# a3 - number when it is written in decimal form Justify your...

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CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING COMP232 MATHEMATICS FOR COMPUTER SCIENCE ASSIGNMENT 3 FALL 2009 1. Give an example of a bijection between Z and Z + . (A bijection is a function that is one-to- one and onto, and hence invertible.) Write down a formula for your function. Also write down a formula for its inverse. 2. Let f ( n, m )=( n + m, m 2 n ). Is f invertible as a function f : Z 2 −→ Z 2 ?I f s o then what is its inverse? 3. Let f ( n, m )=( 3 n +2 m, 4 n +3 m ). Is f invertible as a function f : Z 2 −→ Z 2 ?I f s o then what is its inverse? 4. Let the function f :( R −{ 1 } ) −→ ( R −{ 1 } )beg ivenby f ( x )= x +1 x - 1 . Is f one-to-one? Is f onto? If f invertible? If so then what is the inverse? Also give the graph of f and the graph of its inverse, if it exists. 5. Prove that if n is an odd positive integer then n 2 1(mod8) . 6. Find all integer solutions of 4 x 5(mod 9) . 7. Does there exists an integer x that simultaneously satis±es x 2(mod 6) and x 3(mod 9) ?

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Unformatted text preview: number when it is written in decimal form. Justify your answer. 9. Find all solutions to m 2 − n 2 = 88, for which both m and n are positive integers. 10. Prove that if m, n, d ∈ Z + and m ≡ n (mod d ), then gcd( m, d ) = gcd( n, d ) . 11. Use the Euclidean algorithm to determine whether or not the numbers 1947 and 2009 are relatively prime. 1 12. Use mathematical induction to prove that for all n ∈ Z + we have 21 | (4 n +1 + 5 2 n-1 ) . 13. For n ≥ 1, the n th harmonic number is de±ned as H n = 1 + 1 2 + 1 3 + ··· + 1 n . Use mathematical induction to prove that for all n ≥ 1 we have H 1 + H 2 + H 3 + ··· + H n = ( n + 1) H n − n . 14. The Fibonacci numbers are de±ned as: f 1 = 1 , f 2 = 1 , and f n = f n-1 + f n-2 , for n ≥ 3 . Use a proof by induction to show that the Fibonacci numbers satisfy 3 | f 4 n , for all n ≥ 1 . 2...
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a3 - number when it is written in decimal form Justify your...

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