In the statement $Q(x)\wedge \forall x~P(x)$, the term $x$ occurs free within the statement, even though it also occurs bound to a quantifier.
" When a quantifier is used on the variable x, we say that this occurrence of the variable is bound. An occurrence of a variable that is not bound by a quantifier or set equal to a particular value is said to be free"
The second occurrence of the term $x$ is bound by use of the universal quantifier. However the first occurrence of the term is not bound by this quantifier, and so it occurs free.
"The part of a logical expression to which a quantifier is applied is called the scope of this quantifier. Consequently, a variable is free if it is outside the scope of all quantifiers in the formula that specifies this variable."
The scope of the universal quantifier in the statement contains exactly the predicate $P(x)$. Hence $x$ is free in this statement as it does occur outside this scope; that is within the predicate $Q(x)$.
There is no contradiction.
The term following the quantifier sign indicates that the quantifier binds that term, and immediately following this is the scope of the quantifier (see "order of precedence" for clarification of where the scope lies ). Occurrences of this term within this scope are bound. Occurrences outside the scopes of all quantifiers which bind the term are free.
Also tell me what will be the free and bound variables of the statement $\exists x~(x + y = 1)$ and explain me with the proper definition of free and bound variable.
All occurrences of the terms are within the scope of the existential quantifier, however, it does not bind $y$, it only binds $x$.
So, because $x$ is bound by the existential quantifier and all occurrences of this term do occur within the scope of this quantification; therefore $x$ does not occur free within the statement.
However, $y$ is not bound by any quantifier, so the occurrence of $y$ is outside the scope of all quantifiers which bind $y$; therefore $y$ occurs free within the statement.
