Image of a binary relation

Question

I know it might seem a stupid question but I need some clarification. From my notes:

"a binary relation $R$ associated to each element $x$ of $X$ some elements $y$ of $X$. We denote by $R(x) = \{y \in X : x R y\}$ the image of $x$ through $R$"

but then in an example:
(the relation was $R = \geq, \subseteq \mathbb{N} \times \mathbb{N}$ st $R = \{(x,y) \in \mathbb{N}\times \mathbb{N}: x\geq y \}$
the image is: $R(x) = \{y \in \mathbb{N}: y\geq x \}$

Shouldn't it be $R(x) = \{y \in \mathbb{N}: x\geq y \}$? Why not?

Answer 1

In the example if your binary relation on $\mathbb{N}$ is $xRy\iff x\geq y$, then according your definiton $R(x) = \{y \in X : x R y\}=\{y \in \mathbb{N}: x \geq y\}$. If it is $xRy\iff y\geq x$, then $R(x) = \{y \in X : x R y\}=\{y \in \mathbb{N}: y \geq x\}$