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Exam Solutions - UNIT SIT192 Discrete Mathematics 2008...

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Unformatted text preview: UNIT: SIT192 Discrete Mathematics 2008 Semester 3 1 Attempt all questions . 1. (a) The propositions p , q , r , and s are defined by: Express the following in symbolic form: (i) p ∧ ¬ ( s ∧ r ) (ii) ( q ∨ r ) → ¬ s (iii) p ↔ ( q ∧ r ) (iv) ¬ ( q ∧ s ) → p (b) (c) (i) (ii) L.H.S. ≡ q ∧ ¬ ( ¬ p ∨ q ) ≡ q ∧ ( p ∧ ¬ q ) ≡ ( q ∧ ¬ q ) ∧ p ≡ F ∧ p ≡ F (d) (i) True: Solving ( x- y ) 2 = x 2 + 3 x + y 2 , we get- 2 xy = 3 x , which implies that y =- 3 / 2. (ii) False: Solving ( x- y ) 2 =- 2 xy + y 2 , we get x 2 = 0, which is not true for all x . UNIT: SIT192 Discrete Mathematics 2008 Semester 3 2 Note: in each case, provide a reason for the answer. [6+5+7+6=24 marks] 2. The universal set U is given by U = { x ∈ N : 20 ≤ x ≤ 35 } , where N = { 1 , 2 , 3 , ... } is the set of all natural numbers. Given that A = { x ∈ U : x is a multiple of 4 } ; B = { x ∈ U : x ≤ 25 } ; and C = { x ∈ U : x is a multiple of 6 } , (a) list the elements of the following set: i. A = { 20 , 24 , 28 , 32 } ii. B = { 20 , 21 , 22 , 23 , 24 , 25 } iii. C = { 24 , 30 } iv. B ∩ C = { 24 } v. A- C = { 20 , 28 , 32 } vi. A ∪ B ∪ C = { 20 , 21 , 22 , 23 , 24 , 25 , 28 , 30 , 32 } vii. A ⊕ C = { 20 , 28 , 30 , 32 } viii. ( B- A ) ∩ C = ∅ ix. B ∩ A ∩ C = { 24 } (b) No, because 28 is a number, not a set, so it is not a subset of another set....
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Exam Solutions - UNIT SIT192 Discrete Mathematics 2008...

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