Show that the function f: N * N - > N * N, f(x,y) = (x+y, 2x-3y) is one-to-one.

Question

Show that the function f: N x N - > N x N, f(x,y) = (x+y, 2x-3y) is one-to-one.

(A function f is said to be one -to -one , or injective , if 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.)

Answer 1

Let a = (x,y), b = (w,z) such that a,b ∈ NxN and f(a),f(b) ∈ NxN.

 

f(a) = f(b) implies (x+y, 2x-3y) = (w+z, 2w-3z) implies x+y = w+z, 2x-3y = 2w-3z by the definition of equality of elements in NxN.

 

x+y    =   w+z         ⇒     x-w   =  -(y-z)

2x-3y = 2w-3z              2x-2w = 3y-3z  

 

⇒    x-w   =  -(y-z)   ⇒    -2(y-z) = 3(y-z)

    2(x-w) = 3(y-z)

 

The last equation holds if and only if y-z = 0 implies also x-w = 0 implies y = z and x = w, which implies, if  a = (x,y), b = (w,z), a = b.

 

Note that the domain of ƒ, D =

{ (x,y)∈ NxN | f((x,y)) = (x+y, 2x-3y)∈ NxN } = { (x,y)| 2x > 3y, x,y ∈ N } ⊂ NxN (note ⊂ as opposed to ⊆), but amazingly though as was just shown, these restrictions played no part in determining whether ƒ was an injective function or not.