Question

Let A and B be sets. Show that f : A × B → B × A such that ( a , b ) = ( b , a ) is bijective function.

Answer 1

The given function f:A×B→B×A is defined by ( a,b )=( b,a ).

Let ( a 1 , a 2 ) and ( a 2 , b 2 )∈A×B such that,

f( a 1 , b 1 )=f( a 2 , b 2 ) ⇒( b 1 , a 1 )=( b 2 , a 2 ) ⇒ b 1 = b 2  and  a 1 = a 2 ⇒( a 1 , b 1 )=( a 2 , b 2 )

Hence, f is one-one.

Let, ( b,a )∈B×A be any element.

Then the element ( a,b )∈A×B exists, such that ( a,b )=( b,a ).

So, f is onto.

Thus, f is both one-one and onto and thus, fis bijective.