MHB Example for which the relation does not stand

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The discussion explores the relationship between the ranges of sets under a relation R, specifically addressing the inclusion $R[A \cap B] \subset R[A] \cap R[B]$. It is established that this inclusion holds true, but the reverse inclusion $R[A] \cap R[B] \subset R[A \cap B]$ does not necessarily hold. An example is provided where R is a non-injective function, with distinct elements x1 and x2 mapping to the same output. By defining sets A and B as containing x1 and x2 respectively, the example illustrates the failure of the reverse inclusion. This highlights the complexities of relations in set theory.
evinda
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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)
 
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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 $B=\{x_2\}$.
 

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