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Solved: Show that if x is a real number and n is an

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 52E Chapter 2.3

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 52E

Problem 52E

Show that if x is a real number and n is an integer, then

Step-by-Step Solution:
Step 1 of 3

Solution:

Step1

Given that

We have to show that if x is a real number and n is an integer, then

Step2

Suppose n<x ------------(1)

By using definition of Ceiling function “ it assigns to the real number y the smallest integer that is greater than or equal to y.”

Therefore,

 -----------------(2)

From (1) and (2) we get

Hence, if nx then

----------------(3)

Step3

Suppose

By using definition of  Ceiling Function

If  then

This shows

If then

n< x  -----------------(4)

From (3) and (4) we get

Step4

Suppose x<n ---------(5)

By using definition of floor function, “ The Floor Function assigns to the real number y the largest integer that is less than or equal to y.”

Therefore,

-------------------(6)

From (5) and (6) we get

If ----------------(7)

Step3

Suppose

Using Floor Function definition

If then

This shows that

If then   --------(8)

From (7) and (8) we get

Step 2 of 3

Chapter 2.3, Problem 52E is Solved
Step 3 of 3

Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Solved: Show that if x is a real number and n is an