# Math Help - Equivalence relation and Functions

1. ## Equivalence relation and Functions

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

show that Z is an equivalence relation on N x N

Show that

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

is a function

Show that
$A= \{(([x_{1},y_{1})]_{Z},[(x_{2},y_{2})]_{Z}), [(x_{1}+x_{2},y_{1}+y_{2})]_{Z}): x_{1},x_{2},y_{1},y_{2} \epsilon N \}$
is a function

I'm just having trouble setting these problems up

2. Why don't you start by writing the three properties of an equivalence relation? For example, reflexivity says $\forall p\in\mathbb{N}\times\mathbb{N}\,(p,p)\in Z$, or $\forall x,y\in\mathbb{N}\,((x,y),(x,y))\in Z$. For this particular definition of Z, this means $\forall x,y\in\mathbb{N}\,x+y=x+y$, which is obviously true.