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I am thinking that many example only takes free variable with one occurrence, such as $(\lambda x.x )y$ where $y$ is free.

can we say $y y $ is a $\lambda$-term and two $y$ are same free variable? I did not see such examples so far, just wondering.

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A possible definition (From Lambda-Calculus and Combinators):

Definition 1.11 (Free and bound variables) An occurrence of a variable $x$ in a term $P$ is called:

  • bound if it is in the scope of a $\lambda x$ in $P$

  • bound and binding, iff it is the $x$ in $\lambda x$

  • free otherwise

Let's define the set of free variables $FV$:

  • The free variables of $x$ are just $x$, that is $FV(x) = \{x\}$
  • $FV(\lambda x. P) = FV(P) \setminus \{x\}$ (since $x$ is bound)
  • $FV(st) = FV(s) \cup FV(t) $

And maybe you just need some examples:

  • $y$ is free in $y$

  • $x$ is free in $(\lambda x.x)x$ (Try to work out why)

  • $y$ is free in $(yy)$.

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    $\begingroup$ so I am right is that a single free variable could have many occurrences, such as $yyyyy$ with a single free variable $y$. Thank you! $\endgroup$ – alim Jan 27 '17 at 2:10
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    $\begingroup$ Yes, since $yyyyyy$ is just $((((yy)y)y)y)$. Which makes $FV(yyyyyy) = \{y\}$. $\endgroup$ – Aristu Jan 28 '17 at 0:47

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