Question

If $A{\text{ = }}\left\{ {{\text{a,b}}} \right\}{\text{ }}$ and $B = \left\{ {1,2,3} \right\}$. Find $A \times A$ and $B \times B$

Answer 1

Hint: In order to find the product of $A \times A$ and $B \times B$ , we have to use the definition of Cartesian product. The Cartesian product of two sets $A$ and $B$ , denoted $A \times B$ , is the set of all ordered pairs where $a$ is in $A$ and $b$ is in $B$.

Complete step-by-step answer:
In the above question we have to find $A \times A$ and $B \times B$- Given two sets $A$ and $B$, the Cartesian product is the set of all unique ordered pairs using one element from Set $A$ and one element from set $B$ .
For example-
Suppose, $a = \left\{ {{\text{Dog,Cat}}} \right\}{\text{ }}b = \left\{ {{\text{Meat,Milk}}} \right\}$ then,
$A \times B = \left\{ {\left( {{\text{Dog,Meat}}} \right),\left( {{\text{Cat,Milk}}} \right),\left( {{\text{Dog,Milk}}} \right),\left( {{\text{Cat,Meat}}} \right)} \right\}$
Here, we have -
$
  A = \left\{ {a,b} \right\}{\text{ equation}}\left( 1 \right) \\
  B = {\text{ }}\left\{ {1,2,3} \right\}{\text{ equation}}\left( 2 \right) \\
    \\
 $
Now; we have to find $A \times A$ and $B \times B$ . By using the definition of Cartesian product that is-
$
  A \times A{\text{ = }}\left\{ {a,b} \right\}{\text{ }} \times {\text{ }}\left\{ {a,b} \right\} \\
    \\
 $
$ = \left\{ {\left( {a,a} \right),{\text{ }}\left( {a,b} \right){\text{, }}\left( {b,a} \right),{\text{ }}\left( {b,b} \right)} \right\}$
Similarly,
$B \times B =\left\{ {1,2,3} \right\}{\text{ }} \times {\text{ }}\left\{ {1,2,3} \right\}
=\left\{ {\left( {1,1} \right),{\text{ }}\left( {1,2} \right),{\text{ }}\left( {1,3} \right),{\text{ }}\left( {2,1} \right),{\text{ }}\left( {2,2} \right),{\text{ }}\left( {2,3} \right),{\text{ }}\left( {3,1} \right),{\text{ }}\left( {3,2} \right),{\text{ }}\left( {3,3} \right)} \right\} \\
 $
Note: Whenever we face such type of questions, the key concept is that the Cartesian product of $A$ and $B$ , denoted $A \times B$ , is the set of all possible ordered pairs where the elements of$A$ are first and the elements of $B$ are second. By using the Cartesian product we will get our required answer like we did in the question.