First-order logic exercises

Math 406

2004.11.02

Problem 1. Letting P and Q be unary predicates, determine, from the definition of ⊧, whether the following hold:

(a) ∃x P x → ∃x Qx ⊧ ∀x (P x → Qx);

(b) ∀x P x → ∃x Qx ⊧ ∃x (P x → Qx);

(c) ∃x (P x → Qx) ⊧ ∀x P x → ∃x Qx. Problem 2. Let L = {R}, where R is a binary predicate, and let 𝔄 be the L-structure (ℤ, ⩽). Determine ϕ^{𝔄} if ϕ is:

(a) ∀x_{1} (Rx_{1}x_{0} → Rx_{0}x_{1});

(b) ∀x_{2} (Rx_{2}x_{0} ∨ Rx_{1}x_{2}). Problem 3. Let L be {S, P }, where S and P are binary function-symbols. Then (ℝ, +, ·) is an L-structure. Show that the following sets and relations are definable in this structure:

(a) {0};

(b) {1};

(c) {a ∈ ℝ : 0 < a};

(d) {(a, b) ∈ ℝ^{2} : a < b}. Problem 4. Show that the following sets are definable in (ω, +, ·, ⩽, 0, 1):

(a) the set of even numbers;

(b) the set of prime numbers.