Is the function injective or surjective?

Question

Determine whether the function f: NxN --> Z with f(x,y) = (1-x^2)[y/3] is either:

 

a) injective

b) surjective

 

 

 

Injective function

A function f is said to be one-to-one, or injective, of and only if f(a) = f(b) implies that a = b for all a and b in the domain of f. A function is said to be an injection if it is one-to-one.

 

Surjective function

A function f from A to B is called onto, or surjective if and only if for every element b ∈ B there is an element a ∈ A with f(a) = b. A function f is called a surjection if it is onto.

Answer 1

This function is surjective but not injective:

 

You can attain any z value in the codomain simply by letting x=0 and y=3z for z positive, by letting x,y=1 for when z=0, and by letting x=2 and y=-z for z negative; therefore surjective. 

 

Now let (x1,y1)=(2,3) and (x2,y2)=(-2,3), then we have f(x1,y1)=f(x2,y2) and (x1,y1) not equal to (x2,y2); therefore not injective.