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# If $A = \{ 0,1\} \;$ and $B = \{ 1,2,3\}$, Show that $A \times B \ne B \times A$.

Last updated date: 25th Jun 2024
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Hint: The Cartesian product of two non-empty sets A and B is denoted by A × B. Also, known as the cross-product or the product set of A and B. The ordered pairs (a, b) is such that $a\; \in A$ and $b\; \in B$. So, $A{\text{ }} \times {\text{ }}B{\text{ }} = {\text{ }}\{ \left( {a,b} \right):\;a\; \in A,{\text{ }}b\; \in B\} .$ For example, Consider two non-empty sets A = ${{a_1}, {a_2},{ a_3}}$ and B = ${{b_1}, {b_2}, {b_3}}$.
Cartesian product is $A \times B{\text{ }} = {\text{ }}\left\{ {\left( {{a_1},{b_1}} \right),{\text{ }}\left( {{a_1},{b_2}} \right),{\text{ }}\left( {{a_1},{b_3}} \right),{\text{ }}\left( {{\text{ }}{a_2},{b_1}} \right),{\text{ }}\left( {{a_2},{b_2}} \right),\left( {{a_{2,}}{b_3}} \right),{\text{ }}\left( {{a_3},{b_1}} \right),{\text{ }}\left( {{a_3},{b_2}} \right),{\text{ }}\left( {{a_3},{b_3}} \right)} \right\}.$
We use this to get our answer.

$A = \left\{ {0,1} \right\}$And
$B = \left\{ {1,2,3} \right\}$
The Cartesian product of two non-empty sets A and B is denoted by A × B. Also, known as the cross-product or the product set of A and B. The ordered pairs (a, b) is such that $a\; \in A$ and $b\; \in B$. So, $A{\text{ }} \times {\text{ }}B{\text{ }} = {\text{ }}\{ \left( {a,b} \right):\;a\; \in A,{\text{ }}b\; \in B\} .$
$A \times B = \left\{ {\left( {0,1} \right),\left( {0,2} \right),\left( {0,3} \right),\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right)} \right\}$
$B \times A = \left\{ {\left( {1,0} \right),\left( {2,0} \right),\left( {3,0} \right),\left( {1,1} \right),\left( {2,1} \right),\left( {3,1} \right)} \right\}$
By the definition of equality of ordered pairs, the pair $\left( {0,1} \right)\;$in $\;A \times B\;\;$is not equal to the pair $\left( {1,0} \right)\;$in $B \times A$.
Therefore, $A \times B \ne B \times A$