MATH3066 ALGEBRA AND LOGIC Semester 1 2014 Second Assignment

THE UNIVERSITY OF SYDNEY

MATH3066 ALGEBRA AND LOGIC

Semester 1 Second Assignment 2014

This assignment comprises a total of 60 marks, and is worth 15% of the overall assessment. It should be completed, accompanied by a signed cover sheet, and handed in at the lecture on Wednesday 28 May. Acknowledge any sources or assistance.

Please note that the first question is about the Proposition Calculus (not the Pred- icate Calculus). You should find part (a) straightforward. Part (b) is difficult and optional. Students that complete it successfully may be awarded bonus marks, and there may be a prize for the best correct answer.

1. A positive well-formed formula (positive wff) in the Propositional Calculus is a well-formed formula that avoids all use of the negation symbol ∼ .

(a) Use induction on the length of a wff to prove that if W = W (P1, . . . , Pn) is a positive wff in terms of propositional variables P1, . . . , Pn, then

V (P1) = . . . = V (Pn) = T implies V (W ) = T .

(5 marks)

(b) Prove that if W = W (P1, . . . , Pn) is any wff in the Propositional Calculus such that V (P1) = . . . = V (Pn) = T implies V (W ) = T , then W is logically equivalent to a positive wff.

(optional, bonus marks)

2. Use the rules of deduction in the Predicate Calculus to find formal proofs for the following sequents (without invoking sequent or theorem introduction):

(a) (∃x)(∃y)(∀z) K(y, x, z) ⊢ (∀z)(∃y)(∃x) K(y, x, z)

(b) (∀x)(G(x) ⇒ F (x)) ⊢ (

(∃x) ∼ F (x) )

⇒ (

(∃x) ∼ G(x) )

(c) (∀x)(∀y)(∃z) (

R(x, z) ∧R(y, z) )

⊢ (∀x)(∃y) R(x, y)

(d) (∀x)(∀y)(∀z) [(

R(x, y) ∧R(y, z) )

⇒ R(x, z) ]

,

(∀x)(∀y)(∃z) (

R(x, z) ∧R(z, y) )

⊢ (∀x)R(x, x)

(21 marks)

3. Consider the following well-formed formulae in the Predicate Calculus:

W1 = (∃x)(∃y) R(x, y)

W2 = (∀x)(∀y) [

R(x, y) ⇒ ∼ R(y, x) ]

W3 = (∀x)(∀y) [

R(x, y) ⇒ (∃z) (

R(z, x) ∧R(y, z) )

]

Prove that any model in which W1, W2 and W3 are all true must have at least 3 elements. Find one such model with 3 elements.

(6 marks)

4. Let R = Z[x] and I = 2Z+ xZ[x] ,

the subset of R consisting of polynomials with integer coefficients with even constant terms. Verify that I is an ideal of R. Show that I not a principal ideal.

(8 marks)

5. Let R = Z3[x]/(x 2 − x− 1)Z3[x], so we may write

R = { 0 , 1 , 2 , x , x+ 1 , x+ 2 , 2x , 2x+ 1 , 2x+ 2 } ,

where we identify equivalence classes with remainders after division by the polynomial x2 −x− 1. Then R is a commutative ring with identity. Construct the multiplication table for R and use it to explain why R is a field. Now find a primitive element, that is, an element a ∈ R such that all nonzero elements of R are powers of a.

(8 marks)

6. In each case below, if it helps, you may identify the ring with remainders after division by x2 + x + 1, so that the elements become linear expressions of the form a+ bx where a, b come from Z3 in part (a) or from R in part (b).

(a) Explain why R = Z3[x]/(x 2 + x+ 1)Z3[x] is not a field.

(b) Prove that F = R[x]/(x2 + x + 1)R[x] is isomorphic to C, the field of complex numbers.

(12 marks)