THE UNIVERSITY OF SYDNEY

MATH3066 ALGEBRA AND LOGIC

Semester 1

2014

Second Assignment

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 Predicate 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))

(c)

(∀x)(∀y)(∃z) R(x, z) ∧ R(y, z)

(d)

(∀x)(∀y)(∀z) R(x, y) ∧ R(y, z) ⇒ R(x, z) ,

⊢

(∃x) ∼ F (x) ⇒ (∃x) ∼ G(x)

⊢ (∀x)(∃y) R(x, y)

(∀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

W2

W3

=

=

=

(∃x)(∃y) R(x, y)

(∀x)(∀y) R(x, y) ⇒ ∼ R(y, x)

(∀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]/(x2 − 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]/(x2 + 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)