0 such that a structure A is a model of ϕn iff A contains at least

WISKUNDIGE LOGICA (2016-2017) HOMEWORK 2

• Deadline: Monday, March 14 — at the beginning of class.

• Homework can also be submitted electronically (in a single pdf-file!) to Jolien Oomens).

• Grading is from 0 to 100 points.

• Success!

(1) (20pt) Let A and B be S-structures, and let π : A ∼= B. Let β be an assignment on Aand assume that β^π = π ◦ β. Show that for every S-term t we have π(I(t)) = I^π(t), where I = (A, β) and I^π = (B, β^π).

(2) (20pt)

(a) Write a formula ϕ_n for n ∈ N and n > 0 such that a structure A is a model of ϕn iff A contains at least (!) n points.

(b) Write a formula ψ_n for n ∈ N and n > 0 such that a structure A is a model of ψ_n iff A contains at most (!) n points.

(c) Write a formula χ_n for n ∈ N and n > 0 such that a structure A is a model of χ_n iff A contains exactly (!) n points.

(3) (20pt) A set Φ of sentences is called independent if there is no ϕ ∈ Φ such that Φ \ {ϕ} |= ϕ. Show that the set Φ_eq of axioms (reflexivity, symmetry, transitivity) for equivalence relations is independent.

(4) (20pt) Let S = {E} with E a binary relation symbol. An S-Structure G = (G, E^G) is called a graph if it is a model of

Φ_graph = {∀v₀¬Ev₀v₀, ∀v₀∀v₁(Ev₀v₁ ↔ Ev₁v₀)}

(Intuitively: A graph consists of a set G; two points a 6= b are related iff (a, b) ∈ E^G).

(a) Are the following two graphs isomorphic ?

• • • • •

• • • • • • •

G₁ G₂

(b) Does Gi |= ϕj, for i = 1, 2 and j = 1, 2, 3?

1

2 WISKUNDIGE LOGICA (2016-2017) HOMEWORK 2

(i) ϕ₁ = ∃v₀∃v₁∃v₂((Ev₀v₁∧ Ev₁v₂) ∧ Ev₂v₀) (ii) ϕ₂ = ∀v₀∀v₁(¬Ev₀v₁ → ∃v₂(Ev₀v₂∧ Ev₂v₁))

(iii) ϕ₃ = ∀v₀∃v₁∃v₂∃v₃∃v₄((Ev₀v₁∧ Ev₀v₂) ∧ (Ev₀v₃∧ Ev₀v₄)) (c) Show that the following two graphs are isomorphic:

A • • B •

a

• b

C • E • G •

• F

• H

• D

•

e •

f

• c

• g

• d

• h

(5) (20pt) Let S be a finite symbol set.

(a) Let A be a finite S-structure. Show that there is an S-sentence ϕ_A the models of which are precisely the S-structures isomorphic to A.

(b) We write A ≡ B if A and B satisfy the same S-sentences. If A is finite, does A≡ B imply A ∼= B? Justify your solution.