Question

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

Answer 1

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.

Complete step-by-step answer:
It is given,
\[A = \left\{ {0,1} \right\}\]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, \[A{\text{ }} \times {\text{ }}B{\text{ }} = {\text{ }}\{ \left( {a,b} \right):\;a\; \in A,{\text{ }}b\; \in B\} .\]
In simple language, cartesian products of two sets are defined as, {(terms of the 1st set, terms of the 2nd set)}
So, now,
\[A \times B = \left\{ {\left( {0,1} \right),\left( {0,2} \right),\left( {0,3} \right),\left( {1,1} \right),\left( {1,2} \right),\left( {1,3} \right)} \right\}\]
\[B \times A = \left\{ {\left( {1,0} \right),\left( {2,0} \right),\left( {3,0} \right),\left( {1,1} \right),\left( {2,1} \right),\left( {3,1} \right)} \right\}\]
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, \[A \times B \ne B \times A\]

Note: The Cartesian products of sets mean the product of two non-empty sets in an ordered way. Or, in other words, the collection of all ordered pairs obtained by the product of two non-empty sets. An ordered pair means that two elements are taken from each set.
For two non-empty sets (say A & B), the first element of the pair is from one set A and the second element is taken from the second set B. The collection of all such pairs gives us a Cartesian product.