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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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) ?
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online