Tashkent university of information technologies named after muhammad al-khorazmi, ministry of information technology and communication development of the republic of uzbekistan discrete structure science topic: mi3 Cartesian multiplication of

Cartesian Product Definition

• Cartesian Product Definition
• If C and D are two non-empty sets, then 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. Similar to the other product operations, we use the same multiplication sign × to represent the cartesian product between two sets. Here, we use the notation C × D for the Cartesian product of C and D.
• By using the set-builder notation, we can write the cartesian product as:
• C × D = {(a,b): a ∈ C, b ∈ D}. Here a belongs to set C and b belongs to 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

• Cartesian Product of Sets
• The cartesian products of sets can be considered as the product of two non-empty sets in an ordered way. The final product of the sets will be a collection of all ordered pairs obtained by the product of the two non-empty sets. In an ordered pair, two elements are taken from each of the two sets.
• Finding Cartesian Product
• Consider two non-empty sets C = {x, y, z} and D = {1, 2, 3} as shown in the below image:

The cartesian product, also known as the cross-product or the product set of C and D is obtained by following the below-mentioned steps:

• The cartesian product, also known as the cross-product or the product set of C and D is obtained by following the below-mentioned steps:
• The first element x is taken from the set C {x, y, z} and the second element 1 is taken from the second set D {1, 2, 3}
• Both these elements are multiplied to form the first ordered pair (x,1)
• The same step is repeated for all the other pairs too until all the possible combinations are chosen
• The entire collection of all such ordered pairs gives us a cartesian product C x D = {(x,1), (x,2), (x,3), (y,1), (y,2), (y,3),(z,1), (z,2), (z,3)}.
• Similarly, we can find the cartesian product of D x C.
• Let us find the cartesian product of the two sets C and D, where C = {11,12,13} and D = {7, 8}. After following the steps mentioned above:
• The resultant product C x D will be: {(11,7), (11,8),(12,7),(12,8),(13,7),(13,8)}.
• Similarly, we can find the cartesian product of D and C as D × C = { (7,11),(7,12),(7,13),(8,11),(8,12),(8,13)}.
• The cartesian products C × D and D × C do not contain exactly the same ordered pairs. Hence, in general, C × D ≠ D × C.
• Cartesian Product of Several Sets
• We can extend or define the cartesian product to more than two sets. The cartesian product of several input sets is a larger set that contains every ordered combinations of all the input set elements. The cartesian product of three sets P, Q, and R can be written as:
• P × Q × R = { (a,b,c): a ∈ P, b ∈ Q, c ∈ R }
• Let us consider the example of three sets A, B and C, where A = {2,3} , B = {x,y}, and C = {5,6}. In order to find the cartesian product of A × B × C, let us find the cartesian product of A × B first.
• A × B = {(2,x), (2,y),(3,x),(3,y)}.
• A × B × C = {(2,x,5), (2,x,6), (2,y,5), (2,y,6), (3,x,5), (3,x,6), (3,y,5), (3,y,6)}