TITLE: What does a 'free variable' mean? // Quantifiers
QUESTION [1 upvotes]: I've researched about the Quantifiers but couldn't find any material that explains 'free variables'.
Question:
>  Determine the truth set of:

>                             ∀y (x . y < x) 

>   where x,y are from the universe of real numbers.

Answer:
Answer picture
my question is what is the meaning of 'free' and how this answer came up to empty.

REPLY [0 votes]: I intend to give a heuristic instead of a pedantic illustration, so some terminologies may not be that standard in some logicians eyes.
Sometimes a proper example may explain more than a description. Consider the expression 
$$
\sum_{i=1}^{n}5^{i}.
$$
Here the role of "$i$" is vacuous in the sense that you can replace "$i$" with whatever symbol you want without affecting the value of the sum. However, the role of "$n$" is different from that of "$i$"; for different values of $n$ affect the value of the sum. Here "$n$" is a free variable and "$i$" is a bound variable.
Regarding a logic context, consider the statement "There is some $x > 0$ such that $x=2y$". Here you may have noticed that the role of "$x$" is like that of "$i$" in the sense that you can replace "$i$" with whatever symbol you want without affecting 
the truth value of the statement (assuming $y$ given). However, in that statement the symbol "$y$" is not specified, so the statement could be true if $y = 1$, say, and false if $y < 0$. Since different values of $y$ affect the truth value of the statement, it is a free variable.
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