$($Note: denote the relation, feel free to replace it with R $)$

$xy$ if an only if $x-y$ is positive

$(x,y)(x^{\text{'}},y^{\text{'}})$ if an only if $x+y^{\text{'}}=x^{\text{'}}+y$ is positive

$xy$ if an only if $x+y$ is even

$xy$ if an only if $xy0$

$xy$ if an only if $x+y$ is odd

$xy$ if an only if $x+y$ is positive