hw1a - Copy (2)

hw1a - Copy (2) - MCS4653, Theory of Computation Homework...

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Unformatted text preview: MCS4653, Theory of Computation Homework Assignment 1, Due 9/15/03 Sample Answers From Class Page 1 1. (Sudkamp 1.2) Let X = { a,b,c } and Y = { 1 , 2 } . a) List all the subsets of X . { a,b,c } , { a,b } , { a,c } , { b,c } , { a } , { b } , { c } , {∅} b) List the members of X × Y . [ a, 1], [ a, 2], [ b, 1], [ b, 2], [ c, 1], [ c, 2] c) List all total functions from Y to X . y f ( y ) 1 a 2 a y f ( y ) 1 a 2 b y f ( y ) 1 a 2 c y f ( y ) 1 b 2 a y f ( y ) 1 b 2 b y f ( y ) 1 b 2 c y f ( y ) 1 c 2 a y f ( y ) 1 c 2 b y f ( y ) 1 c 2 c 2. (Sudkamp 1.3) Give functions f : N → N that satisfy a) f is total and one-to-one but not onto. f ( n ) = 2 n b) f is total and onto but not one-to-one. f ( n ) = ‰ if n = 0 n- 1 otherwise c) f is not total, but is onto. f ( n ) = ‰ n/ 2 if n is even ↑ otherwise 3. (Sudkamp 1.17) Prove that the set of real numbers in the interval [0 , 1] is uncountable. Hint: Use the diag- onalization argument on the decimal expansion of real numbers. Be sure that each number is represented byonalization argument on the decimal expansion of real numbers....
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hw1a - Copy (2) - MCS4653, Theory of Computation Homework...

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