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HW9 - → X be an injective function Define a sequence of...

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Name: Math 100 Due Wednesday, December 5, 2:00 pm Homework 9! 1. (3 points) Prove that the following sentence is true: x Z , if x is odd, then y Z , such that xy 1 (mod 16) . Hint: Let R be the set of odd numbers between 1 and 15. Let x be an odd integer, and consider the function from R to R given by: f ( r ) = the remainder left after dividing rx by 16 , for every r R . 2. (2 points) Prove that the following sentence is true: x R , y Z , if y 6 = 1 , then ( y + 1) / ( y - 1) < x. 3. (3 points) Let X be a set. Let f : X
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Unformatted text preview: → X be an injective function. Define a sequence of functions, called g n : X → X , for all n ∈ N , by the following rule: • When n = 0, and x ∈ X , then we define g ( x ) = x . • If n ≥ 1, and x ∈ X , then we define g n ( x ) = f ( g n-1 ( x )). In other words, g n is the function obtained by repeating the function f , n times. Prove that, for every n ∈ N , the function g n is injective....
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