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For $x\in X$ we define

$R\left( x\right)=\left\{ y\in Y:xRy\right\}=\left\{ y\in Y:\left( x,y\right) \in R\right\}$; and for a subset $A\subset X$, define

$R\left( A\right) =\left\{ y\in Y:\exists x\in A, xRy\right\}$.

My question is Why we use symbol of $R(x)$ and $R(A)$? I.e., what is the these mean? I.e., if $x$ is related to $y$ by $R$ then we use $R(x)$ or if $y$ is related to $x$ by $R$ then we use $R(y)$, right? So, $R(A)$?

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    $xRy$ means that "$x$ is related to $y$"; $R(x)$ is the set of all elements of $Y$ that are related to $x$. – 2017-01-10
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    $R(A)$ is the set of elements of $Y$ that are related to some element of $A$. – 2017-01-10
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    if $xRy$ is "$x$ is father of $y$", then $R(x)$ is the set of all sons of $x$. – 2017-01-10

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Let's try it on example. Suppose that the realtion $R$ is $\leq$ and the universe of discourcse is the set of natuarl numbers $\mathbb{N}$. So we can write $2R3$, and we can even write $2R2$. $R(x)$, as stated in the comments means all the possible natural numbers(for our case) y, for which it is true that $xRy$. So for example $R(3) = \{3,4,5,6, ...\}$.

Suppose our subset $A$ is the set of even numbers. Than $R(A)$ is the set of all element which have some value in $leq$ them. So obviously $1$ is not in $R(A)$, But already $2$ is in $R(A)$.

If for example we take $A$ to be all number starting from $10$. Than $R(A)$ will not contain $1,2,3,4 ... ,9$.

Sorry for not writing it in comments. (lengthy)