Understanding First-order Logic 3.3.8 133 Express the following statements in the language of the structure of real numbers LR , using the predicate I (x) to mean “x is an integer.” (a) The number (denoted) z is rational (b) Every rational number can be represented as an irreducible fraction (c) Between every two different real numbers there is a rational number √ (d) is not a rational number (e) Every quadratic polynomial with real coefficients which has a non-zero value has at most two different real zeros 3.3.9 Define by recursion on the inductive definition of formulae, for every formula in A ∈ FOR(L), the set BVAR(A) of all individual variables that are bound in A, that is, have a bound occurrence in A 3.3.10 Determine the scope of each quantifier and the free and bound occurrences of variables in the following formulae where P is a unary and Q a binary predicate (a) ∃x∀z (Q(z, y ) ∨ ¬∀y (Q(y, z ) → P (x))), (b) ∃x∀z (Q(z, y ) ∨ ¬∀x(Q(z, z ) → P (x))), (c) ∃x(∀zQ(z, y ) ∨ ¬∀z (Q(y, z ) → P (x))), (d) ∃x(∀zQ(z, y ) ∨ ¬∀yQ(y, z )) → P (x), (e) ∃x∀z (Q(z, y ) ∨ ¬∀yQ(x, z )) → P (x), (f) ∃x∀zQ(z, y ) ∨ ¬(∀yQ(y, x) → P (x)), (g) ∃x(∀zQ(z, y ) ∨ ¬(∀zQ(y, z ) → P (x))), (h) ∃x(∀x(Q(x, y ) ∨ ¬∀z (Q(y, z ) → P (x)))), (i) ∃x(∀y (Q(x, y ) ∨ ¬∀xQ(x, z ))) → P (x) 3.3.11 Rename the bound variables in each formula above to obtain a clean formula 3.3.12 Show that every formula can be transformed into a clean formula by means of several consecutive renamings of variables 3.3.13 For each of the following formulae (where P is a unary predicate and Q is a binary predicate), determine if the indicated term is free for substitution for the indicated variable If so, perform the substitution (a) Formula: ∃x(∀zP (y ) ∨ ¬∀y (Q(y, z ) → P (x))); term: f (x); variable: z (b) Same formula; term: f (z ); variable: y (c) Same formula; term: f (y ); variable: y (d) Formula: ∀x((¬∀yQ(x, y ) ∨ P (z )) → ∀y ¬∃z ∃xQ(z, y )); term: f (y ); variable: z (e) Same formula; term: g (x, f (z )); variable: z (f) Formula: ∀y (¬(∀x∃z (¬P (z ) ∧ ∃yQ(z, x))) ∧ (¬∀xQ(x, y ) ∨ P (z ))); term f (x); variable: z (g) Same formula; term: f (y ); variable: z (h) Formula: (∀y ∃z ¬P (z ) ∧ ∀xQ(z, x)) → (¬∃yQ(x, y ) ∨ P (z )); term: g (f (z ), y ); variable: z (i) Same formula; term: g (f (z ), y ); variable: x