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Let A and B be two sets. Show that f: Ax...

Let `A` and `B` be two sets. Show that `f: AxxB->BxxA` defined by `f(a ,\ b)=(b ,\ a)` is a bijection.

Text Solution

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In `f: AxxB to BxxA`
`f(a, b) = (b, a) AA (a, b) in A xx B`
Let `" " (a, b), (c, d) in AxxB`
and `" " f(a, b) = f(c,d)`
`rArr (b, a) = (d, c)`
`rArr b =d and a = c `
`rArr (a, b) = (c, d)`
`therefore f ` is one-one.
Again, for each `(a, b) in B xx A, (b, a) in A xx B` is such that
`therefore " " f(b, a) = (a, b)`
`therefore f` is onto.
Therefore, `f` is one-one onto function.
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