DMAT 201

SECOND PUBLIC EXAMINATION

Honour School of Mathematics: Paper B1

Honour School of Mathematics and Computation: Paper B1

Honour School of Mathematics and Philosophy: Paper B1

LOGIC AND SET THEORY

Thursday, 9 June 1994, 2.30 to 5.30

1. Let δ be the formula α ↔ (β ↔ γ), where α, β, and γ are well-formed formulae in thepropositional calculus. Show that δ is logically equivalent to

(α ∧ β ∧ γ) ∨ (α ∧ ¬β ∧ ¬ γ) ∨ (¬α ∧ β ∧ ¬ γ) ∨ (¬α ∧ ¬β ∧ γ).

Hence, or otherwise, show that

(i) any formula obtained from δ by a permutation of α, β, and γ is logically equivalentto δ;

(ii) the formula α↔ (β ↔ β) is logically equivalent to α.

The propositional language L has propositional letters p1, p2, . . . and the single connective↔. For each n = 1, 2, . . . , let Sn be the set of all formulae in L that contain onlypropositional letters drawn from p1, . . . , pn.

Show that each formula in S2 is logically equivalent to one of p1, p2, p1 ↔ p1, p1 ↔ p2.

State how many equivalence classes (with respect to logical equivalence) there are in Snfor n ≥ 1, and give a representative formula for each class.

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2. Let S be a formal system for propositional calculus with the following properties:

(a) modus ponens is the only rule of inference of S;

(b) the deduction theorem holds for S;

(c) the following are theorems of S for any formulae φ and ψ:¬ψ → (ψ → φ), (¬φ→ ¬ψ)→ ((¬φ→ ψ)→ φ), ¬¬φ→ φ.

The set Γ of formulae is consistent if there is no formula θ for which Γ `S θ and Γ `S ¬ θand is inconsistent otherwise. Prove that, if Γ is a consistent set of formulae and φ is anyformula, then

(i) Γ ∪ {¬φ} is inconsistent if and only if Γ `S φ;

(ii) Γ ∪ {φ} is inconsistent if and only if Γ `S ¬φ.

Now suppose that Γ is a maximal consistent set in the sense that, for any formula φ notin Γ, Γ ∪ {φ} is inconsistent. Show that, for formulae φ and ψ,

(iii) exactly one of φ and ¬φ is in Γ;

(iv) Γ `S φ→ ψ if and only if Γ `S ¬φ or Γ `S ψ.

Deduce the existence of a valuation v such that v(γ) = T if and only if γ ∈ Γ.

Suppose that the propositional language used has countably many propositional letters,p1, p2, . . . . Show that there are only countably many maximal consistent sets of formulaethat contain the set Σ, where

Σ = {pi → pj : 1 ≤ i < j}.

3. Write down a formal system S of first-order predicate logic whose only rules of inferenceare modus ponens and generalisation.

State and prove the deduction theorem for S.

Give a proof in S of

(∀z)(P (z, z)→ Q(z))→ (∀x)((∀y)P (x, y)→ Q(x)).

[Properties of propositional calculus may be quoted without proof.]

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4. Let φ be a formula and Γ a set of formulae in a first-order language with equality. Whatis meant by saying φ is logically (universally) valid, and φ is a logical consequence of Γ?[You may assume as understood what is meant by a formula being satisfied by a valuationin an interpretation.]

Show that if ψ is logically valid and φ is a logical consequence of (the set consisting of) ψthen φ is logically valid.

Suppose that the language contains a unary predicate letter P , and that φ is a formulawhich does not involve P . Let A be the formula (∀x)φ, and let B be (∀x)(φ→ P (x))→(∀x)P (x).

(i) Show that B is a logical consequence of A.

(ii) Show that, if φ is the formula (∃y)(¬x = y), then A is not a logical consequence ofB.

(iii) Suppose that φ is a formula such that B is logically valid. Let I be an interpretationand let v be a valuation in I. Suppose further that, for every valuation w in I, P (x)is satisfied by w if and only if w(x) 6= v(x). Show that φ is satisfied by v. Deducethat A is logically valid.

Is A→ B a theorem of predicate calculus? Is B → A? Explain your answers.

5. Determine which of the following statements of set theory are true and which are false.

(i) For any sets a and b, there exists a set c such that x ∈ c if and only if x ∈ a or x ∈ b.(ii) There exists a set a which contains every set.

(iii) There exists a non-empty set a such that, for every x ∈ a, {x} ∈ a.

(iv) There exist sets a and b such that a ∈ b and b ∈ a.

[You may assume without proof the existence of the set of natural numbers with its usualproperties; you should otherwise give a careful statement of each axiom you use.]

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6. What is an inductive set? Describe how the set ω of natural numbers is defined withinset theory. Show that ω is inductive, and state and prove the Principle of MathematicalInduction.

Prove that⋃n+ = n for each natural number n, where n+ denotes the successor of n.

A Peano system is a triple 〈N,S, e 〉, where N is a set, S:N → N is a function, and e isa member of N such that

(i) e does not belong to the range of S;

(ii) S is an injection;

(iii) if e ∈ A ⊆ N and S(a) ∈ A whenever a ∈ A, then A = N .

Show that 〈ω,+, 0 〉 is a Peano system.

Prove that, for any Peano system 〈N,S, e 〉, there exists a bijection h of ω onto N suchthat h(0) = e and h(n+) = S(h(n)) for each natural number n.

7. Let A be a set with a denumerable (countably infinite) subset B, and let C be a denu-merable set with A ∩ C = ∅. Prove that card (A ∪ C) = card (A).

Explain how to construct, given a denumerable sequence (xn)n∈ω of real numbers, a realnumber y such that y 6= xn for each natural number n. Hence show that if A is adenumerable set of real numbers, then there is a denumerable set B such that B ⊆ R \A.

Justifying any statements you make about the cardinalities of specific sets, prove that theset of irrational real numbers has cardinality 2ℵ0 .

8. Let 〈X,<X 〉 be a well-ordered set, and let f : X → X be an order preserving injection.Show that x ≤X f(x) for each x ∈ X and deduce that the range of f is not an initialsegment of X.

Now let 〈Y,<Y 〉 also be a well-ordered set. Prove that either the two sets are isomorphic(as well-ordered sets) or one is isomorphic to an initial segment of the other.

Prove that the Well-Ordering Principle is equivalent to the property (T):

(T) for any cardinals λ and µ, one of λ ≤ µ and µ ≤ λ holds.

[You may assume that, for any set A, there is an ordinal α which is not dominated by A.]

DMAT 201