Composition of sets
Question: Let the set $A$ be defined as $A = \{ a, b, c, d \}$, and let the relations $R$ and $S$ on the set $A$ be defined as
$R = \{(d, a), (a, b), (b, c)\}$, and $S = \{(a, a), (b, d), (d, c)\}$.
Explain why the ordered pair $(a, b)$ is or is not an element of the composition of $S$ and $R$ (denoted $R \circ S$).
I would think $(a,b)$ is not in composition $R\circ S$ because there does not exist an element $x$ of $A$ such that $aSx$ and $xRa$.
Is my thinking correct? How else would you elaborate that it is or is not in composition $R\circ S$?
$\endgroup$2 Answers
$\begingroup$If $(a,b)$ is in $R\circ S$, then there is an $x\in A$, such that $aSx$ and $xRb$ (check if you have a typo in the last bit).
But there is such an $x$. I can take $x=a$. I have that $aSa$, and that $aRb$. Hence $a(R\circ S)b$.
$\endgroup$ $\begingroup$You may think on $R\circ S$ as the relation "first $S$, then $R$". Now, $S$ sends $a$ to $a$ and $R$ sends $a$ to $b$. You can conclude the result now?
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