Let $A=\{c, a, t\}, B=\{f, l, y\}$. 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 is the set of all ordered pairs ( x , y ) where $x
is an element of A and y is an element of B .
We find A × B by pairing each element of A with each element of B .
Thus, A × B = {( c , f ) , ( c , l ) , ( c , y ) , ( a , f ) , ( a , l ) , ( a , y ) , ( t , f ) , ( t , l ) , ( t , y )} .
The final answer is \boxed{\{(c, f), (c, l), (c, y), (a, f), (a, l), (a, y), (t, f), (t, l), (t, y)\}\}
Explanation
Understanding the Cartesian Product We are given two sets, A = { c , a , t } and B = { f , l , y } . We need to find the Cartesian product A × B , which is the set of all ordered pairs ( x , y ) where x is an element of A and y is an element of B .
Calculating A x B To find A × B , we take each element from set A and pair it with each element from set B . This gives us:
( c , f ) , ( c , l ) , ( c , y ) (pairing 'c' with each element of B) ( a , f ) , ( a , l ) , ( a , y ) (pairing 'a' with each element of B) ( t , f ) , ( t , l ) , ( t , y ) (pairing 't' with each element of B)
The Result Therefore, A × B = {( c , f ) , ( c , l ) , ( c , y ) , ( a , f ) , ( a , l ) , ( a , y ) , ( t , f ) , ( t , l ) , ( t , y )} .
Examples
Cartesian products are used in computer science to generate combinations of inputs for testing software. For example, if you have a set of possible usernames and a set of possible passwords, the Cartesian product would give you all possible username-password combinations to test the security of a system. In mathematics, Cartesian products are fundamental in defining relations and functions between sets.
