Constructing Z from N (integers from naturals)

Having some trouble understanding this...

This method uses an equivalence relation defined on NxN such that: (a,b)~(c,d) iff a+d=c+b

Note that we can subtract any k in N from both a and b to get an equivalent pair, providied k < min(a,b) (makes sense)

It follows that every class [a,b] contains a special pair, in which atleast one of the components is equal to 1. Moreover there is only one such pair in each class:

(1,b)=(1,d) => b=d

(a,1)=(b,1) => a=b

(a,1)=(1,b) => a=b=1

What's going on here? Shouldn't the = signs be ~ signs?

The special pairs thus form a transversal,

T={(1,n+1): n in N} union {(1,1)} union {(n+1,1): n in N}

Not sure what's going on here either???:confused: