THE UNIVERSITY OF SYDNEY

MATH3066 ALGEBRA AND LOGIC

Semester 1

First 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 Thursday 17 April. Acknowledge any sources or assistance.

1. Construct truth tables for each of the following wﬀs:

(a)

(P ∨ Q) ∧ R

(b)

(P ∧ R ) ∨ Q

Use your tables to explain brieﬂy why

(P ∨ Q) ∧ R

|=

(P ∧ R ) ∨ Q ,

(P ∧ R ) ∨ Q

|=

(P ∨ Q) ∧ R .

but

(6 marks)

2. Use truth values to determine which one of the following wﬀs is a theorem (in

the sense of always being true).

(a)

(b)

P ⇒ Q⇒R

⇒

P ⇒Q ⇒R

P ⇒Q ⇒R ⇒ P ⇒ Q⇒R

For the one that isn’t a theorem, produce all counterexamples. For the one

that is a theorem, provide a formal proof also using rules of deduction in the

Propositional Calculus (but avoiding derived rules of deduction).

(8 marks)

3. Use the rules of deduction in the Propositional Calculus (but avoiding derived

rules) to ﬁnd formal proofs for the following sequents:

(a)

P ⇒ (Q ⇒ R ) , ∼ R

⊢

(b)

(P ∨ Q) ∧ (P ∨ R )

P ∨ (Q ∧ R )

(c)

P ∨ (Q ∧ R ) ⊢ (P ∨ Q) ∧ (P ∨ R )

⊢

P ⇒∼Q

(12 marks)

4. Let W = W (P1 , . . . , Pn ) be a proposition built from variables P1 , . . . , Pn . Say

that W is even if

W ≡ W ( ∼ P1 , ∼ P2 , . . . , ∼ Pn ) .

Say that W is odd if

W ≡ ∼ W ( ∼ P1 , ∼ P2 , . . . , ∼ Pn ) .

(a) Use truth tables to decide which of the following are even or odd:

(i) W = (P1 ⇔ P2 )

(ii) W = (P1 ⇔ P2 ) ⇔ P3

(b) Use De Morgan’s laws and logical equivalences to explain why the following

proposition is odd:

W=

P1 ∨ P2 ∧ P3 ∨ P1 ∧ P2

(c) Explain why the number of truth tables that correspond to propositions

n

n −1

in variables P1 , . . . , Pn is 22 , and, of those, 22

tables correspond to

2 n −1

tables correspond to odd propositions.

even propositions, and 2

(16 marks)

5. Evaluate each of

in Z11

3

9

10

1

,

,

,

,

5

7

10

9

and Z14 , or explain brieﬂy why the given fraction does not exist.

(8 marks)

6. Prove that the only integer solution to the equation

x2 + y 2 = 3 z 2

is x = y = z = 0.

[Hint: ﬁrst interpret this equation in Zn for an appropriate n.]

(10 marks)

Basic features

- Free title page and bibliography
- Unlimited revisions
- Plagiarism-free guarantee
- Money-back guarantee
- 24/7 support

On-demand options

- Writer’s samples
- Part-by-part delivery
- Overnight delivery
- Copies of used sources
- Expert Proofreading

Paper format

- 275 words per page
- 12 pt Arial/Times New Roman
- Double line spacing
- Any citation style (APA, MLA, Chicago/Turabian, Harvard)

We value our customers and so we ensure that what we do is 100% original..

With us you are guaranteed of quality work done by our qualified experts.Your information and everything that you do with us is kept completely confidential.

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read moreThe Product ordered is guaranteed to be original. Orders are checked by the most advanced anti-plagiarism software in the market to assure that the Product is 100% original. The Company has a zero tolerance policy for plagiarism.

Read moreThe Free Revision policy is a courtesy service that the Company provides to help ensure Customer’s total satisfaction with the completed Order. To receive free revision the Company requires that the Customer provide the request within fourteen (14) days from the first completion date and within a period of thirty (30) days for dissertations.

Read moreThe Company is committed to protect the privacy of the Customer and it will never resell or share any of Customer’s personal information, including credit card data, with any third party. All the online transactions are processed through the secure and reliable online payment systems.

Read moreBy placing an order with us, you agree to the service we provide. We will endear to do all that it takes to deliver a comprehensive paper as per your requirements. We also count on your cooperation to ensure that we deliver on this mandate.

Read more
The price is based on these factors:

Academic level

Number of pages

Urgency