k 3.3.8 Express the following statements in the language of the structure of real num-bersL R, using the predicateIx to mean “xis an integer.” a The number denotedzis rational.. c Betwee
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3.3.8 Express the following statements in the language of the structure of real num-bersL R, using the predicateI(x) to mean “xis an integer.”
(a) The number (denoted)zis 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) √
2 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 inA, that is, have a bound occurrence inA
3.3.10 Determine the scope of each quantifier and the free and bound occurrences of
variables in the following formulae wherePis a unary andQa 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 (whereP is a unary predicate andQis 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)));
termf(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