2.E. 1 = {(, ): (, ) } is a function from B to A if is one-one
function from A to B.Proof. By definition, this set is a function
if (, ) 1 , (, ) 1 implies = .It is true in this case because by
construction (, ) 1 , (, ) 1 implies(, ) , ( , ) and because is
one-one, = F. Proof. Because is a function from A to B and 1 is a
function from B to A, it ispossible to consider both compositions:
1 : , 1 : .Denote ( 1 )() = , we need to prove that = , i.e. : (, )
, (, ) 1 .Indeed, = () is suitable because (, ()) by definition of
and((), ) 1 by definition of 1 , The second part, denote ( 1 )() =
, we need to prove that = , i.e. : (, ) 1 , (, ) . Indeed, = 1 ()
is suitable because (, 1 ()) 1by definition of a function and ( 1
(), ) by definition of inverse function, J. () = 2 , = [1,0], =
[0,1].We see that = {0}, thus ( ) = {(0)} = {0}. From …