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3 - The University of Sydney MATH 1004 Second Semester...

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The University of Sydney MATH 1004 Second Semester Discrete Mathematics 2011 Tutorial 3 Week 4 1. Define f : N N by f ( x ) = x + 1. Determine whether or not f is (a) one-to-one; (b) onto. 2. Each of the following sets of pairs may or may not represent a function from { 1 , 2 , 3 } to { a, b, c, d } . { (1 ,d ) , (2 ,b ) , (3 ,d ) } , { (1 ,c ) , (2 ,a ) , (3 ,b ) } , { (1 ,a ) , (3 ,b ) } { (1 ,a ) , (1 ,c ) , (3 ,d ) } , { (2 ,b ) , (3 ,c ) , (1 ,d ) } ( i ) Identify the sets which represent functions and determine which of these are one-to-one. ( ii ) Explain clearly why each of the sets does or does not represent a function. ( iii ) Explain clearly why each of the sets does or does not represent a one-to- one function. 3. ( i ) Let A = {- 1 , 2 , 3 , 5 , 7 , 11 } and let B = { 1 , 2 ,..., 200 } . Is the function f : A B given by f ( x ) = x 2 one-to-one? ( ii ) Now suppose that A = {- 2 , - 1 , 2 , 3 , 5 , 7 , 11 } and B = { 1 , 2 ,..., 200 } . Is the function f : A B given by f ( x ) = x 2 one-to-one? 4. Use arrow diagrams to write down all the functions from the set { 1 , 2 } to the set { a,b,c } . How many are there? How many one-to-one functions and how many onto functions?
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