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I'm looking to formally define a tree and then extract the leaves from it in a concise way. Does this look ok? What is the best way of doing this?

$ Y = \{a,b,c,d,e,f,g\} \\ R = \{a \mapsto b, a \mapsto d, d \mapsto e, d \mapsto f, f \mapsto g\} \ \text{, where R is a relation on Y.} \\ R^+ = \{a \mapsto b, a \mapsto d, d \mapsto e, d \mapsto f, a \mapsto e, a \mapsto f, a \mapsto g\} \ \text{, where $R^+$ is the transitive closure of R.} \\ leaves = \{x \in range(R^+) \mid \ x \notin dom(R^+) \} $

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  • $\begingroup$ why you need $R^+$? $\endgroup$
    – Inuyasha Yagami
    Commented Aug 20, 2021 at 5:23
  • $\begingroup$ To be honest im not sure why I did that, its redundant $\endgroup$
    – newlogic
    Commented Aug 20, 2021 at 18:48
  • $\begingroup$ Also i made a typo it should be x and not R(x) $\endgroup$
    – newlogic
    Commented Aug 20, 2021 at 18:51

1 Answer 1

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How about this: $leaves = Y \setminus Preimage(R)$

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  • $\begingroup$ Is this the same as $leaves = \{x \in range(R) \mid \ x \notin dom(R) \} $ $\endgroup$
    – newlogic
    Commented Aug 20, 2021 at 18:50
  • $\begingroup$ @newlogic No, it is different. Domain of $R$ is basically the entire set $Y$. Whereas, preimage of $R$ is a subset of the domain; it contains those elements that actually participate in the relation $R$. See this for a formal definition. $\endgroup$
    – Inuyasha Yagami
    Commented Aug 21, 2021 at 4:05

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