Exam_Solutions

Exam_Solutions - UNIT: SIT192 Discrete Mathematics 2010...

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UNIT: SIT192 Discrete Mathematics 2010 Trimester 3 1 Attempt all questions . 1. (a) (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 p ) ( q ∨ ¬ q ) ( q p ) T ( q p ) (d) (i) False: Solving ( x + y ) 2 = x 2 +3 x + y 2 , we get 2 xy = 3 x , which implies that y = - 3 / 2. (Since y = 3 / 2 is not an integers, the statement is false). (ii) False: Solving ( x - y ) 2 = 2 x 2 + y 2 , we get x 2 - 2 xy = 2 x 2 , and therefore y = - 1 2 x . (This cannot be true, and therefore the statement is false). [6+5+7+4=22 marks] 2. The universal set U is given by U = { x N : 25 x 50 } , where N = { 1 , 2 , 3 ,... } is the set of all natural numbers. Given that
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UNIT: SIT192 Discrete Mathematics 2010 Trimester 3 2 A = { x U : x is a multiple of 4 } ; B = { x U : x 30 } ; and C = { x U : x is a multiple of 7 } , (a) list the elements of the following set: i. A = { 28 , 32 , 36 , 40 , 44 , 48 } ii. B = { 25 , 26 , 27 , 28 , 29 , 30 } iii. C = { 28 , 35 , 42 , 49 } iv. B C = { 28 } v. A - ¯ C = { 28 } vi. A B C = { 28 } vii. A C = { 32 , 35 , 36 , 40 , 42 , 44 , 48 , 49 } viii. ( B - A ) C = ix. ( B A ) C = { 28 , 35 , 42 , 49 } (b) A - ¯ C = { 28 } , however { 28 } is a set, hence it cannot be an element of A - ¯ C . (c)
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Exam_Solutions - UNIT: SIT192 Discrete Mathematics 2010...

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