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I've only recently started learning this topic and I'm confused about these definitions:

I found two definitions for each type of variable which seemed to make sense:

Free variable 1: In mathematics, a free variable is a variable that specifies places in an expression where substitution may take place

Free variable 2: a variable, x, is free iff the variable does not occur in the scope of a quantifier.

Bound variable 1:A bound variable is a variable that was previously free, but has been bound to a specific value or set of values called domain of discourse or universe.

Bound variable 2: a variable, x, is bound iff the variable occurs in the scope of a quantifier.

I myself prefer the second definitions. However, there are examples where I cannot decide which type of variable I'm dealing with:

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Based on the first definitions k should be a bound variable and n should be a free variable. But, based on the second ones, am I to say that they are of neither type, since a quantifier hasn't been used (If so, what kind of variable are they?)?

Amin Parvaresh
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    In some sense $\sum$ is a quantifier, since it imposes a restriction on the values $k$ can attain. This makes $k$ a bound variable; you cannot vary $k$ freely. – molarmass Aug 14 '18 at 13:17
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    Instead of quantifier, you could use the term variable-binding operator, which (apart from quantifiers $\forall$ and $\exists$) includes operators such as $\sum$, $\prod$, $\lim$, etc. – molarmass Aug 14 '18 at 13:21
  • Those "quantifier" definitions are for the "lower predicate calculus" which has no symbols like $\sum$. When you move on to less formal mathematics, and include things like $\sum$ or ${;:;}$, then you need a less formal definition as well. – GEdgar Aug 14 '18 at 13:29
  • Have a look at https://math.stackexchange.com/questions/2865409/basic-explanation-and-examples-on-mute-variables/2865458#2865458 – trying Aug 14 '18 at 15:13
  • @molarmass. Can you, please, link to exact citation/definition of term "variable-binding operator" in math. literature? – zkutch May 22 '21 at 19:16

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