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ITS 66204

DISCRETE STRUCTURES

FINAL ASSESSMENT

AUGUST 2022 SEMESTER

SECTION A

Structured questions – 100 marks

Answer all questions.

QUESTION 1 (20 Marks)

a) Rewrite the following statement in if-then form.

i. Achieve a grade of A- in this module is a sufficient condition for it to count towards scholarship. (1 Mark)

ii. Being on time each day is a necessary condition for getting company’s reward. (1 Mark)

iii. Having a large income is not necessary condition for a person to be aloof. (1 Mark)

b) Use Logical Equivalence Theorem to verify the logical equivalences below. ((¬q ∨ ¬p) ∧ (¬q ∨ p)) ∨ (¬p ∧ q) ≡ ¬(p ∧ q) (11 Marks)

c) A set of premises and a conclusion are given below. Use the valid arguments forms to deduce the conclusion from the premises, giving a reason for each step. Assume all variables are statement variables.

p → q

→ t ∨ s

~q ∨

~s ∧ p

∴ t (6 Marks)

QUESTION 2 (20 marks)

a) Use the method of proof by contradiction to prove the following statement.

i. Prove that for any irrational number k, the square root of k is irrational number. (3 Marks)

ii. Prove that ∀ integer p, q and r , if p ∤ qr then p ∤ q . (3 Marks)

iii. Prove that ∀ odd integers r and s , s 2 − r 2 ≠ 6. (5 Marks)

b) Find counterexample to show that the statements are false.

i. For every integer x and y , if x is odd or y is odd then x + y is odd. (3 Marks)

ii. ∀ real number a and b , ⌈a + b⌉ = ⌈a⌉ + ⌈b⌉ . (3 Marks)

iii. ∀ integer x, y and z , if x is a factor of z and y is a factor of z , then xy is a factor of z . (3 Marks)

QUESTION 3 (20 Marks)

a) Prove the following statement by using induction method. For any integer n ≥ 2, ∑j(n)1(1)j(j + 1) = . (7 Marks)

b) Let b0, b1, b2, … . Be the sequence defined by the explicit formula b0 = A ∙ 2n + B for

each integer n ≥ 0, where A and B are real number.

i. Show that for any choice of A and B , bk = 3bk − 1 − 2bk − 2 for every integer

k ≥ 2. (4 Marks)

ii. Find the value of A and B for bk = 3bk − 1 − 2bk − 2 , k ≥ 2 with initial conditions b0 = 1 and b1 = 3. (5 Marks)

c) Let M = {k ∈ ℕ: 4 ≤ k < 12} and N = {k ∈ ℕ: k = 2n} .

i. Find M ∩ N

ii. Find M\N (4 Marks)

QUESTION 4 (20 Marks)

a) Let P be the set of all Malay statements. A relation on I is defined on P as follows: For each a, b ∈ P , aIb ⇔ a → b is true. Determine whether the relation is reflexive, symmetric and transitive. (6 Marks)

b) If a Bank X pays interest at a rate of 2% per year compounded annually and An denotes the amount in the account at the end of year n, then Ak = (2.04)Ak−1 , for each integer k ≥ 1, assuming no deposits or withdrawals during the year. Suppose the initial amount deposited is RM 100,000, and assume that no additional deposits or withdrawals are made.

i. How much will the account be worth at the end of 10 years?

ii. How many years will the account be worth RM 1,000,000? (5 Marks)

c) By using the binomial expansion, find the constant term for the expression (2y − )6 . (9 Marks)