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Determine whether each of these sets is finite, countably

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

Problem 1E Chapter 2.5

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 1E

Problem 1E

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.

a) the negative integers

b) the even integers

c) the integers less than 100

d) the real numbers between 0 and

e) the positive integers less than 1.000.000,000

f) the integers that are multiples of 7

Step-by-Step Solution:

Step 1</p>

We have to check the following sets are finite, countably infinite, or uncountable.

Step 2</p>

a)The negative integers

Set of negative integers   is countably infinite.

Then we can define a one -one function from set of positive integers to set of negative integers as,

 is defined as f(z) = -z

Step 3</p>

b) The even integers

Let E =

      E = { …-4,-2,2,4,6,8,...}

The set of all even integers are countably infinite.

Define a one-one function from the set of positive integers to the set of even integers as

 as f ( z ) = +- 2z

Step 4</p>

c)The integers less than 100.

Let A = { z in Z : z < 100}

      A = { …-2,-1,0,1,2,...,100}

Clearly A is countably infinite.

We define a one-one function from set of positive integers to A as

is defined as f(a) =100-a

Step 5 of 7

Chapter 2.5, Problem 1E is Solved
Step 6 of 7

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

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