Example for which the relation does not stand

[evinda]

Hello! (Smile)
It stands that $R[A \cap B] \subset R[A] \cap R$, since:

$$y \in R[A \cap B] \rightarrow \exists x \in A \cap B: xRy \rightarrow \exists x(x \in A \wedge xRy) \wedge (x \in B: xRy) \rightarrow y \in R[A] \wedge x \in R \rightarrow y \in R[A] \cap R$$

But, it doesn't stand, that: $R[A] \cap R \subset R[A \cap B]$. Could you give me an example, for which the last relation does not stand? (Thinking)

 

[Evgeny.Makarov]

Let $R$ be a non-injective function so that $R(x_1)=R(x_2)$ where $x_1\ne x_2$. Take $A=\{x_1\}$ and $B=\{x_2\}$.