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a1 - MATH 239 Assignment 1 Spring 2007 This assignment is...

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MATH 239 Assignment 1 Spring 2007 This assignment is due at noon on Friday, May 11, 2007, in the drop boxes opposite the Tutorial Centre, MC 4067. 1. A one-to-one correspondence is a function f : A B which is both one-to-one and onto. If a function f : A B has the property that x = y f ( x ) = f ( y ), then we say that f is one-to-one . Function f is said to be onto if, for every z B , there is an x A such that f ( x ) = z . (One-to-one correspondences are an important enumerative technique, but we only touch on them briefly in Section 1.1 of MATH 239.) Determine whether or not each function defined below as one-to-one, or onto, or both. Briefly justify your answer. (a) Let f : Z Z be defined as f ( n ) = - 2 n 2 + 5. (b) Let g : Z Z be defined as g ( n ) = 5 - 2 n . (c) Let B = { b 1 b 2 . . . b 10 : b i ∈ { 0 , 1 } ∀ i, 1 i 10 } be the set of all binary strings of length 10, and let S be all subsets of { 1 , 2 , . . . 10 } . Define h : B S to be h ( b 1 b 2 . . . b 10 ) = { i : b i = 1 } ⊆ { 1 , 2 , . . . , 10 } .
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