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Draw the digraph and the matrix of the relation R= {(1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3,4), (4, 1), (4, 2), (4, 3)} on the set A= {1, 2, 3, 4, 5}. Also decide whether it is reflexive,

whether it is symmetric, whether it is anti symmetric,whether it is transitive.

Construct a formal proof of validity for the following argument. [10]

1. (N v O) → P

2. (P v Q) → R

3. Q v N

4. ¬ Q

∴ R

For propositions p, q, and r, determine whether 𝑝 ⟶ (𝑞 ⟶ 𝑟) and (𝑝 ⟶ 𝑞) ⟶ 𝑟 are logically equivalent.

If R, S and T are relations over the set A, then: Prove that (S∩T)∘R= (S∘R)∩(T∘R).

Let R be a relation from the set A to the set B, then: Prove that Ran (R)=Dom (R-1 ). 

 Let R be a relation over set A, then: Prove that IA∘R=R=R∘IA. 

 If R, S and T are relations over the set A, then: Prove that If R⊆S, then T∘R ⊆ T∘S and R∘T ⊆ S∘T

A bank pays you 4.5% interest per year. In addition, you receive |100 as bonus at

the end of the year (after the interest is paid). Find a recurrence for the amount of

money after n years if you invest |2000.

Assignment 1

Due: 6th April 2021 at 5.00 p.m.

Total: 70 marks

1. Using the theorem divisibility, prove the following

a) If a|b , then a|bc ∀a,b,c∈ℤ ( 5 marks)

b) If a|b and b|c , then a|c (5 marks)

2. Using any programming language of choice (preferably python), implement the following algorithms

a) Modular exponentiation algorithm (10 marks)

b) The sieve of Eratosthenes (10 marks)

3. Write a program that implements the Euclidean Algorithm (10 marks)

4. Modify the algorithm above such that it not only returns the gcd of a and b but also the Bezouts coefficients x and y, such that 𝑎𝑥+𝑏𝑦=1 (10 marks)

5. Let m be the gcd of 117 and 299. Find m using the Euclidean algorithm (5 marks)

6. Find the integers p and q , solution to 1002𝑝 +71𝑞= 𝑚 (5 marks)

7. Determine whether the equation 486𝑥+222𝑦=6 has a solution such that 𝑥,𝑦∈𝑍𝑝 If yes, find x and y. If not, explain your answer. (5 marks)

8. Determine integers x and y such that 𝑔𝑐𝑑(421,11) = 421𝑥 + 11𝑦. (5 marks)

a)   Determine whether  are equivalent without using truth table.

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