Question

2. For each occurrence of a variable in the following formula, say whether that occurrence is free or bound. Is the formula open or closed?

∀xPx → ∃y(Qx∧Ra)

Answer 1

In simple words bound variables are those which are introduced in the formula means they are introduced by some quantifiers and those variables which comes from outside are called free variables.

Now take ∀xPx → ∃y(Qx∧Ra)

here, x is bound by '∀' quantifier and y is bound by '∃' quantifier, hence they are bound variables . But Ra, here the variable 'a' comes from outside and not bound by any quantifier, hence 'a' is free variable.

Now we know that a sentence is closed if it has no free variables.

Here 1 variable is free so it is open sentence.