hw02sol - Math 55: Discrete Mathematics, Fall 2008 Homework...

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Unformatted text preview: Math 55: Discrete Mathematics, Fall 2008 Homework 2 Solutions Problems marked * will be corrected fully, out of 10 points. The others will be checked quickly, for 2 points each. 2.3: 16. The following are not the only possible solutions but they are some of the simplest ones. (a) f ( n ) = n + 1. (b) f (0) = 0, f ( n ) = n- 1 if n > 0. (c) f (0) = 1, f (1) = 0, f ( n ) = n if n > 1. (d) f (0) = 1, f ( n ) = n if n > 0. 40. (a) f- 1 ( S T ) = { a A : f ( a ) S T } = { a A : f ( a ) S or f ( a ) T } = { a A : f ( a ) S } { a A : f ( a ) T } = f- 1 ( S ) f- 1 ( T ). (b) Just change to and or to and in the above solution to part (a). A more laborious but also correct way to do this is to verify in each part that the set on the left-hand side is a subset of the one on the right, and conversely. *[5 pts each part] (C) (i) It is true that if f g is one-to-one, then g is one-to-one. To prove it, suppose g ( x ) = g ( y ). Then f g ( x ) = f g ( y ), which implies x = y since f g is one-to-one. We have shown thatone-to-one....
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hw02sol - Math 55: Discrete Mathematics, Fall 2008 Homework...

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