Solveeit Logo

Question

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

If A=0,1  A = \\{ 0,1\\} \; and B=1,2,3B = \\{ 1,2,3\\} , Show that A×BB×AA \times B \ne B \times A.

Explanation

Solution

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  Aa\; \in A and b  Bb\; \in B. So, A × B = (a,b):  a  A, b  B.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 = a1,a2,a3{{a_1}, {a_2},{ a_3}} and B = b1,b2,b3{{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  Aa\; \in A and b  Bb\; \in B. So, A × B = (a,b):  a  A, b  B.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 (0,1)  \left( {0,1} \right)\;in   A×B    \;A \times B\;\;is not equal to the pair (1,0)  \left( {1,0} \right)\;in B×AB \times A.
Therefore, A×BB×AA \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.