Question

If two sets A and B have 5 elements in common, then the number
of elements common to each of the sets (A x B) and (B x A) is​

Answer 1

To find A × B, we take one element from set A and one from set B.  

Given that 5 elements are common to both set A and set B.

Suppose these common elements are $n_{1} , n_{2}, n_{3}, ..., n_{5}$

Select an ordered pair for A × B such that both are selected out of these common elements.

Examples: $(n_{1}, n_{2}), (n_{3}, n_{5})$

All these will also be elements of B × A.

Hence number of elements common to A × B and B × A is 5 × 5 = 25  

[ First element in ordered pair can be selected in 5 ways; second element can also be selected in 5 ways ]

n[(A × B) ∩ (B × A)] = n[(A ∩ B) ∩ ( B ∩ A)] = (5) (5) = 25

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