Define $\displaystyle Z = \{((x_{1},y_{1}),(x_{2},y_{2})) \epsilon (N \times N) \times (N \times N): x_{1}+y_{2}= y_{1} +x_{2}\} $

show that Z is an equivalence relation on N x N

Show that

$\displaystyle N = \{([(x,y)]_{Z},[(y,x)]_{Z}) :x,y \epsilon N \}$

is a function

Show that

$\displaystyle A= \{(([x_{1},y_{1})]_{Z},[(x_{2},y_{2})]_{Z}), [(x_{1}+x_{2},y_{1}+y_{2})]_{Z}): x_{1},x_{2},y_{1},y_{2} \epsilon N \}$

is a function

I'm just having trouble setting these problems up