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(Solved): 1. Let R be the relation on Z(Z{0}) defined by R={((x1,y1),(x2,y2)):x1y2 ...

$1. Let \( \mathcal{R} \) be the relation on \( \mathbb{Z} \times(\mathbb{Z}-\{0\}) \) defined by \[ \mathcal{R}=\left\{\left($

1. Let

R

be the relation on

Z \times (Z ? {0})

defined by

R = {((x_{1}, y_{1}), (x_{2}, y_{2})) : x_{1} ? y_{2} = x_{2} ? y_{1}}

Prove that

R

is an equivalence relation on

Z \times (Z ? {0})

. Proof:

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x1y1=y1x1?(x1,y1)?(x1,y1)When we focus on left side of the biconditional, the relation ?is

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