Let $A=\{f, o, x\}, B=\{c, o, w\}$. Find $A \times B$ and $B \times A$. $A \times B=\square$ (Type an ordered pair. Use a comma to separate answers as needed.)
Answer (1)
The Cartesian product A × B consists of ordered pairs ( a , b ) where a is from set A and b is from set B .
Given A = { f , o , x } and B = { c , o , w } , we form all possible pairs.
The resulting set A × B is {( f , c ) , ( f , o ) , ( f , w ) , ( o , c ) , ( o , o ) , ( o , w ) , ( x , c ) , ( x , o ) , ( x , w )} .
Therefore, A × B = {( f , c ) , ( f , o ) , ( f , w ) , ( o , c ) , ( o , o ) , ( o , w ) , ( x , c ) , ( x , o ) , ( x , w )} .
Explanation
Understanding the Problem We are given two sets, A = { f , o , x } and B = { c , o , w } . We need to find the Cartesian product of these sets, A × B and B × A . The Cartesian product A × B is the set of all ordered pairs ( a , b ) where a is an element of A and b is an element of B . Similarly, B × A is the set of all ordered pairs ( a , b ) where a is an element of B and b 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:
( f , c ) , ( f , o ) , ( f , w )
( o , c ) , ( o , o ) , ( o , w )
( x , c ) , ( x , o ) , ( x , w )
Therefore, A × B = {( f , c ) , ( f , o ) , ( f , w ) , ( o , c ) , ( o , o ) , ( o , w ) , ( x , c ) , ( x , o ) , ( x , w )} .
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:
( c , f ) , ( c , o ) , ( c , x )
( o , f ) , ( o , o ) , ( o , x )
( w , f ) , ( w , o ) , ( w , x )
Therefore, B × A = {( c , f ) , ( c , o ) , ( c , x ) , ( o , f ) , ( o , o ) , ( o , x ) , ( w , f ) , ( w , o ) , ( w , x )} .
Final Answer for A x B The Cartesian product A × B is the set of all ordered pairs ( a , b ) where a ∈ A and b ∈ B . In our case, A = { f , o , x } and B = { c , o , w } . Thus, A × B = {( f , c ) , ( f , o ) , ( f , w ) , ( o , c ) , ( o , o ) , ( o , w ) , ( x , c ) , ( x , o ) , ( x , w )} .
Examples
Understanding Cartesian products is fundamental in computer science, especially in database management and relational algebra. For instance, if you have a table of customers and a table of products, the Cartesian product would generate all possible combinations of customers and products. While this isn't directly useful, it's a building block for more complex operations like joins, which link related data based on specific criteria. This concept helps in creating comprehensive datasets for analysis and decision-making.
