Open In App

Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Discrete Mathematics and its Applications

Kenneth H. Rosen

$Math Studyset Vaia Explanations$ Math

7th Edition · 808 pages

ISBN: 9780073383095

Chapter 2: Q32SE (page 187)

Show that the set of irrational numbers is an uncountable set.

Short Answer

Expert verified

Set of irrational numbers is uncountable.

Step by step solution

Step: 1

We know that rational number is countable and set of all real numbers is uncountable.

Step: 2

R is union of rational and irrational number and now if irrational number is countable then as rational number is countable so union of two countable set is also countable. But union of rational numbers and irrational numbers is R which is clearly uncountable so a contradiction.

Therefore set of irrational numbers is uncountable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that the function $f (x) = e^{x}$ from the set of real numbers to the set of real numbers is not invertible, but if the co domain is restricted to the set of positive real numbers, the resulting function is invertible.

If f and $f \circ g$ are one-to-one, does it follow that g is one-to-one? Justify your answer.

Question: Determine whether each of these functions from Z to Z is one-to-one. a) $f (n) = n - 1$ b) $f (n) = n^{2} + 1$ c) $f (n) = n^{3}$ d) $f (n) = ⌈ n / 2 ⌉$

Suppose that g is a function from A to B and f is a function from B to C.

Show that if both f and g are one-to-one functions, then $f \circ g$ is also one-to-one.
Show that if both f and g are onto functions, then $f \circ g$ is also onto.

Prove that if x is a positive real number, then

$\begin{aligned} (a) ⌊ \sqrt{⌊ x ⌋} ⌋ = ⌊ \sqrt{x} ⌋ \\ (b) [\sqrt{⌊ x ⌋}] = ⌈ \sqrt{x} ⌉ \end{aligned}$

See all solutions

Recommended explanations on Math Textbooks

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free

Show that the set of irrational numbers is an uncountable set.

Short Answer

Step by step solution

Step: 1

Step: 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Calculus

Geometry

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Company

Product

Help