ECE-103-1011-Final_exam

ECE-103-1011-Final_exam - E&CE103 FINAL EXAMINATION WINTER...

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E&CE103 FINAL EXAMINATION WINTER TERM 2001 QUESTION 1. (a) [3] Write down the negation of (i) x y ( P ( x ) OR Q ( y )). (b) [5] Is the statement (i) from (a) equivalent to (ii) ( x P ( x )) OR ( y Q ( y ))? If so, give a proof. If not, give a conterexample. QUESTION 2: [5] Find all solutions to 10 x + 21 y = 3. QUESTION 3: [6] Find all solutions to the simultaneous congruences x 7 (mod 10) x 5 (mod 21) . QUESTION 4: (a) [5] Recall that, for integers a and b , a | b means there is an integer k such that ak = b . Prove that if a | b and a | c , then for any integers x and y , a | ( bx + cy ). (b) [3] Define GCD ( a, b ). QUESTION 5: Let S denote the set of strings on the symbols a, b, c and let T ⊆ S be the set of strings with no two consecutive a ’s. For example, aabc ∈ S but aabc / ∈ T . (a) [3] How many strings in S have length n ? (b) [6] Let t ( n ) denote the number of strings in T of length n . Explain why t satisfies the recurrence t ( n ) = 2 t ( n - 1) + 2 t ( n - 2). (c) [3] Use the recurrence relation from (b) to compute t (6).
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