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Consider a pure lambda calculus, $L$. Denote by $V$ $L$'s set of variables, and by $C$ $L$'s (possibly empty) set of constants. Let $P$ be a term in $L$ (a.k.a. a $\lambda$-term). Denote by $\text{FV}(P)$ the set consisting of all variables $x\in V$ that are free in $P$.

  1. Suppose $P=a$ for some constant $a\in C$. What is $\text{FV}(P)$? Is it $\emptyset$, or is it $V$?

  2. Denote by $R(P)$ the set consisting of all variables $x\in V$ that occur in $P$. Suppose $x\in V$ is such that $x\notin R(P)$. Is $x\in\text{FV}(P)$?

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    1. $\emptyset$. 2. yes. These properties should be immediate from the recursive definitions of the functions $FV$ and $R$ that you ought to have in view. – 2017-02-09
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    @RobArthan: I'm confused. Isn't 1 a special case of 2, in which case how can $\text{FV}(P)$ be empty in 1 but non-empty in 2? – 2017-02-09
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    Sorry. I misread 2. The answer to 2 is no: $FV(P) \subseteq R(P)$. – 2017-02-09

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