# Math3066 algebra and logic semester 1 2014 second assignment

Don't use plagiarized sources. Get Your Assignment on
Math3066 algebra and logic semester 1 2014 second assignment
Just from \$13/Page

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)

Calculate the price
Make an order in advance and get the best price
Pages (550 words)
\$0.00
*Price with a welcome 15% discount applied.
Pro tip: If you want to save more money and pay the lowest price, you need to set a more extended deadline.
We know how difficult it is to be a student these days. That's why our prices are one of the most affordable on the market, and there are no hidden fees.

Instead, we offer bonuses, discounts, and free services to make your experience outstanding.
How it works
Receive a 100% original paper that will pass Turnitin from a top essay writing service
step 1
Upload your instructions
Fill out the order form and provide paper details. You can even attach screenshots or add additional instructions later. If something is not clear or missing, the writer will contact you for clarification.
Pro service tips
How to get the most out of your experience with Homework Writing Services
One writer throughout the entire course
If you like the writer, you can hire them again. Just copy & paste their ID on the order form ("Preferred Writer's ID" field). This way, your vocabulary will be uniform, and the writer will be aware of your needs.
The same paper from different writers
You can order essay or any other work from two different writers to choose the best one or give another version to a friend. This can be done through the add-on "Same paper from another writer."
Copy of sources used by the writer
Our college essay writers work with ScienceDirect and other databases. They can send you articles or materials used in PDF or through screenshots. Just tick the "Copy of sources" field on the order form.
Testimonials
See why 20k+ students have chosen us as their sole writing assistance provider
Check out the latest reviews and opinions submitted by real customers worldwide and make an informed decision.
Economics
I would like to thank you for your hard work. The essay is clear and clean.
Customer 462687, March 14th, 2022
Retail
Excellent work! This submission was informative and well-structured. We look forward to your future contributions.
Customer 452441, August 16th, 2022
MGMT 221 Introduction to Management Information System
Great job!
Customer 457731, February 25th, 2022
Business
Excellent!
Customer 463469, October 17th, 2022
Other
The writer did an incredible job on the ppt presentation this week. I am disappointed with support for the lack of communication between me and the writer which caused the paper to be late. Overall, the ppt presentation was done and the writer did a phenomenal job.
Customer 452455, February 9th, 2023
Business and administrative studies
Thank you for the corrections. This is what I expected my paper to be.
Customer 460555, November 22nd, 2021
Religious studies
This was well done thank you
Customer 452763, December 10th, 2021
Entertainment & Gaming
Great work.
Customer 452441, April 29th, 2022
Nursing
Excellent work. Had a few issues but tutor stayed in contact and produced a succinct and quality paper.
Customer 453939, April 26th, 2020
Business and administrative studies
Be keen on plagiarism and grammar. Also, always follow the instructions' check-list.
Customer 453509, April 10th, 2022
Business and administrative studies
Good one
Customer 452615, July 16th, 2022
Social Work and Human Services
you are a great group of people.
Customer 463783, February 10th, 2023
11,595
Customer reviews in total
96%
Current satisfaction rate
3 pages
Average paper length
37%
Customers referred by a friend
OUR GIFT TO YOU
15% OFF your first order
Use a coupon FIRST15 and enjoy expert help with any task at the most affordable price.
Claim my 15% OFF Order in Chat