Free and Bound Variables

When a quantifier is used on the variable $x$, we say that this occurrence of the variable is $\text{bound}$

An occurrence of a variable that is not bound by a quantifier or set equal to a particular value is said to be $\text{free}$

If all the variable that occur in a predicate function are $\text{bound}$, then that is called a $\text{sentence}$ of predicate logic

This can be done using a combination of universal quantifiers, existential quantifiers, and value assignments

The part of a logical expression to which a quantifier is applied is called the $\text{scope}$ of this quantifier. Consequently, a variable is free if it is outside the scope of all quantifiers in the formula that specify this variable

For Example

1. $\mathrm{\exists }x(P(x)\wedge Q(x))\vee \mathrm{\forall }xR(x)$
   • The variable x is bound. It occurs within the scope of $\mathrm{\exists }$ in the first part of the disjunction, and within in the scope of $\mathrm{\forall }$ in the second part
2. $\mathrm{\forall }x(P(y)\wedge Q(x))\vee \mathrm{\forall }yR(y)$
   • The variable $x$ is bound. It occurs within the scope of $\mathrm{\exists }$ in the first part of the disjunction. The variable $y$ occurs both as free ( In the first part ) and as bound ( Within the scope of $\mathrm{\forall }$ in the second part )