Prove that the set of irrational numbers in [0,1] is not countable

Question

Prove that the set of irrational numbers in [0,1] is not countable

Question

Answers (1)

Relevant Questions

asked 2021-03-09

Discover prove: Combining Rational and Irrationalnumbers is \(\displaystyle{1.2}+\sqrt{{2}}\) rational or irrational? Is \(\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}\) rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf.
(a) The sum of rational number r and an irrational number t is irrational.
(b) The product of a rational number r and an irrational number t is irrational.

See answers (1)

asked 2021-02-01

Given each set of numbers, list the
a) natural Numbers
b) whole numbers
c) integers
d) rational numbers
e) irrational numbers
f) real numbers
\(\displaystyle{\left\lbrace-{6},\sqrt{{23}},{21},{5.62},{0.4},{3}\frac{{2}}{{9}},{0},-\frac{{7}}{{8}},{2.074816}\ldots\right\rbrace}\)

See answers (1)

asked 2021-02-08

In which set(s) of numbers would you find the number \(\displaystyle\sqrt{{80}}\)
- irrational number
- whole number
- rational number
- integer
- real number
- natural number

See answers (1)

asked 2020-12-14

Writing and Proof: If true prove it, if false give a counterexample. Use contradiction when proving.
(a) For each positive real number x, if x is irrational, then \(\displaystyle{x}^{{2}}\) is irrational.
(b) For every pair of real numbers a nd y, if x+y is irrational, then x if irrational or y is irrational

See answers (1)

asked 2021-01-23

The rational numbers are dense in \(\displaystyle\mathbb{R}\). This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in \(\displaystyle\mathbb{R}\) as well.

See answers (1)

asked 2021-02-13

Determine whether the below given statement is true or false. If the statement is false, make the necessary changes to produce a true statement:
All irrational numbers satisfy |x - 4| > 0.

See answers (1)

asked 2021-02-26

1) Find 100 irrational numbers between 0 and \(\displaystyle\frac{{1}}{{100}}.\)
2) Find 50 rational numbers betwee 1 and 2.
3) Find 50 irrational numbers betwee 1 and 2.

See answers (1)

asked 2020-12-24

True or False?
1) Let x and y real numbers. If \(\displaystyle{x}^{{2}}-{5}{x}={y}^{{2}}-{5}{y}\) and \(\displaystyle{x}\ne{y}\), then x+y is five.
2) The real number pi can be expressed as a repeating decimal.
3) If an irrational number is divided by a nonzero integer the result is irrational.

See answers (1)

asked 2021-01-19

\(\displaystyle{Q}:{\quad\text{if}\quad}{\left\lbrace-\frac{{3}}{{4}},\sqrt{{3}},\frac{{3}}{\sqrt{{3}}},\sqrt{{\frac{{25}}{{5}}}},{20},{1.11222},{2.5015132},\ldots,\frac{{626}}{{262}}\right\rbrace}\),
find the following
1) Rational numbers?
2) Irrational numbers?

See answers (1)

asked 2021-01-06

Write two rational and three irrational numbers that are between 3 and 4 with explanation.

See answers (1)

Answer 1

It is known that the set of rational numbers are countable.
Union of rational numbers and irrational numbers gives the real numbers.
So \(\displaystyle{\left[{0},{1}\right]}={Q}{\left[{0},{1}\right]}\cup{R}{Q}{\left[{0},{1}\right]}\)
Since Q[0,1] is countable and [0,1] is not countable.
So the only possibility is that \(\displaystyle{R}{Q}{\left[{0},{1}\right]}\) is uncountable.
Hence, the set of irrational numbers in [0,1] is not countable.

Didn’t find what you are looking for?

Prove that the set of irrational numbers in [0,1] is not countable

Prove that the set of irrational numbers in [0,1] is not countable

Answers (1)

Relevant Questions