we refer to them as the first and second components. We apply ordered pairs to generate a new set from two provided sets, A and B.
The values p and q create an ordered pair (p, q). For example, (2, 3) and (- 5, 8) are ordered pairs because the numbers are in a specified sequence.
As a consequence, unless p = q, (p, q) = (q,p) (q,p). (p, q) = (s, t) in general if and only if p = s and q = t. For example, (2, 3) is not the same as (3, 2), i.e. (3, 3) (3, 2).
Cartesian Product Definition
Cartesian product X x Y is the set of all ordered pairs (a, b) consisting of one element from X and one element from Y if X and Y are non-empty sets. We use the same multiply sign (‘x’) to indicate the cartesian product between two sets for the other product operations. We use the symbol X x Y for the Cartesian product of X and Q.
The cartesian product may be stated as follows using the set-builder notation:
X x Y = {(a,b): a ∈ X, b ∈ Y}. In this situation, a belongs to set X and b to set Q.
X x Y is considered the cartesian square of the set X and is represented by P2 if both sets are the same, i.e. if X = Q.
X2 = X × X = {(a,b): a ∈ X, b ∈ Y}
Cartesian Product of Sets
As the ordered product of two non-empty sets, the Cartesian product of sets can be thought of as the Cartesian product of sets. All ordered pairs resulting from multiplying the two non-empty sets will be output from the sets. A pair is comprised of two items from both sets.
How can I find a Cartesian Product of Sets?
Consider two sets that aren’t empty. D = {4, 5, 6} and C = {x, y, z}
Following the techniques below, you may acquire the cartesian product set, also known as the cross-product or the product set of C and D:
The first element x comes from the C {x, y, z} set, whereas the second element 1 comes from the D {4, 5, 6} set.
The first ordered pair is created by multiplying both of these items (x,1)
The same approach is continued for the remaining couples until all possible combinations have been identified.
The cartesian product set C x D = (x,4), (x,5), (x,6), (y,4), (y,5), (y,6), (z,4), (z,5), (z,6) is the sum of all such ordered pairs.
Similarly, the cartesian product of D x C may be determined.
Find the cartesian product of the two sets C and D, where C = {21,22,23} and D = {17, 18}, respectively. After you’ve performed the aforementioned steps, you’ll be able to:
The ultimate product C x D will be {(21,17), (21,18), (22,17), (22,18), (23,17), (23,18)}
D x C = {(17,21),(17,22),(17,23),(18,21),(18,22),(18,23)} is the cartesian product of D and C.
The cartesian products C x D and D x C do not have the same ordered pairings as each other. Consequently, in general, C x D = D x C.
Conclusion
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 two sets. Cartesian Products are used in many real-life examples. Hence, it’s necessary that you read and understand this article carefully.
