# Equivalance classes and integer addition

1. Nov 19, 2009

### hitmeoff

1. The problem statement, all variables and given/known data
Prove: If a and b are in N the [(1,1+a)] + [(1,1+b)] = [(1,1+a+b)]

2. Relevant equations
Definition: We define + over Z as follows: if [(a,b)] and [(c,d)] are any two equivalence classes, we define

[(a,b)] + [(c,d)] = [(a+c,b+d)].

3. The attempt at a solution

So the left hand is [(2,2+a+b)] since 2, a, b are all $$\in$$ N does this mean this is equivalent to [(1,1+a+b)]?

2. Nov 20, 2009

### HallsofIvy

Staff Emeritus
What is the equivalence relation?

One way of defining Z from N is to say that two pairs of natural numbers, (a, b) and (c, d) are equivalent if and only if a+ d= b+ c. Is that the equivalence relation you are using?

3. Nov 20, 2009

### hitmeoff

ahh..yes yes it is. i didnt quit understand my notes. i got the solution, thanks for clearing it up