`A` and `B` are two non empty set having `n` element in common ; then prove that `AxxB` and `BxxA` have `n^2` element in common.

Options

A. 2n
B. n
C. `n^(2)`
D. none of these

Answer

C

Explanation

We know that
`(AxxB)nn(CxxD)=(AnnC)xx(BnnD)`
`therefore (AxxB)nn(BxxA)=(AnnB)xx(BnnA)`
`implies (AxxB)nn(BxxA)=(AnnB)xx(AnnB)`
It is given that `AnnB` has n elements.
`therefore (AnnB)xx(AnnB)` has `n^(2)` elements.
But, `(AxxB)nn(BxxA)=(AnnB)xx(AnnB)`
`therefore (AxxB)nn(BxxA)` has `n^(2)` elements in common.