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# a3sfall2009 - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER...

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Unformatted text preview: CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING COMP232 MATHEMATICS FOR COMPUTER SCIENCE ASSIGNMENT 3 SOLUTIONS 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. SOLUTION: An example is the function f ( n ) defined by f ( n ) = 1 if n = 0 , f ( n ) = − 2 n if n < , f ( n ) = 2 n + 1 if n > . Its inverse is f- 1 ( n ) = ( − 1) n +1 b n 2 c . 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? SOLUTION: This function is not onto; for example, there are no integers n and m for which f ( n, m ) = (2 , 1). Since f is not onto it is not invertible. 3. Let f ( n, m ) = (3 n + 2 m, 4 n + 3 m ). Is f invertible as a function f : Z 2 −→ Z 2 ? If so then what is its inverse? SOLUTION: This function is invertible. The inverse is f- 1 ( n, m ) = (3 n − 2 m, − 4 n +3 m ). 4. Let the function f : ( R − { 1 } ) −→ ( R − { 1 } ) be given by 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. SOLUTION: As is best illustrated by a graph, this function is one-to-one and onto, and hence invertible. The inverse is the same as the function, i.e. , f- 1 ( x ) = x +1 x- 1 ....
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a3sfall2009 - CONCORDIA UNIVERSITY DEPARTMENT OF COMPUTER...

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