Your Perfect Assignment is Just a Click Away

Starting at $8 per Page

100% Original, Plagiarism Free, Customized to Your instructions!






Semester 1

First 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 Thursday 17 April. Acknowledge any sources or assistance.

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


(P ∨ Q) ∧ R


(P ∧ R ) ∨ Q

Use your tables to explain briey why

(P ∨ Q) ∧ R


(P ∧ R ) ∨ Q ,

(P ∧ R ) ∨ Q


(P ∨ Q) ∧ R .


(6 marks)

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

the sense of always being true).



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 nd formal proofs for the following sequents:


P ⇒ (Q ⇒ R ) , ∼ R


(P ∨ Q) ∧ (P ∨ R )

P ∨ (Q ∧ R )


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:


P1 ∨ P2 ∧ P3 ∨ P1 ∧ P2

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


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













and Z14 , or explain briey 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: rst interpret this equation in Zn for an appropriate n.]

(10 marks)

"Place your order now for a similar assignment and have exceptional work written by our team of experts, guaranteeing you A results."

Order Solution Now

Our Service Charter

1. Professional & Expert Writers: Eminence Papers only hires the best. Our writers are specially selected and recruited, after which they undergo further training to perfect their skills for specialization purposes. Moreover, our writers are holders of masters and Ph.D. degrees. They have impressive academic records, besides being native English speakers.

2. Top Quality Papers: Our customers are always guaranteed of papers that exceed their expectations. All our writers have +5 years of experience. This implies that all papers are written by individuals who are experts in their fields. In addition, the quality team reviews all the papers before sending them to the customers.

3. Plagiarism-Free Papers: All papers provided by Eminence Papers are written from scratch. Appropriate referencing and citation of key information are followed. Plagiarism checkers are used by the Quality assurance team and our editors just to double-check that there are no instances of plagiarism.

4. Timely Delivery: Time wasted is equivalent to a failed dedication and commitment. Eminence Papers is known for timely delivery of any pending customer orders. Customers are well informed of the progress of their papers to ensure they keep track of what the writer is providing before the final draft is sent for grading.

5. Affordable Prices: Our prices are fairly structured to fit in all groups. Any customer willing to place their assignments with us can do so at very affordable prices. In addition, our customers enjoy regular discounts and bonuses.

6. 24/7 Customer Support: At Eminence Papers, we have put in place a team of experts who answer to all customer inquiries promptly. The best part is the ever-availability of the team. Customers can make inquiries anytime.