Prove that the function H defined analytically in exercise 23b is a one-to-one correspondence.

To prove:

The function H is a one-to-one correspondence.

Given information:

Define a function H:Znonneg×Znonneg by the formula:

H(m,n)=n+(m+n)(m+n+1)2

Proof:

Honto:

Let y∈Znonneg.

Let k be the largest positive integer such that: y−(1+2+3+...+k)≥0 and let p=y−(1+2+3+...+k). Then we will let m=k−p and let n=k−m.

H(m,n)=n+( m+n)( m+n+1)2=n+( m+( k−m ))( m+( k−m )+1)2                       (∵ n=k−m)=n+k( k+1)2=n+∑i=0ki                                                        (∵  ∑ i=0 y i= y( y+1 ) 2 ) =n+∑i=1ki=n+(1+2+3+...+k)=(k−m)+(1+2+3+...+k)                              (∵ n=k−m)=(k−( k−p))+(1+2+3+...+k)                     (∵ m=k−p)

=p+(1+2+3+...+k)                                       (Simplify)=(y−( 1+2+3+