Let $A=\{h, o, g\}, B=\{b, a, t\}$. Find $A \times B$ and $B \times A$.
$A \times B=\{\ \}$ (Type an ordered pair. Use a comma to separate answers as needed.)
Answer (1)
Determine the Cartesian product A × B by pairing each element of set A with each element of set B.
Determine the Cartesian product B × A by pairing each element of set B with each element of set A.
List all ordered pairs in A × B .
The Cartesian product A × B is \boxed{\{(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 ( a , b ) where a ∈ A and b ∈ B . Similarly, B × A is the set of all ordered pairs ( a , b ) where a ∈ B and b ∈ A .
Calculating A x B To find A × B , we take each element of A and pair it with each element of B . This gives us:
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 . This gives us:
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 and in mathematics to define relations and functions. For example, if you have a set of possible shirt sizes S = { S , M , L } and a set of possible colors C = { re d , b l u e , g ree n } , the Cartesian product S × C would give you all possible combinations of shirt sizes and colors: {( S , re d ) , ( S , b l u e ) , ( S , g ree n ) , ( M , re d ) , ( M , b l u e ) , ( M , g ree n ) , ( L , re d ) , ( L , b l u e ) , ( L , g ree n )} . This is a fundamental concept in set theory and has numerous applications in various fields.
