Question

Let R+ denote the set of positive real numbers. Let f : R × R+ → R be given by f(x, y) = x/y. (a) Is f an injective function? Prove your answer. (b) Is f a surjective function? Again, prove your answer. (c) Is f a bijection? Prove your answer.

Answer 1

Let
`f:\Bbb R*\Bbb R^+\to\Bbb R`.

a.
`f` is injective if
`f(x_1,y_1)=f(x_2,y_2)` for any two points
`(x_1,y_1)` and
`(x_2,y_2)`, then the two points must be identical with
`x_1=x_2` and
`y_1=y_2`.

`f` is not injective because we can pick two points for which
`f` gives the same value:

`f(2,1)=\frac21=2`

`f(4,2)=\frac42=2`

b.
`f` is surjective if there exists
`(x,y)` for which every point in the codomain
`\Bbb R` is obtained by
`f(x,y)`.

This should be somewhat obvious if you consider 3 different cases:

1. 
   `f(x,y)<0` and can take on any negative real number for any choice of
   `x<0`
2. 
   `f(x,y)=0` if and only if
   `x=0`
3. 
   `f(x,y)>0` for any choice of
   `x>0`

So
`f` is surjective.

c.
`f` is bijective if it is both injective and surjective. The conclusions above show
`f` is not bijective.