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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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In each of the following, fill in the blank the with the word countable or the word uncountable.

  1. The set of all integer is _______.
  2. The set of all rational numbers is_____
  3. The set of all real numbers between 0 and 1 is______
  4. The set of all real numbers is _____

To determine

(a)

To fill:

The blank in the statement “The set of all integers is _____.”

Explanation

Given information:

Set of all integers.

The set Z of all integers is certainly not finite, so if it is countable, it must be because it is countably infinite.

To show that Z Is countably infinite, find a function from the positive integers Z + to Z that is one-to-one and onto.

Looked at in one light, this contradicts common sense; judging from the diagram below, there appear to be more than twice as many integers as there are positive integers.

Let the first integer be 0, the second 1, the third −1, the fourth 2, the fifth −2, and so forth as shown in figure X, starting at 0 and swinging outward in back-and-forth arcs from positive to negative integers and back again, picking up one additional integer at each swing...

To determine

(b)

To fill:

The blank in the statement “The set of all rational numbers is _____.”

To determine

(c)

To fill:

The blank in the statement “The set of all real numbers between 0 and 1 is _____.”

To determine

(d)

To fill:

The blank in the statement “The set of all real numbers is _____.”

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