Solution:
Here we can find the number of elements common to both A × B and B × A .
We know that
If A and B are any two non-empty sets, then the set of all ordered pairs (a,b) such that a ∈ A and b ∈ B is called the Cartesian product of the set A with set B and is denoted by A × B.
Thus, A × B = {(a, b): a ∈ A and b ∈ B}
In (a,b) a is called the first coordinate and b the second coordinate If A and B have n elements in common, then A × B and B × A will have n2 elements in common.
We have
If A and B are non empty sets having exactly three elements in common.
We know that
If A and B have ‘n’ elements in common , then A × B & B × A will have (n2) elements in common
For example :
A = {1, 2, 3}, B = {5, 2, 1}
(From question exactly three elements in common)
A × B = { (1 ,5), (1, 2), (1, 1), (2, 5), (2, 2), (2, 1), (3, 5), (3, 2), (3, 1)}
B × A = { (5, 1), (5, 2), (5, 3), (2, 1), (2, 2), (2, 3), (1, 1), (1, 2), (1, 3)}
common elements (1, 1), (2, 2), (1, 2), (2, 1)
Now we have ‘3’ elements in common to both A × B and B × A
∴ A × B & B × A will have 32 = 9 elements in common Therefore, the number of elements common to both A × B & B × A is 9.
Hence, if A and B are non empty sets having exactly three elements in common
then the number of elements common to both A × B and B × A, is 9.
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