Let $A = \{h, o, g\}, B = \{b, a, t\}$. Find $A \times B$ and $B \times A$. $A \times B = $
Answer (1)
Identify the elements of sets A and B .
Form ordered pairs ( x , y ) where x in A and y in B to find A × B .
Form ordered pairs ( x , y ) where x in B and y in A to find B × A .
The Cartesian product A × B = {( h , b ) , ( h , a ) , ( h , t ) , ( o , b ) , ( o , a ) , ( o , t ) , ( g , b ) , ( g , a ) , ( g , t )} .
Explanation
Understanding the Problem We are given two sets, A = { h , o , g } and B = { b , a , t } . We need to find the Cartesian products A × B and B × A . The Cartesian product A × B is the set of all ordered pairs ( x , y ) where x is an element of A and y is an element of B . Similarly, B × A is the set of all ordered pairs ( x , y ) where x is an element of B and y is an element of A .
Calculating A x B To find A × B , we take each element of A and pair it with each element of B . So, we have:
A × B = {( h , b ) , ( h , a ) , ( h , t ) , ( o , b ) , ( o , a ) , ( o , t ) , ( g , b ) , ( g , a ) , ( g , t )}
Calculating B x A To find B × A , we take each element of B and pair it with each element of A . So, we have:
B × A = {( b , h ) , ( b , o ) , ( b , g ) , ( a , h ) , ( a , o ) , ( a , g ) , ( t , h ) , ( t , o ) , ( t , g )}
Final Answer for A x B Therefore, A × B = {( h , b ) , ( h , a ) , ( h , t ) , ( o , b ) , ( o , a ) , ( o , t ) , ( g , b ) , ( g , a ) , ( g , t )} .
Examples
Cartesian products are used in computer science to design databases, where each table is a Cartesian product of different domains. For example, if you have a set of users and a set of products, the Cartesian product would give you all possible combinations of users and products, which can be useful for creating a recommendation system or tracking user interactions with products. In mathematics, Cartesian products are fundamental in defining relations and functions between sets, providing a structured way to understand how elements from different sets can be combined.
