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Unformatted text preview: MATH 222 FALL 2011, Assignment One (Answer key) 1. Let f : N → N be defined by f = { ( x, 3 x ) : x ∈ N } . Let S be the set of all even natural numbers. Find the following sets. (a) f ( S ). f ( S ) = { 6 k : k ∈ N } . (b) f 1 ( S ). f 1 ( S ) = S . (c) f 1 (2011). f 1 (2011) = ∅ . (d) S − f ( N ). S − f ( N ) = { 6 k + 2 : k ∈ N } ∪ { 6 k + 4 : k ∈ N } 2. Determine all x ∈ R such that ⌊ x ⌋ + ⌊ x + 1 2 ⌋ = ⌊ 2 x ⌋ . We show that ⌊ x ⌋ + ⌊ x + 1 2 ⌋ = ⌊ 2 x ⌋ for all x ∈ R . First note that for any x ∈ R , x ∈ [ n, n + 1 2 ) or x ∈ [ n + 1 2 , n + 1) for some integer n . If x ∈ [ n, n + 1 2 ), then ⌊ x ⌋ + ⌊ x + 1 2 ⌋ = 2 n = ⌊ 2 x ⌋ ; if x ∈ [ n + 1 2 , n +1), then ⌊ x ⌋ + ⌊ x + 1 2 ⌋ = 2 n +1 = ⌊ 2 x ⌋ . 3. Given an example of a function from N to N that is (a) onetoone but not onto; E.g., f = { ( n, n + 1) : n ∈ N } ....
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This note was uploaded on 01/15/2012 for the course MATH 222 taught by Professor Huangjing during the Spring '11 term at University of Victoria.
 Spring '11
 HuangJing
 Natural Numbers, Sets

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