Explanation Given information: x is any real number. Proof: Proof by division into cases: Let x be a real number. There exists an integer k such that ⌊ x 2 ⌋ = k We know that k is either even or odd. Case 1: k is even By the definition of even, there exists an integer m such that: k = 2m ⌊ x 2 ⌋ = k = 2 m By the definition of the floor function, this then also means that x 2 is equal to 2m or x 2 lies between 2m and 2m + 1. 2 m ≤ x 2 < 2 m + 1 Divide each side of the previous inequality by 2: m ≤ x 4 < m + 1 2 < m + 1 Where m and m + 1 are consecutive integers By the definition of the floor function, the floor of x 4 is then equal to the lower boundary: ⌊ x 4 ⌋ = m = 2 m 2 = k 2 k = 2 m = ⌊ x / 2 ⌋ / 2 = ⌊ ⌊ x / 2 ⌋ / 2 ⌋ The floor of an integer is the integer itself Case 2: k is odd By the definition of odd, there exists an integer m such that: k = 2 m + 1 ⌊ x 2 ⌋ = k = 2 m + 1 By the definition of the floor function, this then also means that x 2 is equal to 2m + 1 or x 2 lies between 2m + 1 and 2m + 2

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ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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Chapter 4.6, Problem 25ES

To determine

To prove:

For every real number x, ⌊⌊x/2⌋/2⌋=⌊x/4⌋.

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Chapter 4.6, Problem 24ES

Chapter 4.6, Problem 26ES

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DISCRETE MATHEMATICS WITH APPLICATION (

Ch. 4.1 - An integer is even if, and only if,_______.Ch. 4.1 - An integer is odd if, and only if,____Ch. 4.1 - An integer n is prime if, and only if,_______Ch. 4.1 - The most common way to disprove a universal...Ch. 4.1 - Prob. 5TY Ch. 4.1 - To use the method of direct proof to prove a...Ch. 4.1 - In 1-4 justify your answer by using the...Ch. 4.1 - In 1-4 justify your answer by using by the...Ch. 4.1 - In 1-4 justify your answers by using the...Ch. 4.1 - In 1-4 justify your answers by using the...

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