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.