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:
my question is what is the meaning of 'free' and how this answer came up to empty.
On
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.
Free variables are variables that are not bound by a quantifier. The opposite of a free variable is a bound variable. Consider the statement "a divides b" in first order logic: $\exists x(a = b \times x)$. The variable $x$ is bound while $a$ and $b$ are free. In a sense the bound variable is really just a dummy variable, they are not key to the mathematical statement. The Mathematical statement $a$ divides $b$ is a statement about $a$ and $b$ as opposed to $x$. Informally you can think of bound variables as tools to help us describe the free or "meaningful" variables. We may change our $x$ to some variable, for example $\exists y(a = b \times y)$ is identical to the previous statement. But if we were to change our free variables the meaning of the statement would be fundamentally different. I would like to answer your second question but I cannot understand what was written on the second line as I do not know if $x$ and $y$ represent real numbers or some other mathematical objects.