Question

Let A and B be sets. Show that `f" ":" "AxxB` ,`BxxA` such that `f" "(a ," "b)" "=" "(b ," "a)` is bijective function.

Answer 1

f is one-one since
`f(a_(1) ,b_(1)) =f(a_(2) , b_(2)) rAr (b_(1) , a_(1)) =(b_(2) ,a_(2))`
`rArr a_(1) = a_(2) " and " b_(1) = b_(2)`
`rArr (a_(1) ,b_(1)) =(a_(2) ,b_(2))`
In order to show that f is onto let (b,a) be arbitrary element of `(Bxx A)`
Then `(b,a) in (B xxA) rArr b in B " and " a in A`
`rArr (a,b) in (A xxB)`
Thus for each (b,a) `in (B xx A) ` there exists (a,b) `in A xxB` such that
` f(a,b) =(b,a)`
`:.` f is onto.
Thus f is one-one onto and hence bijective .